From 7051b19ae5a6bb678f9deb8f8bf682f66e1ee5d9 Mon Sep 17 00:00:00 2001 From: Julien Toulouse Date: Tue, 24 Mar 2020 15:04:08 +0100 Subject: [PATCH] start revised manuscript --- Revised_Manuscript/.gitignore | 1 + Revised_Manuscript/Makefile | 51 + Revised_Manuscript/SI/srDFT_SC-SI.aux | 51 + Revised_Manuscript/SI/srDFT_SC-SI.bbl | 59 + Revised_Manuscript/SI/srDFT_SC-SI.blg | 67 + Revised_Manuscript/SI/srDFT_SC-SI.log | 2065 +++ Revised_Manuscript/SI/srDFT_SC-SI.out | 8 + Revised_Manuscript/SI/srDFT_SC-SI.pdf | Bin 0 -> 354627 bytes Revised_Manuscript/SI/srDFT_SC-SI.tex | 442 + Revised_Manuscript/SI/srDFT_SC-SINotes.bib | 2 + Revised_Manuscript/TODO | 3 + Revised_Manuscript/biblio.bib | 11864 ++++++++++++++ .../C2/DFT_avdzE_relat-eps-converted-to.pdf | Bin 0 -> 13914 bytes .../DFT_avdzE_relat_zoom-eps-converted-to.pdf | Bin 0 -> 15036 bytes .../C2/DFT_avtzE_relat-eps-converted-to.pdf | Bin 0 -> 13890 bytes .../DFT_avtzE_relat_zoom-eps-converted-to.pdf | Bin 0 -> 15187 bytes 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100644 Revised_Manuscript/srDFT_SC.aux create mode 100644 Revised_Manuscript/srDFT_SC.bbl create mode 100644 Revised_Manuscript/srDFT_SC.bib create mode 100644 Revised_Manuscript/srDFT_SC.blg create mode 100644 Revised_Manuscript/srDFT_SC.log create mode 100644 Revised_Manuscript/srDFT_SC.out create mode 100644 Revised_Manuscript/srDFT_SC.tex create mode 100644 Revised_Manuscript/srDFT_SC.tex.bak create mode 100644 Revised_Manuscript/srDFT_SCNotes.bib create mode 100644 Revised_Manuscript/srDFT_SC_jt.tex create mode 100644 Revised_Manuscript/srDFT_SC_old.tex diff --git a/Revised_Manuscript/.gitignore b/Revised_Manuscript/.gitignore new file mode 100644 index 0000000..0a54f0e --- /dev/null +++ b/Revised_Manuscript/.gitignore @@ -0,0 +1 @@ +srDFT_SC.pdf diff --git a/Revised_Manuscript/Makefile b/Revised_Manuscript/Makefile new file mode 100755 index 0000000..73c4ee7 --- /dev/null +++ b/Revised_Manuscript/Makefile @@ -0,0 +1,51 @@ +# * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * +# +# Make file simple pour LaTeX +# +# Compilation simple du document +# Compilation complete du document +# +# * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * + +# fichier principal +TEX=srDFT_SC + +# compilateur latex +CC=pdflatex + +# defaut = aide : +default: simple + +help: + @echo + @echo " Compilation du rapport, regles :" + @echo + @echo " simple : compilation une seule fois" + @echo " all : compile et met la biblio et les references a jour (long)" + @echo " purge : effacement des fichiers autres que .tex et .pdf (.aux .log .out .toc ..." + @echo + +# compilation simple +simple: $(TEX).tex + $(CC) $(TEX) + +# compilation complete, on passe plusieurs fois pour les references croisee plus bibtex pour la +# biblio +all: $(TEX).tex + $(CC) $(TEX) + $(CC) $(TEX) + @bibtex $(TEX) + $(CC) $(TEX) + $(CC) $(TEX) + @echo + @echo + @echo "fin ;)" + @echo + + +# efface les fichiers .log .aux .toc .bbl .blg +.PHONY: purge +purge: + @rm -vf *.aux $(TEX).bbl $(TEX).blg $(TEX).log $(TEX).out $(TEX).toc $(TEX).mtc* $(TEX).lof $(TEX).lot $(TEX).maf + @echo "done" + diff --git a/Revised_Manuscript/SI/srDFT_SC-SI.aux b/Revised_Manuscript/SI/srDFT_SC-SI.aux new file mode 100644 index 0000000..a9c0950 --- /dev/null +++ b/Revised_Manuscript/SI/srDFT_SC-SI.aux @@ -0,0 +1,51 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} +\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined +\global\let\oldcontentsline\contentsline +\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} +\global\let\oldnewlabel\newlabel +\gdef\newlabel#1#2{\newlabelxx{#1}#2} +\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} +\AtEndDocument{\ifx\hyper@anchor\@undefined +\let\contentsline\oldcontentsline +\let\newlabel\oldnewlabel +\fi} +\fi} +\global\let\hyper@last\relax +\gdef\HyperFirstAtBeginDocument#1{#1} +\providecommand\HyField@AuxAddToFields[1]{} +\providecommand\HyField@AuxAddToCoFields[2]{} +\citation{Sav-CP-09} +\newlabel{FirstPage}{{}{1}{}{Doc-Start}{}} +\@writefile{toc}{\contentsline {title}{A density-based basis-set correction for weak and strong correlation}{1}{section*.1}} +\@writefile{toc}{\contentsline {section}{\numberline {I}Size consistency of the basis-set correction}{1}{section*.2}} +\@writefile{toc}{\contentsline {subsection}{\numberline {A}Sufficient condition for size consistency}{1}{section*.3}} +\newlabel{eq:def_ecmdpbebasis}{{1}{1}{}{equation.1.1}{}} +\newlabel{eq:def_ecmdpbebasis}{{2}{1}{}{equation.1.2}{}} +\newlabel{nAB}{{3a}{1}{}{equation.1.3alph1}{}} +\newlabel{zAB}{{3b}{1}{}{equation.1.3alph2}{}} +\newlabel{sAB}{{3c}{1}{}{equation.1.3alph3}{}} +\newlabel{n2AB}{{3d}{1}{}{equation.1.3alph4}{}} +\newlabel{muAB}{{3e}{1}{}{equation.1.3alph5}{}} +\@writefile{toc}{\contentsline {subsection}{\numberline {B}Intensivity of the on-top pair density}{1}{section*.4}} +\newlabel{eq:def_n2}{{5}{1}{}{equation.1.5}{}} +\@writefile{toc}{\contentsline {subsection}{\numberline {C}Property of the local-range separation parameter}{2}{section*.5}} +\newlabel{eq:def_f}{{15}{2}{}{equation.1.15}{}} +\newlabel{eq:def_fa+b}{{16}{2}{}{equation.1.16}{}} +\newlabel{eq:def_faa}{{17}{2}{}{equation.1.17}{}} +\newlabel{eq:def_mur}{{18}{2}{}{equation.1.18}{}} +\newlabel{eq:def_mur}{{19}{2}{}{equation.1.19}{}} +\@writefile{toc}{\contentsline {section}{\numberline {II}Computational considerations}{2}{section*.6}} +\@writefile{toc}{\contentsline {subsection}{\numberline {A}Computation of the on-top pair density for a CASSCF wave function}{2}{section*.7}} +\newlabel{def_n2_good}{{20}{2}{}{equation.2.20}{}} +\newlabel{def_n2_act}{{21}{2}{}{equation.2.21}{}} +\@writefile{toc}{\contentsline {subsection}{\numberline {B}Computation of $\mu _{\Psi _{}^{\mathcal {B}}}({\bf r})$}{2}{section*.8}} +\bibdata{srDFT_SC-SINotes,../srDFT_SC} +\bibcite{Sav-CP-09}{{1}{2009}{{Savin}}{{}}} +\bibstyle{aipnum4-1} +\citation{REVTEX41Control} +\citation{aip41Control} +\newlabel{eq:f_good}{{24}{3}{}{equation.2.24}{}} +\newlabel{LastBibItem}{{1}{3}{}{equation.2.27}{}} +\newlabel{LastPage}{{}{3}{}{}{}} diff --git a/Revised_Manuscript/SI/srDFT_SC-SI.bbl b/Revised_Manuscript/SI/srDFT_SC-SI.bbl new file mode 100644 index 0000000..48d03d5 --- /dev/null +++ b/Revised_Manuscript/SI/srDFT_SC-SI.bbl @@ -0,0 +1,59 @@ +%merlin.mbs aipnum4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked +%Control: key (0) +%Control: author (8) initials jnrlst +%Control: editor formatted (1) identically to author +%Control: production of article title (-1) disabled +%Control: page (0) single +%Control: year (1) truncated +%Control: production of eprint (0) enabled +\begin{thebibliography}{1}% +\makeatletter +\providecommand \@ifxundefined [1]{% + \@ifx{#1\undefined} +}% +\providecommand \@ifnum [1]{% + \ifnum #1\expandafter \@firstoftwo + \else \expandafter \@secondoftwo + \fi +}% +\providecommand \@ifx [1]{% + \ifx #1\expandafter \@firstoftwo + \else 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[0]{\@secondoftwo}% +\providecommand \bibfield [0]{\@secondoftwo}% +\providecommand \translation [1]{[#1]}% +\providecommand \BibitemOpen [0]{}% +\providecommand \bibitemStop [0]{}% +\providecommand \bibitemNoStop [0]{.\EOS\space}% +\providecommand \EOS [0]{\spacefactor3000\relax}% +\providecommand \BibitemShut [1]{\csname bibitem#1\endcsname}% +\let\auto@bib@innerbib\@empty +% +\bibitem [{\citenamefont {Savin}(2009)}]{Sav-CP-09}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont + {Savin}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Chem. + Phys.}\ }\textbf {\bibinfo {volume} {356}},\ \bibinfo {pages} {91} (\bibinfo + {year} {2009})}\BibitemShut {NoStop}% +\end{thebibliography}% diff --git a/Revised_Manuscript/SI/srDFT_SC-SI.blg b/Revised_Manuscript/SI/srDFT_SC-SI.blg new file mode 100644 index 0000000..cb17c0a --- /dev/null +++ b/Revised_Manuscript/SI/srDFT_SC-SI.blg @@ -0,0 +1,67 @@ +This is BibTeX, Version 0.99d (TeX Live 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LaTeX info: "xparse/define-command" +. +. Defining command \absolutevalue with sig. ' l m ' on line 176. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \abs with sig. '' on line 177. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \norm with sig. ' l m ' on line 178. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \order with sig. ' l m ' on line 179. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \evaluated with sig. ' s g d[| d(| ' on line 194. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \eval with sig. '' on line 195. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \poissonbracket with sig. ' l m m ' on line 197. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \pb with sig. '' on line 198. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \commutator with sig. ' l m m ' on line 201. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \comm with sig. '' on line 202. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \anticommutator with sig. ' l m m ' on line 203. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \acommutator with sig. '' on line 204. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \acomm with sig. '' on line 205. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \vectorbold with sig. ' s m ' on line 208. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \vb with sig. '' on line 209. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \vectorarrow with sig. ' s m ' on line 211. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \va with sig. '' on line 212. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \vectorunit with sig. ' s m ' on line 214. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \vu with sig. '' on line 215. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \dotproduct with sig. '' on line 217. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \vdot with sig. '' on line 218. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \crossproduct with sig. '' on line 220. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \cross with sig. '' on line 221. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \cp with sig. '' on line 222. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \gradient with sig. ' g o d() ' on line 232. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \grad with sig. '' on line 233. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \divergence with sig. ' g o d() ' on line 243. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \div with sig. '' on line 246. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \curl with sig. ' g o d() ' on line 256. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \laplacian with sig. ' g o d() ' on line 266. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \principalvalue with sig. 'g' on line 274. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \pv with sig. '' on line 275. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \PV with sig. 'g' on line 276. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \Re with sig. 'g' on line 277. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \Im with sig. 'g' on line 278. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \opbraces with sig. ' m g o d() ' on line 299. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \trigbraces with sig. ' m o d() ' on line 305. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \sin with sig. '' on line 309. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \cos with sig. '' on line 310. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \tan with sig. '' on line 311. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \csc with sig. '' on line 312. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \sec with sig. '' on line 313. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \cot with sig. '' on line 314. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \arcsin with sig. '' on line 316. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \arccos with sig. '' on line 317. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \arctan with sig. '' on line 318. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \arccsc with sig. '' on line 320. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \arcsec with sig. '' on line 322. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \arccot with sig. '' on line 324. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \asin with sig. '' on line 327. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \acos with sig. '' on line 329. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \atan with sig. '' on line 331. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \acsc with sig. '' on line 333. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \asec with sig. '' on line 335. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \acot with sig. '' on line 337. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \sinh with sig. '' on line 339. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \cosh with sig. '' on line 340. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \tanh with sig. '' on line 341. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \csch with sig. '' on line 343. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \sech with sig. '' on line 345. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \coth with sig. '' on line 346. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \exp with sig. '' on line 348. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \log with sig. '' on line 349. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \ln with sig. '' on line 350. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \det with sig. '' on line 351. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \Pr with sig. '' on line 352. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \tr with sig. '' on line 353. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \Tr with sig. '' on line 354. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \Res with sig. '' on line 355. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qqtext with sig. ' s m ' on line 377. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qq with sig. '' on line 378. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qcomma with sig. '' on line 380. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qc with sig. '' on line 381. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qif with sig. 's' on line 383. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qthen with sig. 's' on line 384. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qelse with sig. 's' on line 385. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qotherwise with sig. 's' on line 386. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qunless with sig. 's' on line 387. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qgiven with sig. 's' on line 388. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qusing with sig. 's' on line 389. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qassume with sig. 's' on line 390. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qsince with sig. 's' on line 391. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qlet with sig. 's' on line 392. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qfor with sig. 's' on line 393. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qall with sig. 's' on line 394. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qeven with sig. 's' on line 395. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qodd with sig. 's' on line 396. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qinteger with sig. 's' on line 397. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qand with sig. 's' on line 398. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qor with sig. 's' on line 399. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qas with sig. 's' on line 400. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qin with sig. 's' on line 401. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qcc with sig. 's' on line 402. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \differential with sig. ' o g d() ' on line 415. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \dd with sig. '' on line 416. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \derivative with sig. ' s o m g d() ' on line 435. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \dv with sig. '' on line 436. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \partialderivative with sig. ' s o m g g d() ' on line 460. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \pderivative with sig. '' on line 461. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \pdv with sig. '' on line 462. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \variation with sig. ' o g d() ' on line 474. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \var with sig. '' on line 475. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \functionalderivative with sig. ' s o m g d() ' on line +. 494. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \fderivative with sig. '' on line 495. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \fdv with sig. '' on line 496. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \bra with sig. ' s m t\ket s g ' on line 526. +................................................. +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\ket ' for the command '\bra' should be a single +* character. +************************************************* +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \ket with sig. ' s m ' on line 533. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \innerproduct with sig. ' s m g ' on line 548. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \braket with sig. '' on line 549. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \ip with sig. '' on line 550. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \outerproduct with sig. ' s m g ' on line 565. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \dyad with sig. '' on line 566. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \op with sig. '' on line 567. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \ketbra with sig. '' on line 568. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \expectationvalue with sig. ' s s m g ' on line 587. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \expval with sig. '' on line 588. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \ev with sig. '' on line 589. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \vev with sig. ' m ' on line 590. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \matrixelement with sig. ' s s m m m ' on line 601. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \matrixel with sig. '' on line 602. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \mel with sig. '' on line 603. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \identitymatrix with sig. 'm' on line 633. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \imat with sig. '' on line 634. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \xmatrix with sig. ' s m m m ' on line 659. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \xmat with sig. '' on line 660. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \zeromatrix with sig. ' m g ' on line 662. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \zmat with sig. '' on line 663. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \paulixmatrix with sig. '' on line 665. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \pauliymatrix with sig. '' on line 666. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \paulizmatrix with sig. '' on line 667. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \paulimatrix with sig. 'm' on line 678. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \pmat with sig. '' on line 679. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \diagonalmatrix with sig. 'O{} >{\SplitList {,}}m ' on line +. 681. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \@dmat with sig. 'mmggggggg' on line 714. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \dmat with sig. '' on line 715. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \antidiagonalmatrix with sig. 'O{} >{\SplitList {,}}m ' on +. line 717. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \@admat with sig. 'mmggggggg' on line 750. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \admat with sig. '' on line 751. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \flatfrac with sig. ' m m ' on line 754. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \homework with sig. '' on line 764. +................................................. +) (/usr/share/texlive/texmf-dist/tex/latex/mhchem/mhchem.sty +Package: mhchem 2017/07/24 v4.07 for typesetting chemical formulae + +(/usr/share/texlive/texmf-dist/tex/latex/l3packages/l3keys2e/l3keys2e.sty +Package: l3keys2e 2018/02/21 LaTeX2e option processing using LaTeX3 keys +) +(/usr/share/texlive/texmf-dist/tex/latex/tools/calc.sty +Package: calc 2014/10/28 v4.3 Infix arithmetic (KKT,FJ) +\calc@Acount=\count199 +\calc@Bcount=\count266 +\calc@Adimen=\dimen148 +\calc@Bdimen=\dimen149 +\calc@Askip=\skip60 +\calc@Bskip=\skip61 +LaTeX Info: Redefining \setlength on input line 80. +LaTeX Info: Redefining \addtolength on input line 81. +\calc@Ccount=\count267 +\calc@Cskip=\skip62 +) +(/usr/share/texlive/texmf-dist/tex/latex/chemgreek/chemgreek.sty +Package: chemgreek 2016/12/20 v1.1 interfaceforuprightgreeklettersforuseinchemi +stry (CN) +\l__chemgreek_tmpa_int=\count268 +\g__chemgreek_tmpa_int=\count269 +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \newchemgreekmapping with sig. 'O{}mm' on line 336. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \renewchemgreekmapping with sig. 'O{}mm' on line 339. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \declarechemgreekmapping with sig. 'O{}mm' on line 342. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \newchemgreekmappingalias with sig. 'mm' on line 347. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \renewchemgreekmappingalias with sig. 'mm' on line 350. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \declarechemgreekmappingalias with sig. 'mm' on line 353. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \changechemgreeksymbol with sig. 'mmmm' on line 383. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \chemgreekmappingsymbol with sig. 'mm' on line 477. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \activatechemgreekmapping with sig. 'sm' on line 486. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \selectchemgreekmapping with sig. 'm' on line 491. +................................................. +) +\l__mhchem_ce_distanceFromLastComma_int=\count270 +\__mhchem_arrow_extPgfLength_skip=\skip63 +\mhchem@minispace@tmp=\skip64 +\mhchem@minibackspace@tmp=\skip65 +\mhchem@bondwidth=\skip66 +\mhchem@bondheight=\skip67 +\mhchem@smallbondwidth@tmpA=\skip68 +\mhchem@smallbondwidth@tmpB=\skip69 +\mhchem@smallbondwidth=\skip70 +\mhchem@bondtmp@minussidebearingleft=\skip71 +\mhchem@bondtmp@minussidebearingright=\skip72 +\l__mhchem_option_version_int=\count271 +\mhchem@option@minusmathsidebearingleft=\dimen150 +\mhchem@option@minusmathsidebearingright=\dimen151 +\mhchem@option@minustextsidebearingleft=\dimen152 +\mhchem@option@minustextsidebearingright=\dimen153 + + +Package mhchem Warning: You did not specify a 'version' option for the mhchem +(mhchem) package. 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../../../Manuscript/data/$i + popd + done +done + + + +for i in N2 F2 O2 +do + for j in avdz avtz + do + pushd ../../new/${i}_$j/plots + ./plot.sh + cp *relat*pdf ../../../Manuscript/data/$i + popd + done +done + + diff --git a/Revised_Manuscript/sci.bib b/Revised_Manuscript/sci.bib new file mode 100644 index 0000000..749e4ca --- /dev/null +++ b/Revised_Manuscript/sci.bib @@ -0,0 +1,568 @@ + +@article{BenErn-PhysRev-1969, + Author = {Charles F. Bender and Ernest R. Davidson}, + Doi = {10.1103/physrev.183.23}, + Journal = {Phys. Rev.}, + Month = {jul}, + Number = {1}, + Pages = {23--30}, + Publisher = {American Physical Society ({APS})}, + Title = {Studies in Configuration Interaction: The First-Row Diatomic Hydrides}, + Url = {http://dx.doi.org/10.1103/PhysRev.183.23}, + Volume = {183}, + Year = 1969, + Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRev.183.23}, + Bdsk-Url-2 = {http://dx.doi.org/10.1103/physrev.183.23}} + +@article{WhiHac-JCP-1969, + Author = {J. L. Whitten and Melvyn Hackmeyer}, + Doi = {10.1063/1.1671985}, + Journal = {J. Chem. Phys.}, + Month = {dec}, + Number = {12}, + Pages = {5584--5596}, + Publisher = {{AIP} Publishing}, + Title = {Configuration Interaction Studies of Ground and Excited States of Polyatomic Molecules. I. The {CI} Formulation and Studies of Formaldehyde}, + Url = {https://doi.org/10.1063%2F1.1671985}, + Volume = {51}, + Year = 1969, + Bdsk-Url-1 = {https://doi.org/10.1063%2F1.1671985}, + Bdsk-Url-2 = {https://doi.org/10.1063/1.1671985}} + +@article{HurMalRan-1973, + Author = {B. Huron and J. P. Malrieu and P. Rancurel}, + Doi = {10.1063/1.1679199}, + Journal = {J. Chem. Phys.}, + Month = {jun}, + Number = {12}, + Pages = {5745--5759}, + Publisher = {{AIP} Publishing}, + Title = {Iterative perturbation calculations of ground and excited state energies from multiconfigurational zeroth-order wavefunctions}, + Url = {https://doi.org/10.1063%2F1.1679199}, + Volume = {58}, + Year = 1973, + Bdsk-Url-1 = {https://doi.org/10.1063%2F1.1679199}, + Bdsk-Url-2 = {https://doi.org/10.1063/1.1679199}} + +@article{ShBuPeyChemPhys78, + Author = {Shih, Sk and Butscher, W ans Buenker, Rj and Peyerimhoff, Sd}, + Doi = {{10.1016/0301-0104(78)85076-9}}, + Issn = {{0301-0104}}, + Journal = {{Chemical Physics}}, + Journal-Iso = {{Chem. Phys.}}, + Number = {{1-2}}, + Number-Of-Cited-References = {{37}}, + Pages = {{241-252}}, + Publisher = {{Elsevier Science Bv}}, + Title = {{Calculation of vertical electronic-spectrum of nitrogen molecule using mrd-ci method}}, + Type = {{aRTICLE}}, + Volume = {{29}}, + Year = {{1978}}, + Bdsk-Url-1 = {https://doi.org/10.1016/0301-0104(78)85076-9%7D}} + +@article{BuePeyButMolPhys78, + Author = {Buenker, Rj and Peyerimhoff, Sd and Butscher, W}, + Doi = {{10.1080/00268977800100581}}, + Issn = {{0026-8976}}, + Journal = {{Molecular Physics}}, + Journal-Iso = {{Mol. Phys.}}, + Number = {{3}}, + Pages = {{771-791}}, + Publisher = {{TAYLOR \& FRANCIS LTD}}, + Title = {{ Applicability of multi-reference double-excitation ci (mrd-ci) method to calculation of electronic wavefunctions and comparison with related techniques}}, + Volume = {{35}}, + Year = {{1978}}, + Bdsk-Url-1 = {https://doi.org/10.1080/00268977800100581%7D}} + +@article{EvaDauMal-ChemPhys-83, + Author = {Stefano Evangelisti and Jean-Pierre Daudey and Jean-Paul Malrieu}, + Doi = {10.1016/0301-0104(83)85011-3}, + Journal = {Chemical Physics}, + Month = {feb}, + Number = {1}, + Pages = {91--102}, + Publisher = {Elsevier {BV}}, + Title = {Convergence of an improved {CIPSI} algorithm}, + Url = {https://doi.org/10.1016%2F0301-0104%2883%2985011-3}, + Volume = {75}, + Year = 1983, + Bdsk-Url-1 = {https://doi.org/10.1016%2F0301-0104%2883%2985011-3}, + Bdsk-Url-2 = {https://doi.org/10.1016/0301-0104(83)85011-3}} + + +@article{Cim-JCP-1985, + Author = {Cimiraglia, Renzo}, + Date-Added = {2018-12-01 13:44:03 +0100}, + Date-Modified = {2018-12-01 13:44:03 +0100}, + Doi = {10.1063/1.449362}, + File = {1%2E449362.pdf:/home/scemama/Dropbox/Zotero/storage/52SWQQR4/1%2E449362.pdf:application/pdf;1.449362.pdf:/home/scemama/Dropbox/Zotero/storage/E6WCUH8T/1.449362.pdf:application/pdf;Snapshot:/home/scemama/Dropbox/Zotero/storage/HLPRZTEI/1.html:text/html}, + Issn = {0021-9606, 1089-7690}, + Journal = {J. Chem. Phys.}, + Language = {en}, + Month = aug, + Number = {4}, + Pages = {1746--1749}, + Shorttitle = {Second order perturbation correction to {CI} energies by use of diagrammatic techniques}, + Title = {Second order perturbation correction to {CI} energies by use of diagrammatic techniques: {An} improvement to the {CIPSI} algorithm}, + Url = {http://aip.scitation.org/doi/10.1063/1.449362}, + Urldate = {2017-11-14}, + Volume = {83}, + Year = {1985}, + Bdsk-Url-1 = {http://aip.scitation.org/doi/10.1063/1.449362}, + Bdsk-Url-2 = {http://dx.doi.org/10.1063/1.449362}} + +@article{Cim-JCC-1987, + Author = {Cimiraglia, Renzo and Persico, Maurizio}, + Date-Added = {2018-12-01 13:44:08 +0100}, + Date-Modified = {2018-12-01 13:44:08 +0100}, + File = {28a1e2d5fd9eba4d0cba5227b6acd8463ff941ffaa24949d1437dfca2bc8f3c7.pdf:/home/scemama/Dropbox/Zotero/storage/4L9PPHEJ/28a1e2d5fd9eba4d0cba5227b6acd8463ff941ffaa24949d1437dfca2bc8f3c7.pdf:application/pdf;540080105_ftp.pdf:/home/scemama/Dropbox/Zotero/storage/MZIFQQ9W/540080105_ftp.pdf:application/pdf}, + Journal = {J. Comput. Chem.}, + Number = {1}, + Pages = {39--47}, + Shorttitle = {Recent advances in multireference second order perturbation {CI}}, + Title = {Recent advances in multireference second order perturbation {CI}: {The} {CIPSI} method revisited}, + Volume = {8}, + Year = {1987}} + +@article{IllRubRic-JCP-88, + Author = {Illas, F. and Rubio, J. and Ricart, J. M.}, + Date-Added = {2018-12-01 13:44:13 +0100}, + Date-Modified = {2018-12-01 13:44:13 +0100}, + Doi = {10.1063/1.455405}, + File = {1.455405.pdf:/home/scemama/Dropbox/Zotero/storage/DYD93IM9/1.455405.pdf:application/pdf}, + Issn = {0021-9606, 1089-7690}, + Journal = {J. Chem. Phys.}, + Language = {en}, + Month = nov, + Number = {10}, + Pages = {6376--6384}, + Title = {Approximate natural orbitals and the convergence of a second order multireference many-body perturbation theory ({CIPSI}) algorithm}, + Url = {http://aip.scitation.org/doi/10.1063/1.455405}, + Urldate = {2017-11-14}, + Volume = {89}, + Year = {1988}, + Bdsk-Url-1 = {http://aip.scitation.org/doi/10.1063/1.455405}, + Bdsk-Url-2 = {http://dx.doi.org/10.1063/1.455405}} + +@article{PovRubIll-TCA-92, + Author = {Povill, A. and Rubio, J. and Illas, F.}, + Date-Added = {2018-12-01 13:44:18 +0100}, + Date-Modified = {2018-12-01 13:44:18 +0100}, + Doi = {10.1007/BF01113255}, + File = {BF01113255.pdf:/home/scemama/Dropbox/Zotero/storage/TH3IVJUP/BF01113255.pdf:application/pdf;Snapshot:/home/scemama/Dropbox/Zotero/storage/M8KP3TSQ/BF01113255.html:text/html}, + Journal = {Theor. Chem. Acc.}, + Number = {3}, + Pages = {229--238}, + Title = {Treating large intermediate spaces in the {CIPSI} method through a direct selected {CI} algorithm}, + Volume = {82}, + Year = {1992}, + Bdsk-Url-1 = {https://doi.org/10.1007/BF01113255}} + + + +@article{BunCarRam-JCP-06, + Author = {Bunge, Carlos F. and Carb{\'o}-Dorca, Ramon}, + Date-Added = {2018-12-01 13:44:28 +0100}, + Date-Modified = {2018-12-01 13:44:28 +0100}, + Doi = {10.1063/1.2207621}, + File = {Snapshot:/home/scemama/Dropbox/Zotero/storage/SB8DJGT3/1.html:text/html}, + Issn = {0021-9606}, + Journal = {J. Chem. Phys.}, + Month = jul, + Number = {1}, + Pages = {014108}, + Title = {Select-divide-and-conquer method for large-scale configuration interaction}, + Url = {http://aip.scitation.org/doi/abs/10.1063/1.2207621}, + Urldate = {2017-11-17}, + Volume = {125}, + Year = {2006}, + Bdsk-Url-1 = {http://aip.scitation.org/doi/abs/10.1063/1.2207621}, + Bdsk-Url-2 = {http://dx.doi.org/10.1063/1.2207621}} + +@article{AbrSheDav-CPL-05, + Author = {Abrams, Micah L. and Sherrill, C. David}, + Date-Added = {2018-12-01 13:44:23 +0100}, + Date-Modified = {2018-12-01 13:44:23 +0100}, + Doi = {10.1016/j.cplett.2005.06.107}, + Issn = {0009-2614}, + Journal = {Chem. Phys. Lett.}, + Month = {Aug}, + Number = {1-3}, + Pages = {121--124}, + Publisher = {Elsevier BV}, + Title = {Important configurations in configuration interaction and coupled-cluster wave functions}, + Url = {http://dx.doi.org/10.1016/j.cplett.2005.06.107}, + Volume = {412}, + Year = {2005}, + Bdsk-Url-1 = {http://dx.doi.org/10.1016/j.cplett.2005.06.107}} + + +@article{MusEngels-JCC-06, + Author = {Musch, Patrick and Engels, Bernd}, + Journal = {{Journal of Computational Chemistry}}, + Number = {{10}}, + Year = {{2006}} + } + + + +@article{BytRue-CP-09, + Author = {Laimutis Bytautas and Klaus Ruedenberg}, + Doi = {10.1016/j.chemphys.2008.11.021}, + Journal = {Chemical Physics}, + Month = {feb}, + Number = {1-3}, + Pages = {64--75}, + Publisher = {Elsevier {BV}}, + Title = {A priori identification of configurational deadwood}, + Url = {https://doi.org/10.1016%2Fj.chemphys.2008.11.021}, + Volume = {356}, + Year = 2009, + Bdsk-Url-1 = {https://doi.org/10.1016%2Fj.chemphys.2008.11.021}, + Bdsk-Url-2 = {https://doi.org/10.1016/j.chemphys.2008.11.021}} + +@article{GinSceCaf-CJC-13, + Author = {Giner, Emmanuel and Scemama, Anthony and Caffarel, Michel}, + Date-Added = {2018-11-29 14:22:53 +0100}, + Date-Modified = {2018-11-29 14:22:53 +0100}, + Doi = {10.1139/cjc-2013-0017}, + Issn = {1480-3291}, + Journal = {Can. J. Chem.}, + Month = {Sep}, + Number = {9}, + Pages = {879--885}, + Publisher = {Canadian Science Publishing}, + Title = {Using perturbatively selected configuration interaction in quantum Monte Carlo calculations}, + Url = {http://dx.doi.org/10.1139/cjc-2013-0017}, + Volume = {91}, + Year = {2013}, + Bdsk-Url-1 = {http://dx.doi.org/10.1139/cjc-2013-0017}} + +@article{CafGinScemRam-JCTC-14, + Author = {Caffarel, Michel and Giner, Emmanuel and Scemama, Anthony and Ram{\'\i}rez-Sol{\'\i}s, Alejandro}, + Date-Added = {2018-11-29 14:22:40 +0100}, + Date-Modified = {2018-11-29 14:22:40 +0100}, + Doi = {10.1021/ct5004252}, + Issn = {1549-9626}, + Journal = {J. Chem. Theory Comput.}, + Month = {Dec}, + Number = {12}, + Pages = {5286--5296}, + Publisher = {American Chemical Society (ACS)}, + Title = {Spin Density Distribution in Open-Shell Transition Metal Systems: A Comparative Post-Hartree--Fock, Density Functional Theory, and Quantum Monte Carlo Study of the CuCl$_2$ Molecule}, + Url = {http://dx.doi.org/10.1021/ct5004252}, + Volume = {10}, + Year = {2014}, + Bdsk-Url-1 = {http://dx.doi.org/10.1021/ct5004252}} + + +@article{GinSceCaf-JCP-15, + Author = {Emmanuel Giner and Anthony Scemama and Michel Caffarel}, + Date-Added = {2018-11-29 14:22:53 +0100}, + Date-Modified = {2018-11-29 14:22:53 +0100}, + Doi = {10.1063/1.4905528}, + Issn = {1089-7690}, + Journal = {J. Chem. 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Chem. Phys.}, + Month = {apr}, + Number = {15}, + Pages = {151103}, + Publisher = {{AIP} Publishing}, + Title = {Communication: Toward an improved control of the fixed-node error in quantum Monte Carlo: The case of the water molecule}, + Url = {https://doi.org/10.1063%2F1.4947093}, + Volume = {144}, + Year = 2016, + Bdsk-Url-1 = {https://doi.org/10.1063%2F1.4947093}, + Bdsk-Url-2 = {https://doi.org/10.1063/1.4947093}} + + +@article{SchEva-JCP-16, + Author = {Schriber, Jeffrey B. and Evangelista, Francesco A.}, + Date-Added = {2018-12-01 13:44:38 +0100}, + Date-Modified = {2018-12-01 13:44:38 +0100}, + Doi = {10.1063/1.4948308}, + File = {Full Text PDF:/home/scemama/Dropbox/Zotero/storage/XR99ZTDH/Schriber and Evangelista - 2016 - Communication An adaptive configuration interacti.pdf:application/pdf;Snapshot:/home/scemama/Dropbox/Zotero/storage/6KITP3BL/1.html:text/html}, + Issn = {0021-9606}, + Journal = {J. Chem. Phys.}, + Month = apr, + Number = {16}, + Pages = {161106}, + Shorttitle = {Communication}, + Title = {Communication: {An} adaptive configuration interaction approach for strongly correlated electrons with tunable accuracy}, + Url = {http://aip.scitation.org/doi/abs/10.1063/1.4948308}, + Urldate = {2017-11-17}, + Volume = {144}, + Year = {2016}, + Bdsk-Url-1 = {http://aip.scitation.org/doi/abs/10.1063/1.4948308}, + Bdsk-Url-2 = {http://dx.doi.org/10.1063/1.4948308}} + +@article{LiuHofJCTC-16, + Author = {Wenjian Liu and Mark R. Hoffmann}, + Doi = {10.1021/acs.jctc.5b01099}, + Journal = {J. Chem. 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The MRCI+Q energies and the estimated exact energies have been extracted from Ref.\nobreakspace {}\rev@citealpnum {h10_prx}. }}{8}{figure.1}} +\newlabel{fig:H10}{{1}{8}{Potential energy curves of the H$_{10}$ chain with equally-spaced atoms calculated with MRCI+Q and basis-set corrected MRCI+Q using the cc-pVDZ (top), cc-pVTZ (center), and cc-pVQZ (bottom) basis sets. The MRCI+Q energies and the estimated exact energies have been extracted from Ref.~\onlinecite {h10_prx}}{figure.1}{}} +\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {cc-pVDZ}}}{8}{figure.1}} +\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {cc-pVTZ}}}{8}{figure.1}} +\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {cc-pVQZ}}}{8}{figure.1}} +\citation{HibHumByrLen-JCP-94} +\@writefile{lot}{\contentsline {table}{\numberline {I}{\ignorespaces Atomization energies (in mHa) and associated errors (in square brackets) with respect to the estimated exact values computed at different levels of theory with various basis sets.}}{9}{table.1}} +\newlabel{tab:d0}{{I}{9}{Atomization energies (in mHa) and associated errors (in square brackets) with respect to the estimated exact values computed at different levels of theory with various basis sets}{table.1}{}} +\citation{BecSavSto-TCA-95} +\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Potential energy curves of the \ce {N2} molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets. The estimated exact energies are based on a fit of experimental data and obtained from Ref.\nobreakspace {}\rev@citealpnum {LieCle-JCP-74a}. }}{10}{figure.2}} +\newlabel{fig:N2}{{2}{10}{Potential energy curves of the \ce {N2} molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets. The estimated exact energies are based on a fit of experimental data and obtained from Ref.~\onlinecite {LieCle-JCP-74a}}{figure.2}{}} +\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {aug-cc-pVDZ}}}{10}{figure.2}} +\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {aug-cc-pVTZ}}}{10}{figure.2}} +\@writefile{toc}{\contentsline {section}{\numberline {IV}Conclusion}{10}{section*.21}} +\newlabel{sec:conclusion}{{IV}{10}{}{section*.21}{}} +\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Potential energy curves of the \ce {O2} molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets. The estimated exact energies are based on a fit of experimental data and obtained from Ref.\nobreakspace {}\rev@citealpnum {LieCle-JCP-74a}. }}{11}{figure.3}} +\newlabel{fig:O2}{{3}{11}{Potential energy curves of the \ce {O2} molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets. The estimated exact energies are based on a fit of experimental data and obtained from Ref.~\onlinecite {LieCle-JCP-74a}}{figure.3}{}} +\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {aug-cc-pVDZ}}}{11}{figure.3}} +\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {aug-cc-pVTZ}}}{11}{figure.3}} +\@writefile{toc}{\appendix } +\@writefile{toc}{\contentsline {section}{\numberline {A}Size consistency of the basis-set correction}{11}{section*.22}} +\newlabel{app:sizeconsistency}{{A}{11}{}{section*.22}{}} +\@writefile{toc}{\contentsline {subsection}{\numberline {1}Sufficient condition for size consistency}{11}{section*.23}} +\newlabel{eq:def_ecmdpbebasisAnnex}{{A1}{11}{}{equation.A.1}{}} +\citation{Sav-CP-09} +\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Potential energy curves of the \ce {F2} molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets. The estimated exact energies are based on a fit of experimental data and obtained from Ref.\nobreakspace {}\rev@citealpnum {LieCle-JCP-74a}. The estimated exact energies are based on a fit of the non-relativistic valence-only CEEIS data extracted from Ref.\nobreakspace {}\rev@citealpnum {BytNagGorRue-JCP-07}. }}{12}{figure.4}} +\newlabel{fig:F2}{{4}{12}{Potential energy curves of the \ce {F2} molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets. The estimated exact energies are based on a fit of experimental data and obtained from Ref.~\onlinecite {LieCle-JCP-74a}. 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Alavi,\ and\ Tew}}} +\bibcite{TouColSav-PRA-04}{{32}{2004}{{Toulouse, Colonna,\ and\ Savin}}{{}}} +\bibcite{GorSav-PRA-06}{{33}{2006}{{Gori-Giorgi\ and\ Savin}}{{}}} +\bibcite{FraMusLupTou-JCP-15}{{34}{2015}{{Franck\ \emph {et~al.}}}{{Franck, Mussard, Luppi,\ and\ Toulouse}}} +\bibcite{AngGerSavTou-PRA-05}{{35}{2005}{{\'Angy\'an\ \emph {et~al.}}}{{\'Angy\'an, Gerber, Savin,\ and\ Toulouse}}} +\bibcite{GolWerSto-PCCP-05}{{36}{2005}{{Goll, Werner,\ and\ Stoll}}{{}}} +\bibcite{TouGerJanSavAng-PRL-09}{{37}{2009}{{Toulouse\ \emph {et~al.}}}{{Toulouse, Gerber, Jansen, Savin,\ and\ \'Angy\'an}}} +\bibcite{JanHenScu-JCP-09}{{38}{2009}{{Janesko, Henderson,\ and\ Scuseria}}{{}}} +\bibcite{TouZhuSavJanAng-JCP-11}{{39}{2011}{{Toulouse\ \emph {et~al.}}}{{Toulouse, Zhu, Savin, Jansen,\ and\ \'Angy\'an}}} +\bibcite{MusReiAngTou-JCP-15}{{40}{2015}{{Mussard\ \emph {et~al.}}}{{Mussard, Reinhardt, \'Angy\'an,\ and\ Toulouse}}} +\bibcite{KalTou-JCP-18}{{41}{2018}{{Kalai\ and\ Toulouse}}{{}}} +\bibcite{KalMusTou-JCP-19}{{42}{2019}{{Kalai, Mussard,\ and\ Toulouse}}{{}}} +\bibcite{LeiStoWerSav-CPL-97}{{43}{1997}{{Leininger\ \emph {et~al.}}}{{Leininger, Stoll, Werner,\ and\ Savin}}} +\bibcite{FroTouJen-JCP-07}{{44}{2007}{{Fromager, Toulouse,\ and\ Jensen}}{{}}} +\bibcite{FroCimJen-PRA-10}{{45}{2010}{{Fromager, Cimiraglia,\ and\ Jensen}}{{}}} +\bibcite{HedKneKieJenRei-JCP-15}{{46}{2015}{{Hedeg{\r a}rd\ \emph {et~al.}}}{{Hedeg{\r a}rd, Knecht, Kielberg, Jensen,\ and\ Reiher}}} +\bibcite{HedTouJen-JCP-18}{{47}{2018}{{Hedeg{\r a}rd, Toulouse,\ and\ Jensen}}{{}}} +\bibcite{FerGinTou-JCP-18}{{48}{2019}{{Fert{\'e}, Giner,\ and\ Toulouse}}{{}}} +\bibcite{MusTou-MP-17}{{49}{2017}{{Mussard\ and\ Toulouse}}{{}}} +\bibcite{GinPraFerAssSavTou-JCP-18}{{50}{2018}{{Giner\ \emph {et~al.}}}{{Giner, Pradines, Fert\'e, Assaraf, Savin,\ and\ Toulouse}}} +\bibcite{LooPraSceTouGin-JCPL-19}{{51}{2019{}}{{Loos\ \emph {et~al.}}}{{Loos, Pradines, Scemama, Toulouse,\ and\ Giner}}} +\bibcite{GinSceTouLoo-JCP-19}{{52}{2019}{{Giner\ \emph {et~al.}}}{{Giner, Scemama, Toulouse,\ and\ Loos}}} +\bibcite{LooPraSceGinTou-JCTC-20}{{53}{2020}{{Loos\ \emph {et~al.}}}{{Loos, Pradines, Scemama, Giner,\ and\ Toulouse}}} +\bibcite{Lev-PNAS-79}{{54}{1979}{{Levy}}{{}}} +\bibcite{Lie-IJQC-83}{{55}{1983}{{Lieb}}{{}}} +\bibcite{Kat-CPAM-57}{{56}{1957}{{Kato}}{{}}} +\bibcite{TouGorSav-TCA-05}{{57}{2005}{{Toulouse, Gori-Giorgi,\ and\ Savin}}{{}}} +\bibcite{Tou-THESIS-05}{{58}{2005}{{Toulouse}}{{}}} +\bibcite{PerBurErn-PRL-96}{{59}{1996}{{Perdew, Burke,\ and\ Ernzerhof}}{{}}} +\bibcite{PazMorGorBac-PRB-06}{{60}{2006}{{Paziani\ \emph {et~al.}}}{{Paziani, Moroni, Gori-Giorgi,\ and\ Bachelet}}} +\bibcite{CarTruGag-JPCA-17}{{61}{2017}{{Carlson, Truhlar,\ and\ Gagliardi}}{{}}} +\bibcite{GritMeePer-PRA-18}{{62}{2018}{{Gritsenko, van Meer,\ and\ Pernal}}{{}}} +\bibcite{MosSan-PRA-91}{{63}{1991}{{Moscard\'o\ and\ San-Fabi\'an}}{{}}} +\bibcite{BecSavSto-TCA-95}{{64}{1995}{{Becke, Savin,\ and\ Stoll}}{{}}} +\bibcite{Sav-INC-96a}{{65}{1995}{{Savin}}{{}}} +\bibcite{Sav-INC-96}{{66}{1996}{{Savin}}{{}}} +\@writefile{toc}{\contentsline {subsection}{\numberline {2}Computation of the local range-separation function}{14}{section*.27}} +\newlabel{eq:f_good}{{B3}{14}{}{equation.B.3}{}} +\bibcite{MieStoSav-MP-97}{{67}{1997}{{Miehlich, Stoll,\ and\ Savin}}{{}}} +\bibcite{TakYamYam-CPL-02}{{68}{2002}{{Takeda, Yamanaka,\ and\ Yamaguchi}}{{}}} +\bibcite{TakYamYam-IJQC-04}{{69}{2004}{{Takeda, Yamanaka,\ and\ Yamaguchi}}{{}}} +\bibcite{GraCre-MP-05}{{70}{2005}{{Gr\"afenstein\ and\ Cremer}}{{}}} +\bibcite{TsuScuSav-JCP-10}{{71}{2010}{{Tsuchimochi, Scuseria,\ and\ Savin}}{{}}} +\bibcite{LimCarLuoMaOlsTruGag-JCTC-14}{{72}{2014}{{{G. Li Manni, R. K. Carlson, S. Luo, D. Ma, J. Olsen, D. G. Truhlar, and L. 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Phys.}, + Pages = {14405}, + Title = {Density functional theory for strongly-interacting electrons: Perspectives for Physics and Chemistry }, + Volume = {12}, + Year = {2010}} + +@book{AngDobJanGou-BOOK-20, + Author = {Janos \'Angy\'an and John Dobson and Georg Jansen and Tim Gould}, + Date-Added = {2020-01-23 09:37:59 +0100}, + Date-Modified = {2020-01-23 09:40:07 +0100}, + Publisher = {Royal Society of Chemistry}, + Series = {Theoretical and Computational Chemistry Series}, + ISBN = {978-1-78262-045-7}, + Title = {London Dispersion Forces in Molecules, Solids and Nano-structures: An Introduction to Physical Models and Computational Methods}, + Year = {2020}} + +@article{IrmGru-JCP-2019, + Author = {A. Irmler and A. Gruneis}, + Date-Added = {2020-01-06 14:14:58 +0100}, + Date-Modified = {2020-01-06 14:18:41 +0100}, + Doi = {10.1063/1.5110885}, + Journal = {J. Chem. 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Lett.}, + Pages = {142}, + Title = {A density functional method for degenerate spin-multiplet components}, + Volume = {340}, + Year = {2001}} + +@article{LooPraSceGinTou-JCTC-20, + Author = {P.-F. Loos and B. Pradines and A. Scemama and E. Giner and J. Toulouse}, + Date-Added = {2019-12-12 03:48:12 +0100}, + Date-Modified = {2020-01-06 14:17:24 +0100}, + Journal = {J. Chem. Theory Comput.}, + Pages = {in press}, + Title = {A density-based basis-set incompleteness correction for GW methods}, + Year = {2020}} + +@article{KalMusTou-JCP-19, + Author = {C. Kalai and B. Mussard and J. Toulouse}, + Date-Added = {2019-12-12 03:47:01 +0100}, + Date-Modified = {2019-12-12 03:47:01 +0100}, + Doi = {10.1063/1.5108536}, + Journal = {J. Chem. Phys.}, + Pages = {074102}, + Title = {Range-separated double-hybrid density-functional theory with coupled-cluster and random-phase approximations}, + Volume = {151}, + Year = {2019}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.5108536}} + +@article{BarLoo-JCP-17, + Author = {Barca, Giuseppe MJ and Loos, Pierre-Fran{\c c}ois}, + title = {Three- and four-electron integrals involving Gaussian geminals: Fundamental integrals, upper bounds, and recurrence relations}, + Journal = {J. Chem. Phys.}, + volume = {147}, + number = {2}, + pages = {024103}, + year = {2017}, + doi = {10.1063/1.4991733}, + URL = {https://doi.org/10.1063/1.4991733}, + eprint = {https://doi.org/10.1063/1.4991733} +} + + +@article{PerRuzTaoStaScuCso-JCP-05, + Author = {J. P. Perdew and A. Ruzsinszky and J. Tao and V. N. Staroverov and G. E. Scuseria and G. I. Csonka}, + Date-Added = {2019-04-03 22:17:53 +0200}, + Date-Modified = {2019-04-03 22:18:33 +0200}, + Journal = {J. Chem. 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Acta}, + Pages = {445}, + Volume = {68}, + Year = {1985}} + +@article{malrieu_cipsi, + Author = {Huron, B. and Malrieu, J. P. and Rancurel, P.}, + Journal = {J. Chem. Phys.}, + Pages = {5745}, + Title = {Iterative perturbation calculations of ground and excited state energies from multiconfigurational zeroth‐order wavefunctions}, + Volume = {58}, + Year = {1973}} + +@article{malrieu, + Author = {B. Huron and J.P. Malrieu and P. Rancurel}, + Doi = {10.1063/1.1679199}, + Journal = {J. Chem. Phys.}, + Pages = {5745}, + Volume = {58}, + Year = {1973}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.1679199}} + +@article{bender, + Author = {C. F. Bender and E. R. Davidson}, + Journal = {Phys. Rev.}, + Pages = {23}, + Volume = {183}, + Year = {1969}} + +@article{buenker1, + Author = {R. J. Buenker and S. D. Peyerimholf}, + Journal = {Theor. Chim. Acta}, + Pages = {33}, + Volume = {35}, + Year = {1974}} + +@book{buenker-book, + Address = {Dordrecht}, + Author = {R. J. Buenker and S. D. 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Becke}, + Issue = {12}, + Journal = {The Journal of Chemical Physics}, + Keywords = {DENSITY FUNCTIONAL METHOD; EXCHANGE INTERACTIONS; CORRELATION FUNCTIONS; SPIN DENSITY; CORRECTIONS; SURVEYS; IONIZATION POTENTIAL; ATOMS; MOLECULES; AFFINITY; PROTONS; ATOMIZATION; THERMOCHEMICAL PROCESSES; DATA BASE MANAGEMENT}, + Page = {9173-9177}, + Publisher = {AIP}, + Title = {Density-functional thermochemistry. II. The effect of the Perdew--Wang generalized-gradient correlation correction}, + Url = {http://link.aip.org/link/?JCP/97/9173/1}, + Volume = {97}, + Year = {1992}, + Bdsk-Url-1 = {http://link.aip.org/link/?JCP/97/9173/1}} + +@article{Bec-JCP-93a, + Author = {Axel D. Becke}, + Journal = {J. Chem. Phys.}, + Pages = {1372}, + Title = {A new mixing of Hartree--Fock and local density-functional theories}, + Volume = {98}, + Year = {1993}} + +@article{Bec-JCP-93, + Author = {Axel D. Becke}, + Journal = {J. Chem. Phys.}, + Pages = {5648}, + Title = {Density-functional thermochemistry. III. The role of exact exchange}, + Volume = {98}, + Year = {1993}} + +@article{Bec-JCP-96, + Author = {Axel D. Becke}, + Journal = {J. Chem. Phys.}, + Pages = {1040}, + Title = {Density-functional thermochemistry. IV. A new dynamical correlation functional and implications for exact-exchange mixing}, + Volume = {104}, + Year = {1996}} + +@article{Bec-JCP-97, + Author = {Axel D. Becke}, + Journal = {J. Chem. Phys.}, + Number = {20}, + Pages = {8554}, + Volume = {107}, + Year = {1997}} + +@article{Bec-JCP-98, + Author = {Axel D. Becke}, + Journal = {J. Chem. Phys.}, + Number = {6}, + Pages = {2092}, + Volume = {109}, + Year = {1998}} + +@article{BecJoh-JCP-05, + Author = {A. D. Becke and E. R. Johnson}, + Journal = {J. Chem. Phys.}, + Pages = {154104}, + Volume = {122}, + Year = {2005}} + +@article{BecJoh-JCP-07, + Author = {A. D. Becke and E. R. Johnson}, + Journal = {J. Chem. Phys.}, + Pages = {154108}, + Volume = {127}, + Year = {2007}} + +@article{Bec-PRA-88, + Author = {A. D. Becke}, + Journal = {Phys. Rev. A}, + Pages = {3098}, + Volume = {{38}}, + Year = {1988}} + +@article{BecSavSto-TCA-95, + doi = {10.1007/BF01114982}, + Author = {A. D. Becke and A. Savin and H. Stoll}, + Journal = {Theoret. Chim. Acta}, + Pages = {147}, + Volume = {{91}}, + Year = {1995}} + +@article{BecStaBurBla-CP-08, + Author = {E. V. Beck and E. A. Stahlberg and L. W. Burggraf and J.-P. Blaudeau}, + Journal = {Chem. Phys.}, + Pages = {158}, + Volume = {349}, + Year = {2008}} + +@article{BenDisLocChaHea-JPCA-08, + Author = {{T. Benighaus, R. A. DiStasio Jr., R. C. Lochan, J.-D. Chai and M. Head-Gordon}}, + Journal = {J. Phys. Chem. A}, + Pages = {2702}, + Volume = {112}, + Year = {2008}} + +@article{BenMay-CPL-98, + Author = {N. Ben Amor and D. Maynau}, + Journal = {Chem. Phys. Lett.}, + Pages = {211}, + Volume = {{286}}, + Year = {1998}} + +@book{BenOrs-BOOK-99, + Address = {New York}, + Author = {C. M. Bender and S. A. Orszag}, + Publisher = {Springer}, + Title = {Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory}, + Year = {1999}} + +@article{BenPal-JCP-80, + Author = {M. B\'enard and J. Paldus}, + Journal = {J. Chem. Phys.}, + Pages = {6546}, + Volume = {72}, + Year = {1980}} + +@article{BerGriHie-PRA-06, + Author = {R. A. Bertlmann and W. Grimus and B. C. Hiesmayr}, + Journal = {Phys. Rev. A}, + Pages = {054101}, + Volume = {73}, + Year = {2006}} + +@article{BerLinOlsFul-MP-99, + Author = {A. Bernhardsson and R. Lindh and J. Olsen and M. F\"ulscher}, + Journal = {Mol. Phys.}, + Pages = {617}, + Volume = {{96}}, + Year = {1999}} + +@article{BerShaKry-JCP-12, + Author = {Y. A. Bernard and Y. Shao and A. I. Krylov}, + Journal = {J. Chem. Phys.}, + Pages = {204103}, + Title = {General formulation of spin-flip time-dependent density functional theory using non-collinear kernels: Theory, implementation, and benchmarks}, + Volume = {136}, + Year = {2012}} + +@article{Bes-JCP-06, + Author = {Nicholas A. Besley}, + Journal = {J. Chem. Phys.}, + Pages = {074104}, + Volume = {125}, + Year = {2006}} + +@book{BetSal-BOOK-57, + Address = {Berlin}, + Author = {H. A. Bethe and E. E. Salpeter}, + Publisher = {Springer}, + Title = {Quantum Mechanics of one- and two-electron atoms}, + Year = {1957}} + +@article{BisLuh-PRB-82, + Author = {R. F. Bishop and K. H. L\"{u}hrmann}, + Journal = {Phys. Rev. B}, + Pages = {5523}, + Volume = {{26}}, + Year = {1982}} + +@article{Bis-TCA-91, + Author = {R.F. Bishop}, + Journal = {Theoret. Chim. Acta}, + Pages = {95}, + Volume = {{80}}, + Year = {1991}} + +@article{BlaAtt-APL-11, + Author = {X. Blase and C. Attaccalite}, + Journal = {Appl. Phys. Lett.}, + Pages = {171909}, + Volume = {99}, + Year = {2011}} + +@article{BlaYouRoo-JACS-37, + Author = {F. E. Blacet and W. G. Young and J. G. Roof}, + Journal = {J. Am. Chem. Soc.}, + Pages = {608}, + Volume = {59}, + Year = {1937}} + +@article{BleHesGor-JCP-13, + Author = {P. Bleiziffer and A. He{\ss}elmann and A. G\"orling}, + Journal = {J. Chem. Phys.}, + Pages = {084113}, + Volume = {139}, + Year = {2013}} + +@article{BloGraBau-JACS-84, + Author = {C. E. Blom and G. Grassi and A. Bauder}, + Journal = {J. Am. Chem. Soc.}, + Pages = {7427}, + Volume = {106}, + Year = {1984}} + +@incollection{BobGod-INC-77, + Address = {New York}, + Author = {{F. W. Bobrowicz and W. A. Goddard III}}, + Booktitle = {Methods of Electronic Structure Theory, Modern Theoretical Chemistry}, + Editor = {{H. F. Schaeffer III}}, + Pages = {79}, + Publisher = {Plenum}, + Volume = {3}, + Year = {1977}} + +@article{BoeHan-JCP-02, + Author = {A. Daniel Boese and Nicholas C. Handy}, + Journal = {J. Chem. Phys.}, + Number = {22}, + Pages = {9559}, + Volume = {116}, + Year = {2002}} + +@article{BoeMar-JCP-04, + Author = {A. Daniel Boese and Jan M. L. Martin}, + Journal = {J. Chem. Phys.}, + Number = {8}, + Pages = {3405}, + Volume = {121}, + Year = {2004}} + +@article{BohPin-PR-51, + Author = {D. Bohm and D. Pines}, + Journal = {Phys. Rev.}, + Pages = {625}, + Volume = {82}, + Year = {1951}} + +@article{BohPin-PR-53, + Author = {D. Bohm and D. Pines}, + Journal = {Phys. Rev.}, + Pages = {609}, + Title = {A Collective Description of Electron Interactions: III. Coulomb Interactions in a Degenerate Electron Gas}, + Volume = {{92}}, + Year = {1953}} + +@article{Bon-CPL-84, + Author = {V. E. Bondybey}, + Journal = {Chem. Phys. Lett.}, + Pages = {436}, + Volume = {109}, + Year = {1984}} + +@article{BonEng-JCP-84, + Author = {V. E. Bondybey and J. H. English}, + Journal = {J. Chem. Phys.}, + Pages = {568}, + Volume = {80}, + Year = {1984}} + +@article{BooGruKreAla-Nat-13, + Author = {G. H. Booth and A. Gr\"uneis and G. Kresse and A. Alavi}, + Journal = {Nature}, + Pages = {365}, + Title = {Towards an exact description of electronic wavefunctions in real solids}, + Volume = {493}, + Year = {2013}} + +@article{BooThoAla-JCP-09, + Author = {G. H. Booth and A. J. W. Thom and A. Alavi}, + Journal = {J. Chem. Phys.}, + Number = {{5}}, + Title = {{Fermion Monte Carlo without fixed nodes: A game of life, death, and annihilation in Slater determinant space}}, + Volume = {{131}}, + Year = {{2009}}} + +@article{BorJorNicNac-TCA-98, + Author = {P. Borowski and K. D. Jordan and J. Nichols and P. Nachtigall}, + Journal = {Theor. Chem. Acc.}, + Pages = {135}, + Volume = {99}, + Year = {1998}} + +@article{BouBenMayCaf-JCP-09, + Author = {T. Bouab\c{c}a and N. Ben Amor and D. Maynau and M. Caffarel}, + Journal = {J. Chem. Phys.}, + Pages = {114107}, + Volume = {130}, + Year = {2009}} + +@article{BouBraCaf-JCP-10, + Author = {T. Bouab\c{c}a and B. Bra\"ida and M. Caffarel}, + Journal = {J. Chem. Phys.}, + Pages = {044111}, + Title = {Multi-Jastrow trial wavefunctions for electronic structure calculations with quantum Monte Carlo}, + Volume = {133}, + Year = {2010}} + +@article{BouChiLegDucBlaJac-JCTC-14, + Author = {{P. Boulanger, S. Chibani, B. Le Guennic, I. Duchemin, X. Blase, and D. Jacquemin}}, + Journal = {J. Chem. Theory Comput.}, + Pages = {4548}, + Volume = {10}, + Year = {2014}} + +@article{BouJacDucBla-JCTC-14, + Author = {P. Boulanger and D. Jacquemin and I. Duchemin and X. Blase}, + Journal = {J. Chem. Theory Comput.}, + Pages = {1212}, + Volume = {10}, + Year = {2014}} + +@article{BowSugAld-PRB-94, + Author = {C. Bowen and G. Sugiyama and B. J. Alder}, + Journal = {Phys. Rev. B}, + Pages = {14838}, + Volume = {{50}}, + Year = {1996}} + +@article{BoyCou-JPB-73, + Author = {R. J. Boyd and C. A. Coulson}, + Journal = {J. Phys. B}, + Pages = {782}, + Volume = {6}, + Year = {1973}} + +@article{BoySarUga-JPB-88, + Author = {R. J. Boyd and C. Sarasola and J. M. Ugalde}, + Journal = {J. Phys. B}, + Pages = {2555}, + Volume = {21}, + Year = {1988}} + +@article{BraHib-02, + Author = {B. Braida and L. Thogersen and W. Wu and P. C. Hiberty}, + Journal = {J. Am. Chem. Soc.}, + Pages = {11781}, + Volume = {124}, + Year = {2002}} + +@article{BraHib-04, + Author = {B. Braida and P. C. Hiberty}, + Journal = {J. Am. Chem. Soc.}, + Pages = {14890}, + Volume = {126}, + Year = {2004}} + +@article{BraHib-08, + Author = {B. Braida and P. C. Hiberty}, + Journal = {J. Phys. Chem. A}, + Pages = {13045}, + Volume = {112}, + Year = {2008}} + +@article{BraTouCafUmr-JCP-11, + Author = {B. Bra\"ida and J. Toulouse and M. Caffarel and C. J. Umrigar}, + Journal = {J. Chem. Phys.}, + Pages = {084108}, + Title = {Quantum Monte Carlo with Jastrow-valence-bond wave functions}, + Volume = {134}, + Year = {2011}} + +@misc{BraTouCafUmr-JJJ-XX-note2, + Note = {In our BOVB wave functions, we use different sets of orbital coefficients, but a single set of basis exponents shared by all orbital sets.}} + +@misc{BraTouCafUmr-JJJ-XX-note3, + Note = {Bouab\c{c}a {\it et al.}~\cite{BouBraCaf-JCP-10} have introduced a wave function with several Jastrow factors attached to individual molecular orbitals. With such a wave function, the correlation effects can be treated differently in atomic and binding regions. In the case of the FH molecule, no atomic core Jastrow was used and two different valence Jastrow factors (one for the lone pairs paying a role in the bond and the other one for the $\sigma$-bond) were introduced. The resulting binding energy was essentially exact within error bars. Combining the various VB wave functions discussed in this work with this multi-Jastrow approach is presently under investigation.}} + +@misc{BraTouCafUmr-JJJ-XX-note, + Note = {In the condensed-matter community, ``stricly localized orbitals'' often refers to orbitals that vanish exactly at some finite distance. In the present work, ``stricly localized orbitals'' is employed in the sense usually used in the quantum chemistry community, i.e. orbitals expanded on Gaussian or Slater basis functions centered on a single atom. Thus, these orbitals vanish exactly only at infinite distance.}} + +@article{BreAda-JCP-11, + Author = {E. Br\'emond and C. Adamo}, + Journal = {J. Chem. Phys.}, + Pages = {024106}, + Volume = {135}, + Year = {2011}} + +@article{BreBou-PRB-96, + Author = {J. L. Bretonnet and M. Boulahbak}, + Journal = {Phys. Rev. B}, + Pages = {6859}, + Volume = {{53}}, + Year = {1996}} + +@incollection{BreCepRey-INC-01, + Author = {D. Bressanini and D. M. Ceperley and P. Reynolds}, + Booktitle = {Recent Advances in Quantum Monte Carlo Methods, II}, + Editor = {S. Rothstein}, + Publisher = {World Scientific}, + Title = {What do we know about wave function nodes?}, + Year = {2001}} + +@article{BreMelMor-PRA-98, + Author = {D. Bressanini and M. Mella and G. Morosi}, + Journal = {Phys. Rev. A}, + Pages = {1678}, + Volume = {57}, + Year = {1998}} + +@article{BreMorTar-JCP-05, + Author = {D. Bressanini and G. Morosi and S. Tarasco}, + Journal = {J. Chem. Phys.}, + Pages = {204109}, + Volume = {123}, + Year = {2005}} + +@article{BroDevLem-PRB-80, + Author = {F. Brosens and J. T. Devreese and L. F. Lemmens}, + Journal = {Phys. Rev. B}, + Pages = {1363}, + Volume = {{21}}, + Year = {1980}} + +@article{BroPanSto-PRL-09, + Author = {Christian Brouder and Gianluca Panati and Gabriel Stoltz}, + Journal = {Phys. Rev. Lett.}, + Pages = {230401}, + Volume = {103}, + Year = {2009}} + +@article{BroTraLopNee-JCP-07, + Author = {M. D. Brown and J. R. Trail and P. L\'opez R\'ios and R. J. Needs}, + Journal = {J. Chem. Phys.}, + Pages = {224110}, + Title = {Energies of the first row atoms from quantum Monte Carlo}, + Volume = {126}, + Year = {2007}} + +@article{BroYinLor-SR-13, + Author = {V. Brosco and Z.-J. Ying and J. Lorenzana}, + Journal = {Sci. Rep.}, + Pages = {2172}, + Volume = {3}, + Year = {2013}} + +@article{BruHamNea-JCP-15, + Author = {F. Bruneval and S. M. Hamed and J. B. Neaton}, + Journal = {J. Chem. Phys.}, + Pages = {244101}, + Volume = {142}, + Year = {2015}} + +@article{BruProKroBre-JCP-17, + Author = {Iulia Emilia Brumboiu and Georgia Prokopiou and Leeor Kronik and Barbara Brena}, + Journal = {J. Chem. Phys.}, + Pages = {044301}, + Volume = {147}, + Year = {2017}} + +@phdthesis{Bru-THESIS-05, + Author = {F. Bruneval}, + School = {Ecole Polytechnique}, + Year = {2005}} + +@article{BuiBae-MP-02, + Author = {M. A. Buijse and E. J. Baerends}, + Journal = {Mol. Phys.}, + Pages = {401}, + Volume = {100}, + Year = {2002}} + +@article{BurCruLam-JCP-98, + Author = {K. Burke and F. G. Cruz and K.-C. Lam}, + Journal = {J. Chem. Phys}, + Pages = {8161}, + Volume = {109}, + Year = {1998}} + +@article{BurErnPer-CPL-97, + Author = {K. Burke and M. Ernzerhof and J. P. Perdew}, + Journal = {Chem. Phys. Lett.}, + Pages = {115}, + Volume = {265}, + Year = {1997}} + +@article{BurFilDol-JCP-07, + Author = {M. Burkatzki and C. Filippi and M. Dolg}, + Journal = {J. Chem. Phys.}, + Pages = {234105}, + Volume = {126}, + Year = {2007}} + +@article{Bur-JCP-12, + Author = {K. Burke}, + Journal = {J. Chem. Phys.}, + Pages = {150901}, + Title = {{Perspective on density functional theory}}, + Volume = {136}, + Year = {2012}} + +@article{BurPerErn-JCP-98, + Author = {K. Burke and J. P. Perdew and M. Ernzerhof}, + Journal = {J. Chem. Phys}, + Pages = {3760}, + Volume = {109}, + Year = {1998}} + +@article{BurPer-IJQC-95, + Author = {K. Burke and J. P. Perdew}, + Journal = {Int. J. Quantum. Chem.}, + Pages = {199}, + Volume = {{56}}, + Year = {1995}} + +@article{BurPerLan-PRL-94, + Author = {K. Burke and J. P. Perdew and D. C. Langreth}, + Journal = {Phys. Rev. Lett.}, + Pages = {1283}, + Volume = {73}, + Year = {1994}} + +@article{BurPerLev-PRA-96, + Author = {K. Burke and J. P. Perdew and M. Levy}, + Journal = {Phys. Rev. A}, + Pages = {R2915}, + Volume = {53}, + Year = {1996}} + +@inbook{BurPerWan-INC-98, + Address = {NY}, + Author = {Kieron Burke and John P. Perdew and Y. Wang}, + Booktitle = {Electronic Density Functional Theory: Recent Progress and New Directions}, + Date-Modified = {2013-02-12 00:16:04 +0000}, + Editor = {J. F. Dobson and G. Vignale and M. P. Das}, + Organization = {Plenum}, + Pages = {81}, + Pub-Num = {34}, + Publisher = {Plenum}, + Title = {Derivation of a generalized gradient approximation: The PW91 density functional}, + Url = {http://link.springer.com/chapter/10.1007%2F978-1-4899-0316-7_7}, + Year = {1997}, + Bdsk-Url-1 = {http://link.springer.com/chapter/10.1007%2F978-1-4899-0316-7_7}} + +@article{BygAllMan-JCP-12, + Author = {P. J. Bygrave and N. L. Allan and F. R. Manby}, + Journal = {J. Chem. Phys.}, + Pages = {164102}, + Volume = {137}, + Year = {2012}} + +@article{BylKle-PRB-90, + Author = {D. M. Bylander and L. Kleinman}, + Journal = {Phys. Rev. B}, + Pages = {7868}, + Volume = {41}, + Year = {1990}} + +@article{ByrCotMon-JCP-11, + Author = {{J. N. Byrd, R. C\^ot\'e, and J. A. Montgomery, Jr}}, + Journal = {J. Chem. Phys.}, + Pages = {244307}, + Volume = {135}, + Year = {2011}} + +@article{BytRue-JCP-05, + Author = {L. Bytautas and K. Ruedenberg}, + Journal = {J. Chem. Phys.}, + Pages = {154110}, + Volume = {122}, + Year = {2005}} + +@misc{BytRue-JCP-05-note, + Note = {Note that this estimate of the exact well depth differs from the one used in Ref.~\onlinecite{UmrTouFilSorHeg-JJJ-XX} where we used instead the scalar-relativistic, valence-corrected estimate of Ref.~\onlinecite{BytRue-JCP-05} since calculations were performed with a relativistic pseudopotential.}} + +@article{BytNagGorRue-JCP-07, + Author = {Bytautas,Laimutis and Nagata,Takeshi and Gordon,Mark S. and Ruedenberg,Klaus}, + Journal = {J. Chem. Phys.}, + Number = {16}, + Pages = {164317}, + Title = {Accurate ab initio potential energy curve of F2. I. Nonrelativistic full valence configuration interaction energies using the correlation energy extrapolation by intrinsic scaling method}, + Url = {https://doi.org/10.1063/1.2800017}, + Volume = {127}, + Year = {2007}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.2800017}} + +@article{CadWah-ADNDT-74, + Author = {P. E. Cade and A. C. Wahl}, + Journal = {At. Data Nucl. Data Tables}, + Pages = {340}, + Volume = {13}, + Year = {1974}} + +@article{Caf-ActNum-98, + Author = {R. E. Caflisch}, + Journal = {Acta Numerica}, + Pages = {1}, + Title = {{Monte Carlo and quasi-Monte Carlo methods}}, + Volume = {1998}, + Year = {1998}} + +@article{CafCla-JCP-88, + Author = {M. Caffarel and P. Claverie}, + Doi = {10.1063/1.454228}, + Journal = {J. Chem. Phys.}, + Keywords = {QUANTUM STATISTICAL MECHANICS; MONTE CARLO METHOD; FERMIONS; FEYNMAN PATH INTEGRAL; SCHROEDINGER EQUATION}, + Number = {2}, + Pages = {1100-1109}, + Publisher = {AIP}, + Title = {{Development of a pure diffusion quantum Monte Carlo method using a full generalized Feynman--Kac formula. II. Applications to simple systems}}, + Volume = {88}, + Year = {1988}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.454228}} + +@article{CaiRei-JCP-00, + Author = {Z.-L. Cai and J. R. Reimers}, + Journal = {J. Chem. Phys.}, + Pages = {527}, + Volume = {112}, + Year = {2000}} + +@article{CaiZanKitKocKreScr-PRA-05, + Author = {J. Caillat and J. Zanghellini and M. Kitzler and O. Koch and W. Kreuzer and A. Scrinzi}, + Journal = {Phys. Rev. A}, + Pages = {012712}, + Volume = {71}, + Year = {2005}} + +@article{CalSor-PRB-98, + Author = {{M. Calandra Buonaura and S. Sorella}}, + Journal = {Phys. Rev. B}, + Pages = {11446}, + Title = {{Numerical study of the two-dimensional Heisenberg model using a Green function Monte Carlo technique with a fixed number of walkers}}, + Volume = {57}, + Year = {1998}} + +@article{CanBoyTha-JCP-93, + Author = {N. M. Cann and R. J. Boyd and A. J. Thakkar}, + Journal = {J. Chem. 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Parr}, + Journal = {The Journal of Chemical Physics}, + Keywords = {MOLECULES; MOLECULAR STRAUCTURE; FUNCTIONALS; BORN–OPPENHEIMER APPROXIMATION; GROUND STATES; HAMILTONIANS; ELECTRON DENSITY; NUCLEI; ELECTRIC FIELDS; EXCITED STATES; BOUND STATE; USES}, + Number = {1}, + Pages = {568-573}, + Publisher = {AIP}, + Title = {Non-Born--Oppenheimer density functional theory of molecular systems}, + Url = {http://link.aip.org/link/?JCP/76/568/1}, + Volume = {76}, + Year = {1982}, + Bdsk-Url-1 = {http://link.aip.org/link/?JCP/76/568/1}} + +@article{CarGod-JCP-88, + Author = {{E. A. Carter and W. A. Goddard III}}, + Journal = {J. Chem. Phys.}, + Pages = {3132}, + Volume = {88}, + Year = {1988}} + +@article{CarSilMet-JCP-79, + Author = {D. P. Carroll and H. J. Silverstone and R. M. Metzger}, + Journal = {J. Chem. Phys.}, + Pages = {4142}, + Volume = {71}, + Year = {1979}} + +@article{CarTruGag-JCTC-15, + Author = {R. K. Carlson and D. G. Truhlar and L. Gagliardi}, + Journal = {J. Chem. 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Lett.}, + Pages = {983}, + Volume = {2}, + Year = {2011}} + +@article{EshYarFur-JCP-10, + Author = {Henk Eshuis and Julian Yarkony and Filipp Furche}, + Journal = {J. Chem. Phys.}, + Pages = {234114}, + Volume = {132}, + Year = {2010}} + +@article{EvaShuTul-JPCA-13, + Author = {F. A. Evangelista and P. Shushkov and J. C. Tully}, + Journal = {J. Phys. Chem. A}, + Pages = {7378}, + Volume = {117}, + Year = {2013}} + +@article{Ewa-AP-21, + Author = {P. P. Ewald}, + Journal = {Ann. Phys.}, + Pages = {253}, + Volume = {{64}}, + Year = {1921}} + +@misc{exp_2nd_excited_state, + Note = {Le r{\'e}sultat exp{\'e}rimental dans l'article de Alijah et Duxbury (1.411 eV) a {\'e}t{\'e} corrig{\'e} pour enlever l'{\'e}nergie de point z{\'e}ro en utilisant les calculs FCI/TZ2P de Sherrill et al. (0.004 eV).}} + +@article{FabBouDucAttBla-JCP-13, + Author = {C. Faber and P. Boulanger and I. Duchemin and C. Attaccalite and X. Blase}, + Journal = {J. Chem. Phys.}, + Pages = {194308}, + Volume = {139}, + Year = {2013}} + +@article{FabDel-JCP-07, + Author = {{E. Fabiano and F. Della Sala}}, + Journal = {J. Chem. Phys.}, + Pages = {214102}, + Volume = {126}, + Year = {2007}} + +@article{FabDel-TCA-12, + Author = {{E. Fabiano and F. Della Sala}}, + Journal = {Theor. Chem. Acc.}, + Pages = {1278}, + Volume = {131}, + Year = {2012}} + +@article{FabDucDeuBla-PRB-12, + Author = {C. Faber and I. Duchemin and T. Deutsch and X. Blase}, + Journal = {Phys. Rev. B}, + Pages = {155315}, + Volume = {86}, + Year = {2012}} + +@article{FacEngSchDre-PRL-01, + Author = {{A. Facco Bonetti, E. Engel, R. N. Schmid, and R. M. Dreizler}}, + Journal = {Phys. Rev. Lett.}, + Pages = {2241}, + Volume = {86}, + Year = {2001}} + +@article{FacEngSchDre-PRL-03, + Author = {{A. Facco Bonetti, E. Engel, R. N. Schmid, and R. M. Dreizler}}, + Journal = {Phys. Rev. Lett.}, + Pages = {219302}, + Volume = {90}, + Year = {2003}} + +@incollection{Fah-INC-99, + Address = {Dordrecht}, + Author = {S. Fahy}, + Booktitle = {Quantum Monte Carlo Methods in Physics and Chemistry}, + Editor = {M. P. Nightingale and C. J. Umrigar}, + Pages = {101}, + Publisher = {Kluwer}, + Series = {NATO ASI Ser. C 525}, + Year = {1999}} + +@article{FarHeiEngRob-PRB-93, + Author = {B. Farid and V. Heine and G. E. Engel and I. J. Robertson}, + Journal = {Phys. Rev. B}, + Pages = {11602}, + Volume = {{48}}, + Year = {1993}} + +@article{FasCorSanTru-JPCA-99, + Author = {P. L. Fast and J. Corchado and M. L. Sanchez and D. G. Truhlar}, + Journal = {J. Phys. Chem. A}, + Pages = {3139}, + Volume = {103}, + Year = {1999}} + +@article{FasSanTru-JCP-99, + Author = {P. L. Fast and M. L. S\'anchez and D. G. Truhlar}, + Journal = {J. Chem. Phys.}, + Pages = {2921}, + Volume = {111}, + Year = {1999}} + +@article{Fel-JCC-96, + Author = {D. Feller}, + Journal = {J. Comput. Chem.}, + Pages = {1571}, + Volume = {17}, + Year = {1996}} + +@article{Fel-JCP-13, + Author = {D. Feller}, + Journal = {J. Chem. Phys.}, + Pages = {074103}, + Volume = {138}, + Year = {2013}} + +@article{Fel-JCP-92, + Author = {D. Feller}, + Journal = {J. Chem. Phys.}, + Pages = {6104}, + Volume = {96}, + Year = {1992}} + +@article{Fel-JCP-93, + Author = {D. Feller}, + Journal = {J. Chem. Phys.}, + Pages = {7059}, + Volume = {98}, + Year = {1993}} + +@article{FelPetHil-JCP-10, + Author = {D. Feller and K. A. Peterson and J. G. Hill}, + Journal = {J. Chem. Phys.}, + Pages = {184102}, + Volume = {133}, + Year = {2010}} + +@article{FerAss-JCP-02, + Author = {N. Ferr\'e and X. Assfeld}, + Journal = {J. Chem. Phys.}, + Pages = {4119}, + Volume = {117}, + Year = {2002}} + +@article{FerBalLop-JCTC-15, + Author = {R. G. Fernando and M. C. Balhoff and K. Lopata}, + Journal = {J. Chem. Theory Comput.}, + Pages = {646}, + Volume = {11}, + Year = {2015}} + +@article{FerLopAguEmaRam-IJQC-01, + Author = {{J. Fern\'andez Rico, R. L\'opez, A. Aguado, I. Ema, and G. Ram\'irez}}, + Journal = {Int. J. Quantum Chem.}, + Pages = {148}, + Volume = {81}, + Year = {2001}} + +@article{Fer-PR-57, + Author = {R. A. Ferrell}, + Journal = {Phys. Rev.}, + Pages = {450}, + Volume = {{107}}, + Year = {1957}} + +@article{FerRamLopFer-CCCC-88, + Author = {{J. Fern\'andez Rico, G. Ram\'irez, R. L\'opez and J. I. Fern\'andez-Alonso}}, + Journal = {Collection Czechoslovak Chem. Commun.}, + Pages = {2250}, + Volume = {{53}}, + Year = {1988}} + +@book{FetWal-BOOK-03, + Author = {A. L. Fetter and J. D. Walecka}, + Publisher = {Dover}, + Title = {Quantum Theory of Many-Particle Systems}, + Year = {2003}} + +@article{FilAssMor-JCP-16, + Author = {C. Filippi, R. Assaraf and S. Moroni}, + Journal = {J. Chem. Phys.}, + Pages = {194105}, + Title = {{Simple formalism for efficient derivatives and multi-determinant expansions in quantum Monte Carlo}}, + Volume = {144}, + Year = {2016}} + +@article{FilFah-JCP-00, + Author = {Claudia Filippi and Stephen Fahy}, + Journal = {J. Chem. Phys.}, + Pages = {3523}, + Volume = {112}, + Year = {2000}} + +@incollection{FilGonUmr-INC-96, + Address = {Amsterdam}, + Author = {Claudia Filippi and Xavier Gonze and C. J. Umrigar}, + Booktitle = {Recent Developments of Modern Density Functional Theory}, + Editor = {J. M. Seminario}, + Pages = {295-326}, + Publisher = {Elsevier}, + Title = {Generalized gradient approximations to density functional theory: comparison with exact results}, + Year = {1996}} + +@misc{Fil-PRIV-XX, + Author = {C. Filippi}, + Note = {private communication}} + +@article{FilSha-CPL-00, + Author = {Michael Filatov and Sason Shaik}, + Journal = {Chem. Phys. Lett.}, + Pages = {409}, + Volume = {332}, + Year = {1999}} + +@article{FilSha-CPL-99, + Author = {M. Filatov and S. Shaik}, + Journal = {Chem. Phys. Lett.}, + Pages = {429}, + Title = {{A spin-restricted ensemble-referenced Kohn-Sham method and its application to diradicaloid situations}}, + Volume = {304}, + Year = {1999}} + +@article{FilSha-JPCA-00, + Author = {Michael Filatov and Sason Shaik}, + Journal = {J. Phys. Chem. A}, + Pages = {6628}, + Volume = {104}, + Year = {2000}} + +@article{FilSha-JPCA-99, + Author = {Michael Filatov and Sason Shaik}, + Journal = {J. Phys. Chem. A}, + Pages = {8885}, + Volume = {103}, + Year = {1999}} + +@article{FilShaWoeGriPey-CPL-00, + Author = {Michael Filatov and Sason Shaik and M. Woeller and S. Grimme and S.D. Peyerimhoff}, + Journal = {Chem. Phys. Lett.}, + Pages = {135}, + Volume = {316}, + Year = {2000}} + +@misc{FilTouUmr-JJJ-XX, + Author = {Claudia Filippi and Julien Toulouse and C. J. Umrigar}, + Title = {in preparation}} + +@article{FilUmrGon-JCP-97, + Author = {C. Filippi and C. J. Umrigar and X. Gonze}, + Journal = {J. Chem. Phys.}, + Pages = {9994}, + Title = {{Excitation energies from density functional perturbation theory}}, + Volume = {107}, + Year = {1997}} + +@article{FilUmrGon-PRA-96, + Author = {C. Filippi and C. J. Umrigar and X. Gonze}, + Journal = {Phys. Rev. A}, + Pages = {4810}, + Volume = {54}, + Year = {1996}} + +@article{FilUmr-JCP-96, + Author = {C. Filippi and C. J. Umrigar}, + Journal = {J. Chem. Phys.}, + Pages = {213}, + Title = {Multiconfiguration wave functions for quantum Monte Carlo calculations of first-row diatomic molecules}, + Volume = {{105}}, + Year = {1996}} + +@article{Fil-WIR-14, + Author = {M. Filatov}, + Journal = {WIREs Comput. Mol. Sci.}, + Pages = {146}, + Title = {{Spin-restricted ensemble-referenced Kohn-Sham method: basic principles and application to strongly correlated ground and excited states of molecules}}, + Volume = {5}, + Year = {2015}} + +@article{FilZacBud-JCTC-09, + Author = {C. Filippi and M. Zaccheddu and F. Buda}, + Journal = {J. Chem. Theory Comput.}, + Pages = {2074}, + Title = {Absorption Spectrum of the Green Fluorescent Protein Chromophore: A Difficult Case for ab Initio Methods?}, + Volume = {5}, + Year = {2009}} + +@article{FilZacBud-JJJ-XX, + Author = {Claudia Filippi and Maurizio Zaccheddu and Francesco Buda}, + Note = {unpublished}} + +@incollection{FlaCafSav-INC-97, + Address = {Singapore}, + Author = {H.-J. Flad and M. Caffarel and A. Savin}, + Booktitle = {Recent advances in quantum Monte Carlo methods}, + Editor = {W. A. Lester}, + Pages = {73}, + Publisher = {World Scientific}, + Title = {Quantum Monte Carlo calculations with multi-reference trial wave functions}, + Year = {1997}} + +@misc{Fla-PRIV-XX, + Author = {H. J. Flad}, + Title = {private communication}} + +@article{FlaSav-JCP-95, + Author = {H.-J. Flad and A. Savin}, + Journal = {J. Chem. Phys.}, + Pages = {691}, + Volume = {103}, + Year = {1995}} + +@article{FlaSavPre-JCP-92, + Author = {H.-J. Flad and A. Savin and H. Preuss}, + Journal = {J. Chem. Phys.}, + Pages = {459}, + Volume = {97}, + Year = {1992}} + +@article{FlaSavScuNicPre-CPL-94, + Author = {H.-J. Flad and A. Savin and M. Schultheiss and A. Nicklass and H. Preuss}, + Journal = {Chem. Phys. Lett.}, + Pages = {274}, + Volume = {222}, + Year = {1994}} + +@article{FloGda-JCP-05, + Author = {J. R. Flores and R. J. Gdanitz}, + Journal = {J. Chem. Phys.}, + Pages = {144316}, + Volume = {123}, + Year = {2005}} + +@article{Foc-ZP-30, + Author = {V. Fock}, + Journal = {Z. Phys.}, + Pages = {855}, + Volume = {63}, + Year = {1930}} + +@article{FouMitNeeRaj-RMP-01, + Author = {W. M. C. Foulkes and L. Mitas and R. J. Needs and G. Rajagopal}, + Journal = {Rev. Mod. Phys.}, + Pages = {33}, + Title = {{Quantum Monte Carlo simulations of solids}}, + Volume = {{73}}, + Year = {2001}} + +@article{FraDurMes-JCP-97, + Author = {X. Fradera and M. Duran and J. Mestres}, + Journal = {J. Chem. Phys.}, + Pages = {3576}, + Volume = {107}, + Year = {1997}} + +@article{FraFilAmo-JCTC-12, + Author = {F. Fracchia and C. Filippi and C. Amovilli}, + Journal = {J. Chem. Theory Comput.}, + Pages = {1943}, + Title = {Size-extensive wave functions for quantum Monte Carlo: A linear scaling generalized valence bond approach}, + Volume = {8}, + Year = {2012}} + +@misc{FraLupTou-JJJ-XX, + Author = {O. Franck and E. Luppi and J. Toulouse}, + Note = {unpublished}} + +@article{FraMusLupTou-JCP-15, + Author = {O. Franck and B. Mussard and E. Luppi and J. Toulouse}, + Journal = {J. Chem. Phys.}, + Pages = {074107}, + Title = {Basis convergence of range-separated density-functional theory}, + Volume = {142}, + doi = {10.1063/1.4907920}, + Year = {2015}} + +@misc{FraMusLupTou-JJJ-XX, + Author = {O. Franck and B. Mussard and E. Luppi and J. Toulouse}, + Note = {Basis convergence of range-separated density-functional theory, arXiv:1412.2613, to appear in J. Chem. Phys.}} + +@misc{FraMusLupTou-JJJ-XX-note1, + Note = {Note that the partial-wave expansion of any {\it odd} power of $r_{12}$ contains an infinite number of terms. In contrast, the partial-wave expansion of any {\it even} power of $r_{12}$ terminates at a finite $\ell$.}} + +@misc{FraMusLupTou-JJJ-XX-note2, + Note = {For a discussion of angular and radial correlation and the shape of the correlation hole in the He atom with the long-range interaction, see Ref.~\onlinecite{PolColLeiStoWerSav-IJQC-03}.}} + +@misc{FraMusLupTou-JJJ-XX-note3, + Note = {For $\mu=1$ the difference between the long-range MP2 correlation energies for $X=5$ and $X=6$ is still as small as about 0.2 mhartree.}} + +@misc{FraMusLupTou-JJJ-XX-note4, + Note = {For $r_1=r_2$, $f\lr_\ell >0$ for $\ell=0$ and $f\lr_\ell <0$ for $\ell \geq 1$.}} + +@article{FreHuxMor-PRA-84, + Author = {{D. E. Freund, B. D. Huxtable, and J. D. Morgan III}}, + Journal = {Phys. Rev. A}, + Pages = {980}, + Volume = {29}, + Year = {1984}} + +@article{FreLev-JCP-82, + Author = {Karl F. Freed and Mel Levy}, + Journal = {The Journal of Chemical Physics}, + Keywords = {ALGORITHMS; ELECTRON DENSITY; FUNCTIONALS; SERIES EXPANSION; WAVE FUNCTIONS; GROUND STATES}, + Number = {1}, + Pages = {396-398}, + Publisher = {AIP}, + Title = {Direct first principles algorithm for the universal electron density functional}, + Url = {http://link.aip.org/link/?JCP/77/396/1}, + Volume = {77}, + Year = {1982}, + Bdsk-Url-1 = {http://link.aip.org/link/?JCP/77/396/1}} + +@article{Fre-PRB-77, + Author = {D. L. Freeman}, + Journal = {Phys. Rev. B}, + Pages = {5512}, + Volume = {{15}}, + Year = {1977}} + +@article{FroCimJen-PRA-10, + Author = {E. Fromager and R. Cimiraglia and H. J. Aa. Jensen}, + Journal = {Phys. Rev. A}, + doi = {10.1103/PhysRevA.81.024502}, + Pages = {024502}, + Volume = {81}, + Year = {2010}} + +@article{Fro-JCP-11, + Author = {E. Fromager}, + Journal = {J. Chem. Phys.}, + Pages = {244106}, + Volume = {135}, + Year = {2011}} + +@article{FroJen-JCP-11, + Author = {E. Fromager and H. J. Aa. Jensen}, + Journal = {J. Chem. Phys.}, + Pages = {034116}, + Volume = {135}, + Year = {2011}} + +@article{FroJen-PRA-08, + Author = {E. Fromager and H. J. Aa. Jensen}, + Journal = {Phys. Rev. A}, + Pages = {022504}, + Volume = {78}, + Year = {2008}} + +@misc{FroKneJen-ARX-12, + Author = {Emmanuel Fromager and Stefan Knecht and H. J. Aa. Jensen}, + Note = {\url{http://fr.arxiv.org/abs/1211.4829v1}}} + +@article{FroKneJen-JCP-13, + Author = {E. Fromager and S. Knecht and H. J. Aa. Jensen}, + Journal = {J. Chem. Phys.}, + Pages = {084101}, + Title = {{Multi-configuration time-dependent density-functional theory based on range separation}}, + Volume = {138}, + Year = {2013}} + +@article{FroReaWahWahJen-JCP-09, + Author = {E. Fromager and F. R\'eal and P. W{\aa}hlin and U. Wahlgren and H. J. Aa. Jensen}, + Journal = {J. Chem. Phys.}, + Pages = {054107}, + Volume = {131}, + Year = {2009}} + +@article{FroTouJen-JCP-07, + Author = {E. Fromager and J. Toulouse and H. J. Aa. Jensen}, + Journal = {J. Chem. Phys.}, + Pages = {074111}, + doi = {10.1063/1.2566459}, + Title = {On the universality of the long-/short-range separation in multiconfigurational density-functional theory}, + Volume = {126}, + Year = {2007}} + +@article{FucGon-PRB-02, + Author = {M. Fuchs and X. Gonze}, + Journal = {Phys. Rev. B}, + Pages = {235109}, + Volume = {65}, + Year = {2002}} + +@article{FucNiqGonBur-JCP-05, + Abstract = {We show that density functional theory within the RPA (random phase + approximation for the exchange-correlation energy) provides a correct + description of bond dissociation in H2 in a spin-restricted Kohn-<96>Sham + formalism, i.e., without artificial symmetry breaking. We present + accurate adiabatic connection curves both at equilibrium and beyond + the Coulson-Fisher point. The strong curvature at large bond length + implies important static (left-<96>right) correlation, justifying modern + hybrid functional constructions but also demonstrating their limitations. + Although exact at infinite separation and accurate near the equilibrium + bond length, the RPA dissociation curve displays unphysical repulsion + at larger but finite bond lengths. Going beyond the RPA by including + the exact exchange kernel (RPA + X), we find a similar repulsion. + We argue that this deficiency is due to the absence of double excitations + in adiabatic linear response theory. Further analyzing the H2 dissociation + limit we show that the RPA + X is not size consistent, in contrast + to the RPA.}, + Author = {Fuchs, M. and Niquet, Y. M. and Gonze, X. and Burke, K.}, + Doi = {10.1063/1.1858371}, + Journal = {J. Chem. Phys.}, + Pages = {094116}, + Title = {Describing static correlation in bond dissociation by Kohn-Sham density functional theory}, + Volume = {122}, + Year = {2005}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.1858371}} + +@article{FukIwaSaw-PR-64, + Author = {N. Fukuda and F. Iwamoto and K. Sawada}, + Journal = {Phys. Rev.}, + Pages = {A932}, + Volume = {135}, + Year = {1964}} + +@book{Ful-BOOK-93, + Address = {Berlin}, + Author = {P. Fulde}, + Publisher = {Springer}, + Title = {Electron Correlations in Molecules and Solids}, + Year = {1993}} + +@article{Fur-JCP-01, + Author = {F. Furche}, + Journal = {J. Chem. Phys.}, + Pages = {5982}, + Volume = {114}, + Year = {2001}} + +@article{Fur-JCP-08, + Author = {F. Furche}, + Journal = {J. Chem. Phys.}, + Pages = {114105}, + Volume = {129}, + Year = {2008}} + +@article{FurPer-JCP-06, + Author = {F. Furche and J. P. Perdew}, + Journal = {J. Chem. Phys.}, + Pages = {044103}, + Volume = {124}, + Year = {2006}} + +@article{Fur-PRA-04, + Author = {Filipp Furche}, + Eid = {022514}, + Journal = {Physical Review A (Atomic, Molecular, and Optical Physics)}, + Keywords = {density functional theory; HF calculations; boson systems; electron pairs; ground states; Schrodinger equation}, + Number = {2}, + Numpages = {10}, + Pages = {022514}, + Publisher = {APS}, + Title = {Towards a practical pair density-functional theory for many-electron systems}, + Url = {http://link.aps.org/abstract/PRA/v70/e022514}, + Volume = {70}, + Year = {2004}, + Bdsk-Url-1 = {http://link.aps.org/abstract/PRA/v70/e022514}} + +@article{Fur-PRB-01, + Author = {F. Furche}, + Journal = {Phys. Rev. B}, + Pages = {195120}, + Volume = {64}, + Year = {2001}} + +@incollection{FurRap-INC-05, + Address = {Amsterdam}, + Author = {F. Furche and D. Rappoport}, + Booktitle = {Computational Photochemistry}, + Editor = {M. Olivucci}, + Pages = {93-128}, + Publisher = {Elsevier}, + Series = {Theoretical and Computational Chemistry, Vol. 16}, + Title = {Density functional methods for excited states: equilibrium structure and electronic spectra}, + Year = {2005}} + +@article{FurVoo-JCP-05, + Author = {F. Furche and T. Van Voorhis}, + Journal = {J. Chem. Phys.}, + Pages = {164106}, + Title = {Fluctuation-dissipation theorem density-functional theory}, + Volume = {122}, + Year = {2005}} + +@article{GagTruLiCarHoyBa-ACR-17, + Author = {{L. Gagliardi, D. G. Truhlar, G. Li Manni, R. K. Carlson, C. E. Hoyer, and J. Lucas Bao}}, + Date-Modified = {2020-01-27 08:44:02 +0100}, + Doi = {10.1021/acs.accounts.6b00471}, + Journal = {Acc. Chem. Res.}, + Pages = {66}, + Volume = {50}, + Year = {2017}} + +@book{Gal-BOOK-02, + Author = {G. A. Gallup}, + Publisher = {Cambridge University Press}, + Title = {Valence Bond Methods: Theory and applications}, + Year = {2002}} + +@article{GalBueSar-CPL-03a, + Author = {F. J. G\'alvez and E. Buend\'ia and A. Sarsa}, + Journal = {Chem. Phys. Lett.}, + Pages = {330}, + Volume = {378}, + Year = {2003}} + +@article{GalBueSar-CPL-03, + Author = {F. J. G\'alvez and E. Buend\'ia and A. Sarsa}, + Journal = {Chem. Phys. Lett.}, + Pages = {327}, + Volume = {370}, + Year = {2003}} + +@article{GalBueSar-JCP-01, + Author = {F. J. G\'alvez and E. Buend\'ia and A. Sarsa}, + Journal = {J. Chem. Phys.}, + Pages = {1166}, + Volume = {115}, + Year = {2001}} + +@article{GalBueSar-JCP-02, + Author = {F. J. G\'alvez and E. Buend\'ia and A. Sarsa}, + Journal = {J. Chem. Phys.}, + Pages = {6071}, + Volume = {117}, + Year = {2002}} + +@article{GalBueSar-JCP-03, + Author = {F. J. G\'alvez and E. Buend\'ia and A. Sarsa}, + Journal = {J. Chem. Phys.}, + Pages = {6858}, + Volume = {118}, + Year = {2003}} + +@article{GalBueSar-JCP-05a, + Author = {F. J. G\'alvez and E. Buend\'ia and A. Sarsa}, + Journal = {J. Chem. Phys.}, + Pages = {034302}, + Volume = {123}, + Year = {2005}} + +@article{GalBueSar-JCP-05, + Author = {F. J. G\'alvez and E. Buend\'ia and A. Sarsa}, + Journal = {J. Chem. Phys.}, + Pages = {154307}, + Volume = {122}, + Year = {2005}} + +@article{GalBueSar-JCP-99, + Author = {F. J. G\'alvez and E. Buend\'ia and A. Sarsa}, + Journal = {J. Chem. Phys.}, + Pages = {10903}, + Volume = {111}, + Year = {1999}} + +@article{GalBueSar-PRA-00, + Author = {F. J. G\'alvez and E. Buend\'ia and A. Sarsa}, + Journal = {Phys. Rev. A}, + Pages = {052505}, + Volume = {61}, + Year = {2000}} + +@article{GalHanCohCha-CPL-05, + Author = {P. T. A. Galek and N. C. Handy and A. J. Cohen and G. K.-L. Chan}, + Journal = {Chem. Phys. Lett.}, + Pages = {156}, + Volume = {404}, + Year = {2005}} + +@article{GalHanLes-MP-06, + Author = {{P. T. A. Galek, N. C. Handy, and W. A. Lester Jr}}, + Journal = {Mol. Phys.}, + Pages = {3069}, + Volume = {104}, + Year = {2006}} + +@article{GamCatGra-PRC-09, + Author = {D. 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Phys.}, + Pages = {1985}, + Volume = {8}, + Year = {2006}} + +@article{KadVoo-JCP-10, + Author = {{B. Kaduk and T. Van Voorhis}}, + Journal = {J. Chem. Phys.}, + Pages = {061102}, + Volume = {133}, + Year = {2010}} + +@article{KalTou-JCP-18, + Author = {C. Kalai and J. Toulouse}, + Doi = {10.1063/1.5025561}, + Journal = {J. Chem. Phys.}, + Pages = {164105}, + Title = {A general range-separated double-hybrid density-functional theory}, + Volume = {148}, + Year = {2018}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.5025561}} + +@misc{KalTou-JJJ-XX-note1, + Note = {The geometries are available in the Minnesota Database at http://comp.chem.umn.edu/db/.}} + +@misc{KalTou-JJJ-XX-note2, + Note = {The term ``range-separated double hybrid (RSDH)'' has already been used in Ref.~\onlinecite{SanCivUsvTouShaMas-JCP-15} to designate the RSH+MP2 method and in Refs.~\onlinecite{CorStoJenFro-PRA-13,CorFro-IJQC-14} to generically designate several kinds of range-separated MP2/DFT hybrids. In the present work, we use RSDH to designate the specific scheme corresponding to Eq.~(\ref{EmulRSDH}).}} + +@misc{KalTou-JJJ-XX-note3, + Note = {We use the AE49 set of Fast {\it et al.}~\cite{FasCorSanTru-JPCA-99} to facilitate comparisons with previous related works~\cite{ShaTouSav-JCP-11,SouShaTou-JCP-14,MusReiAngTou-JCP-15} where this set was used. We do not expect any special difficulties with the six molecules that have been dropped from the G2-1 set.}} + +@article{KamSekTsuHir-JCP-05, + Author = {M. Kamiya and H. Sekino and T. Tsuneda and K. Hirao}, + Journal = {J. Chem. Phys.}, + Pages = {234111}, + Volume = {122}, + Year = {2005}} + +@article{KamTsuHir-JCP-02, + Author = {M. Kamiya and T. Tsuneda and K. Hirao}, + Journal = {J. Chem. Phys.}, + Pages = {6010}, + Volume = {{117}}, + Year = {2002}} + +@article{KanBec-JCTC-10, + Author = {F. O. Kannemann and Axel D. Becke}, + Journal = {J. Chem. Theory Comput.}, + Pages = {1081}, + Title = {{van der Waals interactions in density-functional theory: Intermolecular complexes}}, + Volume = {6}, + Year = {2010}} + +@article{Kar-JCP-03, + Author = {V. V. Karasiev}, + Journal = {J. Chem. Phys.}, + Pages = {8576}, + Volume = {118}, + Year = {2003}} + +@article{KarKroKum-JCP-13, + Author = {A. Karolewski and L. Kronik and S. K\"ummel}, + Journal = {J. Chem. Phys.}, + Pages = {204115}, + Volume = {138}, + Year = {2013}} + +@article{KarMar-JCP-11, + Author = {Amit Karton and J. M. L. Martin}, + Journal = {J. Chem. Phys.}, + Pages = {144119}, + Volume = {135}, + Year = {2011}} + +@article{KarTarLamSchMar-JPCA-08, + Author = {Amir Karton and Alex Tarnopolsky and Jean-Francois Lam\`ere and George C. Schatz and Jan M. L. Martin}, + Journal = {J. Phys. Chem. A}, + Pages = {12868}, + Volume = {112}, + Year = {2008}} + +@article{Kat-CPAM-57, + Author = {T. Kato}, + Journal = {Comm. Pure Appl. 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Assaraf}, + Note = {unpublished.}} + +@article{KolTho-NL-12, + Author = {Kolb, Brian and Thonhauser, T.}, + Doi = {10.1142/S1793984412300063}, + Journal = {Nano Life}, + Number = {02}, + Pages = {1230006}, + Title = {Molecular biology at the quantum level: can modern density functional theory forge the path?}, + Volume = {02}, + Year = {2012}, + Bdsk-Url-1 = {https://doi.org/10.1142/S1793984412300063}} + +@article{KopPet-JPCA-02, + Author = {Jacek Koput and Kirk A. Peterson}, + Journal = {J. Phys. Chem. A}, + Pages = {9595}, + Volume = {106}, + Year = {2002}} + +@article{KorBouDucBlaMarBot-JCTC-14, + Author = {S. K\"orbel and P. Boulanger and I. Duchemin and X. Blase and M. A. L. Maarques and S. Botti}, + Journal = {J. Chem. Theory Comput.}, + Pages = {3934}, + Volume = {10}, + Year = {2014}} + +@article{KorLucGri-JPCA-08, + Author = {M. Korth and A. L\"uchow and S. Grimme}, + Journal = {J. Phys. Chem. 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Rice}, + Journal = {Chem. Phys. Lett.}, + Pages = {406}, + Volume = {150}, + Year = {1988}} + +@phdthesis{Lee-THESIS-94, + Address = {Amsterdam}, + Author = {{R. van Leeuwen}}, + School = {Vrije Universitei}, + Title = {Kohn-Sham potentials in density functional theory}, + Type = {{PhD thesis}}, + Year = {1994}} + +@article{LeeYanPar-PRB-88, + Author = {C. Lee and W. Yang and R. G. Parr}, + Journal = {Phys. Rev. B}, + Pages = {785}, + Volume = {37}, + Year = {1988}} + +@article{LefWya-JCP-83, + Author = {Claude Leforestier and Robert E. Wyatt}, + Journal = {J. Chem. Phys.}, + Pages = {2334}, + Volume = {78}, + Year = {1983}} + +@article{LeiDobGro-JCC-99, + Author = {M. Lein and J. F. Dobson and E. K. U. Dobson}, + Journal = {J. Comput. Chem.}, + Pages = {12}, + Volume = {20}, + Year = {1999}} + +@article{LeiGroPer-PRB-00, + Author = {Manfred Lein and E. K. U. Gross and John P. Perdew}, + Journal = {Phys. Rev. B}, + Pages = {13431}, + Volume = {{61}}, + Year = {2000}} + +@article{LeiStoWerSav-CPL-97, + Author = {T. Leininger and H. Stoll and H.-J. Werner and A. Savin}, + Journal = {Chem. Phys. Lett.}, + Pages = {151}, + doi = {10.1016/S0009-2614(97)00758-6}, + Title = {Combining long-range configuration interaction with short-range density functionals}, + Volume = {{275}}, + Year = {1997}} + +@article{LeiTou-AC-14, + Author = {T. Leininger and J. Toulouse}, + Journal = {L'Actualit\'e Chimique}, + Pages = {13-21}, + Title = {{Relever le d\'efi de la r\'esolution de l'\'equation de Schr\"odinger}}, + Volume = {382-383}, + Year = {2014}} + +@article{LemRogChe-PRA-95, + Author = {A. Lembarki and F. Rogemond and H. Chermette}, + Journal = {Phys. Rev. A}, + Pages = {3704}, + Volume = {52}, + Year = {1995}} + +@article{LenBal-JCP-83, + Author = {J. H. van Lenthe and G. G. Balint-Kurti}, + Journal = {J. Chem. Phys.}, + Pages = {5699}, + Volume = {78}, + Year = {1983}} + +@article{Len-JCP-80, + Author = {{B. H. Lengsfield III}}, + Journal = {J. Chem. Phys.}, + Pages = {382}, + Volume = {73}, + Year = {1980}} + +@article{LenLiu-JCP-81, + Author = {{B. H. Lengsfield III and B. Liu}}, + Journal = {J. Chem. Phys.}, + Pages = {478}, + Volume = {75}, + Year = {1981}} + +@article{LenLiu-JCP-81_fr, + Author = {{B. H. Lengsfield III et B. Liu}}, + Journal = {J. Chem. Phys.}, + Pages = {478}, + Volume = {75}, + Year = {1981}} + +@article{LenVerPul-MP-91, + Author = {J. H. van Lenthe and J. Verbeek and P. Pulay}, + Journal = {Mol. Phys.}, + Pages = {1159}, + Volume = {73}, + Year = {1991}} + +@article{LepMalPel-PRA-89, + Author = {M. B. Lepetit and J. P. Malrieu and M. Pelissier}, + Journal = {Phys. Rev. A}, + Pages = {981}, + Volume = {39}, + Year = {1989}} + +@article{LesKra-JCP-66, + Author = {W. A. Lester and M. Krauss}, + Journal = {J. Chem. Phys.}, + Pages = {207}, + Volume = {44}, + Year = {1966}} + +@article{LevBer-IJQC-68, + Author = {B. Levy and G. Berthier}, + Journal = {Int. J. Quantum Chem.}, + Pages = {307}, + Volume = {2}, + Year = {1968}} + +@incollection{Lev-INC-95, + Address = {New York}, + Author = {M. Levy}, + Booktitle = {Density Functional Theory}, + Editor = {E.K.U. Gross and R.M. Dreizler}, + Publisher = {Plenum Press}, + Title = {Coordinate scaling requirements for approximating exchange and correlation}, + Year = {1995}} + +@article{LevMarHan-JCP-96, + Author = {M. Levy and N. H. March and N. C. Handy}, + Journal = {J. Chem. Phys.}, + Pages = {1989}, + Volume = {104}, + Year = {1996}} + +@article{LevNag-PRL-99, + Author = {M. Levy and A. Nagy}, + Journal = {Phys. Rev. Lett.}, + Pages = {4361}, + Volume = {83}, + Year = {1999}} + +@incollection{LevPer-INC-85, + Address = {New York}, + Author = {M. Levy and J. P. Perdew}, + Booktitle = {Density Functional Methods in Physics}, + Editor = {R. M. Dreizler and J. da Providencia}, + Publisher = {Plenum}, + Year = {1985}} + +@article{LevPer-PRA-85, + Author = {M. Levy and J. P. Perdew}, + Journal = {Phys. Rev. A}, + Pages = {2010}, + Volume = {32}, + Year = {1985}} + +@article{LevPer-PRB-93, + Author = {M. Levy and J. P. Perdew}, + Journal = {Phys. Rev. B}, + Pages = {11638}, + Volume = {48}, + Year = {1993}} + +@article{LevPerSah-PRA-84, + Author = {M. Levy and J. P. Perdew and V. Sahni}, + Journal = {Phys. Rev. A}, + Pages = {2745}, + Volume = {30}, + Year = {1984}} + +@article{Lev-PNAS-79, + Author = {M. Levy}, + Journal = {Proc. Natl. Acad. Sci. U.S.A.}, + doi = {10.1073/pnas.76.12.6062}, + Pages = {6062}, + Volume = {76}, + Year = {1979}} + +@article{Lev-PRA-82, + Author = {M. Levy}, + Journal = {Phys. Rev. A}, + Pages = {1200}, + Volume = {26}, + Year = {1982}} + +@article{Lev-PRA-91, + Author = {M. Levy}, + Journal = {Phys. Rev. A}, + Pages = {4637}, + Volume = {43}, + Year = {1991}} + +@article{LevSov-PRA-84, + Author = {Z. H. Levine and P. Soven}, + Journal = {Phys. Rev. A}, + Pages = {625}, + Volume = {29}, + Year = {1984}} + +@article{LevYanPar-JCP-85, + Author = {M. Levy and W. Yang and R. G. Parr}, + Journal = {J. Chem. Phys.}, + Pages = {2334}, + Volume = {83}, + Year = {1985}} + +@article{LieCle-JCP-74a, + Author = {George C. Lie and Enrico Clementi}, + Journal = {J. Chem. Phys.}, + Pages = {1275}, + doi = {https://doi.org/10.1063/1.1681192}, + Title = {Study of the electronic structure of molecules. XXI. Correlation energy corrections as a functional of the Hartree-Fock density and its application to the hydrides of the second row atoms}, + Volume = {60}, + Year = {1974}} + +@article{LieCle-JCP-74b, + Author = {George C. Lie and Enrico Clementi}, + Journal = {J. Chem. Phys.}, + Pages = {1288}, + Title = {Study of the electronic structure of molecules. XXII. Correlation energy corrections as a functional of the Hartree-Fock type density and its application to the homonuclear diatomic molecules of the second row atoms}, + Volume = {60}, + Year = {1974}} + +@article{Lie-IJQC-83, + Author = {E. H. Lieb}, + Journal = {Int. J. Quantum Chem.}, + doi= {10.1002/qua.560240302}, + Pages = {24}, + Volume = {{24}}, + Year = {1983}} + +@article{LieOxf-IJQC-81, + Author = {E. H. Lieb and S. Oxford}, + Journal = {Int. J. Quantum. Chem.}, + Pages = {427}, + Volume = {{19}}, + Year = {1981}} + +@misc{lifetimes-note1, + Note = {For $\ell=0$, $R_2(r)$ only diverges as $1/r$ at $r=0$ which is normalizable, i.e. $\int_0^\infty \text{d}r r^2 |R_2(r)|^2 < \infty$. However, even in this case, the expectation value of the Hamiltonian operator is infinite. For $\ell \geq 1$, $R_2(r)$ is not even normalizable.}} + +@misc{lifetimes-note2, + Note = {The general solution of the Schr\"odinger equation given in Eqs.~(\ref{solR})-(\ref{solR2}) is valid for $Z\not=0$. For $Z=0$, $R_1(r)$ in Eq.~(\ref{solR1}) becomes identically zero, and should be replaced by a function which has the same expression as $R_2(r)$ with the substitution $\sqrt{-2E} \to -\sqrt{-2E}$. The continuum scattering states are then obtained as a linear combination of these two functions. However, our analysis of complex-energy states remains unchanged. Thus, our procedure for determining the lifetimes is still valid for $Z=0$.}} + +@misc{lifetimes-note3, + Note = {Because we do not impose spherical symmetry in our calculations, we do not have exactly degenerate s, p, and d states.}} + +@article{LikMorSen-PRB-97, + Author = {C. N. Likos and S. Moroni and G. Senatore}, + Journal = {Phys. Rev. B}, + Pages = {8867}, + Volume = {{55}}, + Year = {1997}} + +@article{LilTavRotSeb-PRL-04, + Author = {O. A. von Lilienfeld and I. Tavernelli and U. Rothlisberger}, + Journal = {Phys. Rev. Lett.}, + Pages = {153004}, + Volume = {93}, + Year = {2004}} + +@article{LimCarLuoMaOlsTruGag-JCTC-14, + Author = {{G. Li Manni, R. K. Carlson, S. Luo, D. Ma, J. Olsen, D. G. Truhlar, and L. Gagliardi}}, + Journal = {J. Chem. Theory Comput.}, + doi = {https://doi.org/10.1021/ct500483t}, + Pages = {3669}, + Volume = {10}, + Year = {2014}} + +@article{LinRyuLiu-JCP-91, + Author = {R. Lindh and U. Ryu and B. Liu}, + Journal = {J. Chem. Phys.}, + Pages = {5889}, + Volume = {{95}}, + Year = {1991}} + +@article{LinSal-AQC-03, + Author = {Ingvar Lindgren and Sten Salomonson}, + Journal = {Adv. Quantum Chem.}, + Pages = {95-117}, + Title = {Differentiability in density-functional theory}, + Volume = {43}, + Year = {2003}} + +@article{LinTsaLiCha-JCP-12, + Author = {Y.-S. Lin and C.-W. Tsai and G.-D. Li and J.-D. Chai}, + Journal = {J. Chem. Phys.}, + Pages = {154109}, + Volume = {136}, + Year = {2012}} + +@article{LinZhaRap-JCP-00, + Author = {Xi Lin and Hongkai Zhang and Andrew M. Rappe}, + Journal = {J. Chem. Phys.}, + Pages = {2650}, + Volume = {112}, + Year = {2000}} + +@article{LiuEggRefKroNea-JCP-17, + Author = {Zhen-Fei Liu and David A. Egger and Sivan Refaely-Abramson and Leeor Kronik and Jeffrey B. Neaton}, + Journal = {J. Chem. Phys.}, + Pages = {092326}, + Volume = {146}, + Year = {2017}} + +@article{LiuParNag-PRA-95, + Author = {S. Liu and R. G. Parr and A. Nagy}, + Journal = {Phys. Rev. A}, + Pages = {2645}, + Volume = {52}, + Year = {1995}} + +@article{LivBae-PCCP-07, + Author = {E. Livshits and R. Baer}, + Journal = {Phys. Chem. Chem. Phys.}, + Pages = {2932}, + Volume = {9}, + Year = {2007}} + +@article{LjuKovFerFoeSan-PRB-15, + Author = {M. P. Ljungberg and P. Koval and F. Ferrari and D. Foerster and D. S\'anchez-Portal}, + Journal = {Phys. Rev. B}, + Pages = {075422}, + Volume = {92}, + Year = {2015}} + +@article{LocHea-JCP-07, + Author = {Rohini C. Lochan and Martin Head-Gordon}, + Journal = {J. Chem. Phys.}, + Pages = {164101}, + Volume = {126}, + Year = {2007}} + +@article{Lon-DFS-65, + Author = {H. C. Longuet-Higgins}, + Journal = {Discuss. Faraday Soc.}, + Pages = {7}, + Volume = {40}, + Year = {1965}} + +@article{Lon-ZP-30, + Author = {F. London}, + Journal = {Z. Physik}, + Pages = {245}, + Volume = {63}, + Year = {1930}} + +@article{LooGil-WIRES-16, + Author = {Pierre-Francois Loos and Peter M. W. Gill}, + Journal = {WIREs Comput. Mol. Sci.}, + Note = {doi: 10.1002/wcms.1257}, + Pages = {410}, + Title = {The uniform electron gas}, + Volume = {6}, + Year = {2016}} + +@article{LopGov-JCTC-11, + Author = {K. Lopata and N. Govind}, + Journal = {J. Chem. Theory Comput.}, + Pages = {1344}, + Title = {Modeling Fast Electron Dynamics with Real-Time Time-Dependent Density Functional Theory: Application to Small Molecules and Chromophores}, + Volume = {7}, + Year = {2011}} + +@article{LopGov-JCTC-13, + Author = {K. Lopata and N. Govind}, + Journal = {J. Chem. Theory Comput.}, + Pages = {4939}, + Volume = {9}, + Year = {2013}} + +@article{LopMaDruTowNee-PRE-06, + Author = {{P. L\'opez Rios, A. Ma, N. D. Drummond, M. D. Towler, and R. J. Needs}}, + Journal = {Phys. Rev. E}, + Pages = {066701}, + Title = {{Inhomogeneous backflow transformations in quantum Monte Carlo calculations}}, + Volume = {74}, + Year = {2006}} + +@article{LopRamGarFer-JCP-87, + Author = {{R. L\'opez, G. Ram\'irez, J. M. Garc\'ia de la Vega and J. Fern\'andez Rico}}, + Journal = {J. chim. phys.}, + Pages = {695}, + Volume = {{84}}, + Year = {1987}} + +@article{LosFdeAguMar-JPC-07, + Author = {{A. M. Losa, I. Fdez.-Galv\'an, M. A. Aguilar, and M. E. Mart\'in}}, + Journal = {J. Phys. Chem. B}, + Pages = {9864}, + Volume = {111}, + Year = {2007}} + +@article{LotBar-JCP-11, + Author = {V. Lotrich and R. J. Bartlett}, + Journal = {J. Chem. Phys.}, + Pages = {184108}, + Volume = {134}, + Year = {2011}} + +@article{Low-AP-56, + Author = {P.-O. L\"owdin}, + Journal = {Adv. Phys.}, + Pages = {1}, + Volume = {5}, + Year = {1956}} + +@article{Low-JCP-50, + Author = {P.-O. L\"owdin}, + Journal = {J. Chem. Phys.}, + Pages = {365}, + Volume = {18}, + Year = {1950}} + +@article{Low-RMP-63, + Author = {P.-O. L\"owdin}, + Journal = {Rev. Mod. Phys.}, + Pages = {496}, + Volume = {35}, + Year = {1963}} + +@article{LowShu-PR-56, + Author = {P.-O. L\"owdin and H. Shull}, + Journal = {Phys. Rev.}, + Pages = {1730}, + Volume = {101}, + Year = {1956}} + +@article{LufRefPacResRamKroPus-PRB-14, + Author = {D. L\"uftner and Sivan Refaely-Abramson and Michael Pachler and Roland Resel and Michael G. Ramsey and Leeor Kronik and Peter Puschnig}, + Journal = {Phys. Rev. B}, + Pages = {075204}, + Volume = {90}, + Year = {2014}} + +@article{LuLiRocGal-PRL-09, + Author = {Deyu Lu and Yan Li and Dario Rocca and Giulia Galli}, + Journal = {Phys. Rev. Lett.}, + Pages = {206411}, + Volume = {102}, + Year = {2009}} + +@article{LuNguGal-JCP-10, + Author = {Deyu Lu and Huy-Viet Nguyen and Giulia Galli}, + Journal = {J. Chem. Phys.}, + Pages = {154110}, + Volume = {133}, + Year = {2010}} + +@article{LupHea-JCP-13, + Author = {E. Luppi and M. Head-Gordon}, + Journal = {J. Chem. Phys.}, + Pages = {164121}, + Title = {{The role of Rydberg and continuum levels in computing high harmonic generation spectra of the hydrogen atom using time-dependent configuration interaction}}, + Volume = {139}, + Year = {2013}} + +@article{Lut-ZPC-23, + Author = {A. Luthy}, + Journal = {Z. Phys. Chem., Stoechiom. Verwandtschaftsl.}, + Pages = {284}, + Volume = {107}, + Year = {1923}} + +@article{LynTru-JPCA-03, + Author = {B. J. Lynch and D. G. Truhlar}, + Journal = {J. Phys. Chem. A}, + Pages = {8996}, + Title = {Small Representative Benchmarks for Thermochemical Calculations}, + Volume = {107}, + Year = {2003}} + +@misc{LynZhaTru-JJJ-XX, + Note = {The geometries are available in the Minnesota Databases for Chemistry and Solid-State Physics at http://comp.chem.umn.edu/db/.}} + +@article{MaBru-PR-68, + Author = {S.-K. Ma and K. A. Brueckner}, + Journal = {Phys. Rev.}, + Pages = {165}, + Volume = {{165}}, + Year = {1968}} + +@article{MacVos-JPC-79, + Author = {A. H. MacDonald and S. H. Vosko}, + Journal = {J. Phys. C}, + Pages = {2977}, + Volume = {12}, + Year = {1978}} + +@article{MaDruTowNee-PRE-05, + Author = {A. Ma and N. D. Drummond and M. D. Towler and R. J. Needs}, + Journal = {Phys. Rev. E}, + Pages = {066704}, + Volume = {71}, + Year = {2005}} + +@book{Mah-BOOK-09, + Address = {Princeton}, + Author = {Gerald D. Mahan}, + Publisher = {Princeton University Press}, + Title = {Quantum Mechanics in a Nutshell}, + Year = {2009}} + +@article{MaiZhaCavBur-JCP-04, + Author = {N. T. Maitra and F. Zhang and R. J. Cave and K. Burke}, + Journal = {J. Chem. Phys.}, + Pages = {5932}, + Volume = {120}, + Year = {2004}} + +@article{MalMcd-CPL-98, + Author = {N. O. J. Malcolm and J. J. W. McDouall}, + Journal = {Chem. Phys. Lett.}, + Pages = {121}, + Volume = {282}, + Year = {1998}} + +@article{MalMcd-JPC-96, + Author = {N. O. J. Malcolm and J. J. W. McDouall}, + Journal = {J. Phys. Chem.}, + Pages = {10131}, + Volume = {100}, + Year = {1996}} + +@article{MalPenYanBalHol-TCA-14, + Author = {A. Malek and D. Peng and W. Yang and R. Balawender and A. Holas}, + Journal = {Theor. Chem. Acc.}, + Pages = {1559}, + Volume = {133}, + Year = {2014}} + +@article{ManLuc-JCP-01, + Author = {S. Manten and A. L\"uchow}, + Journal = {J. Chem. Phys.}, + Pages = {5362}, + Volume = {115}, + Year = {2001}} + +@article{MarAzaCasSor-JCP-09, + Author = {M. Marchi and S. Azadi and M. Casula and S. Sorella}, + Journal = {J. Chem. Phys.}, + Pages = {154116}, + Title = {Correlated geminal wave function for molecules: An efficient resonating valence bond approach}, + Volume = {131}, + Year = {2009}} + +@article{MarCasMalMulBot-JCP-07, + Author = {M. A. L. Marques and A. Castro and G. Malloci and G. Mulas and S. Botti}, + Journal = {J. Chem. Phys.}, + Pages = {014107}, + Volume = {127}, + Year = {2007}} + +@article{Mar-CPL-96, + Author = {J. M. L. Martin}, + Journal = {Chem. Phys. Lett.}, + Pages = {669}, + Volume = {259}, + Year = {1996}} + +@article{MarGarRub-PRL-06, + Abstract = {We present a first-principles description of anisotropic materials + characterized by having both weak (dispersionlike) and strong covalent + bonds, based on the adiabatic-connection fluctuation-dissipation + theorem with density functional theory. For hexagonal boron nitride + the in-plane and out-of-plane bonding as well as vibrational dynamics + are well described both at equilibrium and when the layers are pulled + apart. Bonding in covalent and ionic solids is also described. The + formalism allows us to ping down the deficiencies of common exchange-correlation + functionals and provides insight toward the inclusion of dispersion + interactions into the correlation functional.}, + Author = {Marini, A. and Garc\'ia-Gonz\'alez, P. and Rubio, A.}, + Journal = {Phys. Rev. Lett.}, + Pages = {136404}, + Title = {First-Principles Description of Correlation Effects in Layered Materials}, + Volume = {96}, + Year = {2006}} + +@article{MarHeiKle-WIRES-18, + Author = {Christel M. Marian and Adrian Heil and Martin Kleinschmidt}, + Journal = {WIREs Comput. Mol. Sci.}, + Note = {doi: 10.1002/wcms.1394}, + Pages = {e1394}, + Title = {The DFT/MRCI method}, + Year = {2018}} + +@article{MarLosFdeAgu-JCP-04, + Author = {{M. E. Mart\'in, A. M. Losa, I. Fdez.-Galv\'an, and M. A. Aguilar}}, + Journal = {J. Chem. Phys.}, + Pages = {3710}, + Volume = {121}, + Year = {2004}} + +@article{MarMea-JCP-78, + Author = {D. J. Margoliash and W. J. Meath}, + Journal = {J. Chem. Phys.}, + Pages = {1426}, + Volume = {68}, + Year = {1978}} + +@article{MaRohMol-JCTC-10, + Author = {Y. Ma and M. Rohlfing and C. Molteni}, + Journal = {J. Chem. Theory Comput.}, + Pages = {257}, + Volume = {6}, + Year = {2010}} + +@article{MaRohMol-PRB-09, + Author = {Y. Ma and M. Rohlfing and C. Molteni}, + Journal = {Phys. Rev. B}, + Pages = {241405}, + Volume = {80}, + Year = {2009}} + +@article{Mar-PR-58, + Author = {N. H. March}, + Journal = {Phys. Rev.}, + Pages = {604}, + Volume = {{110}}, + Year = {1958}} + +@article{MarWer-JPCA-09, + Author = {O. Marchetti and H.-J. Werner}, + Journal = {J. Phys. Chem. 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Mattson}, + Journal = {Science}, + Pages = {759}, + Volume = {298}, + Year = {2002}} + +@article{MccStrWis-PRA-91, + Author = {C. W. McCurdy and C. K. Stroud and M. K. Wisinski}, + Journal = {Phys. Rev. A}, + Pages = {5980}, + Volume = {43}, + Year = {1991}} + +@article{Mcd-MP-03, + Author = {J. J. W. McDouall}, + Journal = {Mol. Phys.}, + Pages = {361}, + Title = {Combining two-body density functionals with multiconfigurational wavefunctions: diatomic molecules}, + Volume = {101}, + Year = {2003}} + +@article{MclBal-RMP-64, + Author = {McLachlan, A. D. and Ball, M. A.}, + Doi = {10.1103/RevModPhys.36.844}, + Journal = {Rev. Mod. Phys.}, + Pages = {844-855}, + Title = {Time-dependent Hartree-Fock Theory for Molecules}, + Volume = {36}, + Year = {1964}, + Bdsk-Url-1 = {https://doi.org/10.1103/RevModPhys.36.844}} + +@book{McW-BOOK-92, + Address = {London}, + Author = {R. McWeeny}, + Publisher = {Academic Press}, + Title = {Methods of Molecular Quantum Mechanics. 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B.}, + Pages = {245103}, + Title = {Total energy of solids: An exchange and random-phase approximation correlation study}, + Volume = {66}, + Year = {2002}} + +@book{Moi-BOOK-11, + Address = {Cambridge}, + Author = {Nimrod Moiseyev}, + Publisher = {Cambridge University Press}, + Title = {Non-Hermitian Quantum Mechanics}, + Year = {2011}} + +@article{MolPle-PR-34, + Author = {C. M{\o}ller and M. S. Plesset}, + Journal = {Phys. Rev.}, + Pages = {618}, + Volume = {46}, + Year = {1934}} + +@misc{Molpro-PROG-02, + Address = {Birmingham, UK}, + Author = {R. D. Amos and A. Bernhardsson and A. Berning and P. Celani and D. L. Cooper and M. J. O. Deegan and A. J. Dobbyn and F. Eckert and C. Hampel and G. Hetzer and P. J. Knowles and T. Korona and R. Lindh and A. W. Lloyd and S. J. {McNicholas} and F. R. Manby and W. Meyer and M. E. Mura and A. Nicklass and P. Palmieri and R. Pitzer and G. Rauhut and M. {Sch\"{u}tz} and U. Schumann and H. Stoll and A. J. Stone and R. Tarroni and T. Thorsteinsson and H.-J. Werner}, + Title = {MOLPRO, a package of ab initio programs designed by {H.-J.~Werner} and {P.~J. Knowles}, version 2002.2}, + Year = {2002}} + +@misc{Molpro-PROG-10, + Address = {Cardiff, UK}, + Author = {H.-J. Werner and P. J. Knowles and G. Knizia and F. R. Manby and M. {Sch\"{u}tz} and P. Celani and T. Korona and R. Lindh and A. Mitrushenkov and G. Rauhut and K. R. Shamasundar and T. B. Adler and R. D. Amos and A. Bernhardsson and A. Berning and D. L. Cooper and M. J. O. Deegan and A. J. Dobbyn and F. Eckert and E. Goll and C. Hampel and A. Hesselmann and G. Hetzer and T. Hrenar and G. Jansen and C. K\"oppl and Y. Liu and A. W. Lloyd and R. A. Mata and A. J. May and S. J. McNicholas and W. Meyer and M. E. Mura and A. Nicklass and D. P. O'Neill and P. Palmieri and K. Pfl\"uger and R. Pitzer and M. Reiher and T. Shiozaki and H. Stoll and A. J. Stone and R. Tarroni and T. Thorsteinsson and M. Wang and A. Wolf}, + Note = {see http://www.molpro.net}, + Title = {MOLPRO, version 2010.1, a package of ab initio programs}, + Year = 2010} + +@misc{Molproshort-PROG-02, + Address = {Birmingham, UK}, + Title = {MOLPRO, a package of ab initio programs designed by {H.-J.~Werner} and {P.~J. Knowles}, version 2002.2}, + Year = {2002}} + +@misc{Molproshort-PROG-08, + Address = {Cardiff, UK}, + Author = {H.-J. Werner and P. J. Knowles and R. Lindh and F. R. Manby and M. {Sch\"{u}tz} and others}, + Note = {see www.molpro.net}, + Title = {MOLPRO, version 2008.2, a package of ab initio programs}, + Year = {2008}} + +@misc{Molproshort-PROG-10, + Author = {{H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby, M. {Sch\"{u}tz}, and others}}, + Note = {Cardiff, UK, 2010, see \url{http://www.molpro.net}}, + Title = {MOLPRO, version 2010.1, a package of ab initio programs}} + +@misc{Molproshort-PROG-12, + Author = {{H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby, M. {Sch\"{u}tz}, and others}}, + Note = {Cardiff, UK, 2012, see \url{http://www.molpro.net}}, + Title = {MOLPRO, version 2012.1, a package of ab initio programs}} + +@misc{Molproshort-PROG-15, + Author = {{H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby, M. {Sch\"{u}tz}, and others}}, + Note = {{Cardiff, UK, 2015, see \url{http://www.molpro.net}}}, + Title = {{MOLPRO, version 2015.1, a package of ab initio programs}}} + +@misc{Molproweb-PROG-XX, + Note = {\url{http://www.molpro.net}}, + Title = {MOLPRO, a package of ab initio programs}} + +@article{MomShi-JCP-87, + Author = {T. Momose and T. Shida}, + Journal = {J. Chem. Phys.}, + Pages = {2832}, + Volume = {87}, + Year = {1987}} + +@article{MorCepSen-PRL-95, + Author = {S. Moroni and D. M. Ceperley and G. Senatore}, + Journal = {Phys. Rev. Lett.}, + Pages = {689}, + Volume = {75}, + Year = {1995}} + +@article{MorCohYan-JCP-06a, + Author = {Paula Mori-S\'anchez and Aron J. Cohen and Weitao Yang}, + Journal = {J. Chem. Phys.}, + Pages = {091102}, + Volume = {124}, + Year = {2006}} + +@article{MorCohYan-JCP-06, + Author = {P. Mori-S{\'a}nchez and A. J. Cohen and W. Yang}, + Journal = {J. Chem. Phys.}, + Pages = {201102}, + Volume = {125}, + Year = {2006}} + +@article{MorCohYan-PRA-12, + Author = {P. Mori-S{\'a}nchez and A. J. Cohen and W. Yang}, + Journal = {Phys. Rev. A}, + Pages = {042507}, + Volume = {85}, + Year = {2012}} + +@article{MorCohYan-PRL-08, + Author = {Paula Mori-S\'anchez and Aron J. Cohen and Weitao Yang}, + Journal = {Phys. Rev. Lett.}, + Pages = {146401}, + Volume = {100}, + Year = {2008}} + +@article{MorCohYan-PRL-09, + Author = {Mori-S\'anchez, Paula and Cohen, Aron and Yang, Weitao}, + Journal = {Phys. Rev. Lett.}, + Pages = {066403}, + Title = {Discontinuous Nature of the Exchange-Correlation Functional in Strongly Correlated Systems}, + Volume = {102}, + Year = {2009}} + +@article{MorMcmClaKimScu-JCTC-12, + Author = {M. A. Morales and J. McMinis and B. K. Clark and J. 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This study is completed by illustrating the well-known isomorphism between the exchange hole and the square of the dominant localized orbital lying in the space region of the reference point of the exchange hole function. 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G. \'Angy\'an}, + Doi = {http://dx.doi.org/10.1007/s00214-015-1751-2}, + Journal = {Theor. Chem. Acc.}, + Pages = {148}, + Volume = {134}, + Year = {2015}, + Bdsk-Url-1 = {http://dx.doi.org/10.1007/s00214-015-1751-2}} + +@incollection{MusCocAssOttUmrTou-AQC-18, + Author = {B. Mussard and E. Coccia and R. Assaraf and M. Otten and C. J. Umrigar and J. Toulouse}, + Booktitle = {Novel Electronic Structure Theory: General Innovations and Strongly Correlated Systems}, + Doi = {10.1016/bs.aiq.2017.05.005}, + Editor = {P. E. Hoggan}, + Pages = {255-270}, + Publisher = {Academic Press}, + Series = {Advances in Quantum Chemistry Vol. 76}, + Title = {Time-dependent linear-response variational Monte Carlo}, + Year = {2018}, + Bdsk-Url-1 = {https://doi.org/10.1016/bs.aiq.2017.05.005}} + +@article{MusReiAngTou-JCP-15, + Author = {B. Mussard and P. Reinhardt and J. G. \'Angy\'an and J. Toulouse}, + doi = {10.1063/1.4918710}, + Journal = {J. Chem. Phys.}, + Note = {Erratum: J. Chem. Phys. {\bf 142}, 219901 (2015)}, + Pages = {154123}, + Volume = {142}, + Year = {2015}} + +@misc{MusReiAngTou-JJJ-XX-note, + Note = {See the supplementary material for the detailed results of the calculations on the AE49 and DBH24/08 datasets.}} + +@article{MusRocJanAng-JCTC-16, + Author = {B. Mussard and D. Rocca and G. Jansen and J. G. \'Angy\'an}, + Doi = {10.1021/acs.jctc.5b01129}, + Journal = {J. Chem. Theory Comput.}, + Pages = {2191-2202}, + Title = {Dielectric Matrix Formulation of Correlation Energies in the Random Phase Approximation: Inclusion of Exchange Effects}, + Volume = {12}, + Year = {2016}, + Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.5b01129}} + +@article{MusRocJanAng-JJJ-XX, + Author = {B. Mussard and D. Rocca and G. Jansen and J. G. \'Angy\'an}, + Note = {unpublished}} + +@article{MusSzaAng-JCTC-14, + Author = {B. Mussard and P. G. Szalay and J. G. \'Angy\'an}, + Journal = {J. Chem. Theory Comput.}, + Pages = {1968}, + Volume = {10}, + Year = {2014}} + +@misc{MusTou-JJJ-XX, + Author = {B. Mussard and J. Toulouse}, + Note = {Fractional-charge and fractional-spin errors in range-separated density-functional theory, Mol. Phys., to appear (2016); preprint arXiv:1607.03621}} + +@article{MusTou-MP-17, + Author = {Bastien Mussard and Julien Toulouse}, + Journal = {Mol. Phys.}, + Pages = {161-173}, + doi = {10.1080/00268976.2016.1213910}, + Title = {Fractional-charge and fractional-spin errors in range-separated density-functional theory}, + Volume = {115}, + Year = {2017}} + +@article{Nag-IJQC-04, + Author = {A. Nagy}, + Journal = {Int. J. Quantum Chem.}, + Pages = {256}, + Volume = {99}, + Year = {2004}} + +@article{Nag-IJQC-98, + Author = {A. Nagy}, + Journal = {Int. J. Quantum Chem.}, + Pages = {681}, + Volume = {70}, + Year = {1998}} + +@article{Nag-JCP-06, + Author = {\'A. Nagy}, + Journal = {J. Chem. 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Phys.}, + Note = {[In Eq.~(70) of this paper the minimization over potentials should actually be a maximization.]}, + Pages = {399-407}, + Publisher = {AIP}, + Title = {Legendre transforms and Maxwell relations in density functional theory}, + Volume = {77}, + Year = {1982}} + +@article{NalPar-JCP-fr-82, + Author = {Roman F. Nalewajski and Robert G. Parr}, + Journal = {J. Chem. Phys.}, + Note = {[Dans l'\'equation~(70) de cet article la minimisation sur le potentiel est en r\'ealit\'e une maximisation.]}, + Pages = {399-407}, + Publisher = {AIP}, + Title = {Legendre transforms and Maxwell relations in density functional theory}, + Volume = {77}, + Year = {1982}} + +@article{NeeSchKosSchGri-JCTC-09, + Author = {F. Neese and T. Schwabe and S. Kossmann and B. Schirmer and S. Grimme}, + Journal = {J. Chem. Theory Comput.}, + Pages = {3060}, + Volume = {5}, + Year = {2009}} + +@article{NemTowNee-JCP-10, + Author = {N. Nemec and M. D. Towler and R. J. Needs}, + Journal = {J. Chem. 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Lett.}, + Pages = {1562}, + Volume = {60}, + Year = {1988}} + +@article{NigMel-PRL-01, + Author = {M. P. Nightingale and V. Melik-Alaverdian}, + Journal = {Phys. Rev. Lett.}, + Pages = {043401}, + Title = {{Optimization of ground- and excited-state wave functions and van der Waals clusters}}, + Volume = {87}, + Year = {2001}} + +@book{NigUmr-BOOK-99, + Address = {Dordrecht}, + Editor = {M. P. Nightingale and C. J. Umrigar}, + Publisher = {Kluwer}, + Series = {NATO ASI Ser. C 525}, + Title = {Quantum Monte Carlo Methods in Physics and Chemistry}, + Year = {1999}} + +@incollection{NigUmr-INC-98, + Address = {NY}, + Author = {M. P. Nightingale and C. J. Umrigar}, + Booktitle = {Monte Carlo Methods in Chemistry}, + Editor = {D. M. Ferguson and J. I. Siepmann and D. G. Truhlar}, + Pages = {Chapter 4}, + Publisher = {Wiley}, + Series = {Advances in Chemical Physics Vol. 105}, + Year = {1998}} + +@article{Nik-PRB-74, + Author = {G. Niklasson}, + Journal = {Phys. Rev. B.}, + Pages = {3052}, + Volume = {{10}}, + Year = {1974}} + +@article{NiqFucGon-JCP-03, + Author = {Y. M. Niquet and M. Fuchs and X. Gonze}, + Journal = {J. Chem. Phys.}, + Pages = {9504}, + Volume = {118}, + Year = {2003}} + +@article{NiqFucGon-PRL-03, + Author = {Y. M. Niquet and M. Fuchs and X. Gonze}, + Journal = {Phys. Rev. Lett.}, + Pages = {219301}, + Volume = {90}, + Year = {2003}} + +@article{NiqGon-PRB-04, + Author = {Y. M. Niquet and X. Gonze}, + Journal = {Phys. Rev. B}, + Pages = {245115}, + Volume = {70}, + Year = {2004}} + +@misc{Nis-BOOK-05, + Title = {NIST Chemistry WebBook, NIST Standard Reference Database Number 69, June 2005 Release}, + Url = {http://webbook.nist.gov/chemistry York}, + Bdsk-Url-1 = {http://webbook.nist.gov/chemistry%20York}} + +@book{NIST-BOOK-05, + Address = {Gaithersburg}, + Editor = {P. J. Linstrom and W. G. Mallard}, + Publisher = {NIST}, + Title = {NIST Chemistry WebBook, NIST Standard Reference Database Number 69}, + Year = {June 2005}} + +@misc{NIST-BOOK-15, + Author = {{NIST Computational Chemistry Comparison and Benchmark Database}}, + Editor = {{Russell D. Johnson III}}, + Title = {{NIST Standard Reference Database Number 101, Release 17b}}, + Url = {http://cccbdb.nist.gov/}, + Year = {September 2015}, + Bdsk-Url-1 = {http://cccbdb.nist.gov/}} + +@article{NogHiyAkiKog-JCP-14, + Author = {Y. Noguchi and M. Hiyama and H. Akiyama and N. Koga}, + Journal = {J. Chem. Phys.}, + Pages = {044309}, + Volume = {141}, + Year = {2014}} + +@article{NooShaMuk-MP-05, + Author = {M. Nooijen and K. R. Shamasundary and D. Mukherjee}, + Journal = {Mol. Phys.}, + Pages = {2277}, + Volume = {103}, + Year = {2005}} + +@article{NorJieSer-JCP-03, + Author = {P. Norman and A. Jiemchooroj and Bo E. Sernelius}, + Journal = {J. Chem. Phys.}, + Pages = {9167}, + Volume = {118}, + Year = {2003}} + +@article{NozPin-PR-58, + Author = {P. Nozi\`eres and D. Pines}, + Journal = {Phys. Rev.}, + Pages = {442}, + Volume = {111}, + Year = {1958}} + +@article{Odd-AQC-78, + Author = {Jens Oddershede}, + Journal = {Adv. Quantum Chem.}, + Pages = {275}, + Title = {Polarization Propagator Calculations}, + Volume = {11}, + Year = {1978}} + +@article{OddJor-JCP-77, + Author = {J. Oddershede and P. J{\o}rgensen}, + Journal = {J. Chem. Phys.}, + Pages = {1541}, + Volume = {66}, + Year = {1977}} + +@article{OlePalOniDel-PRB-99, + Author = {Valerio Olevano and Maurizia Palummo and Giovanni Onida and Rodolfo Del Sole}, + Journal = {Phys. Rev. B}, + Pages = {14224}, + Volume = {{60}}, + Year = {1999}} + +@article{OliBotMar-PCCP-11, + Author = {M. J. T. Oliveira and S. Botti and M. A. L. Marques}, + Journal = {Phys. Chem. Chem. Phys.}, + Pages = {15055}, + Volume = {13}, + Year = {2011}} + +@article{OliPer-PRA-79, + Author = {G. L. Oliver and J. P. Perdew}, + Journal = {Phys. Rev. A}, + Pages = {397}, + Volume = {20}, + Year = {1979}} + +@article{OniReiRub-RMP-02, + Author = {G. Onida and L. Reining and A. Rubio}, + Journal = {Rev. Mod. Phys.}, + Pages = {601}, + Title = {{Electronic excitations: density-functional versus many-body Green's-function approaches}}, + Volume = {74}, + Year = {2002}} + +@article{OrtBal-PRB-94, + Author = {G. Ortiz and P. Ballone}, + Journal = {Phys. Rev. B}, + Pages = {1391}, + Volume = {{50}}, + Year = {1994}} + +@article{OrtBal-PRB-err-97, + Author = {G. Ortiz and P. Ballone}, + Journal = {Phys. Rev. B}, + Pages = {9970}, + Volume = {{56}}, + Year = {1997}} + +@misc{Ort-PROG-XX, + Author = {P. Reinhardt}, + Note = {Different programs to perform ab-initio calculations within highly localized orbitals (Toulouse-Dresden-Paris, 1998--), unpublished}} + +@article{OsiGisSniBae-JCP-97, + Author = {{V. P. Osinga, S. J. A. van Gisbergen, J. G. Snijders, and E. J. Baerends}}, + Journal = {J. Chem. Phys.}, + Pages = {5091}, + Volume = {106}, + Year = {1997}} + +@article{OstKar-CPL-71, + Author = {N. Ostlund and M. Karplus}, + Journal = {Chem. Phys. Lett.}, + Pages = {450}, + Volume = {11}, + Year = {1971}} + +@misc{OthMonMar-ARX-16, + Author = {{A. A. Othman, M. de Montigny, and F. Marsiglio}}, + Note = {\url{http://arxiv.org/abs/1612.06706}}, + Title = {The Coulomb potential in quantum mechanics revisited}} + +@article{Ove-CJP-95, + Author = {A. W. Overhauser}, + Journal = {Can. J. Phys.}, + Pages = {683}, + Volume = {{73}}, + Year = {1995}} + +@article{PacBye-JCP-66, + Author = {{R. T. Pack and W. Byers-Brown}}, + Journal = {J. Chem. Phys.}, + Pages = {556}, + Title = {{Cusp conditions for molecular wavefunctions}}, + Volume = {45}, + Year = {1966}} + +@article{PaiJanHenScuGruKre-JCP-10, + Author = {J. Paier and B. G. Janesko and T. M. Henderson and G. E. Scuseria and A. Gr\"uneis and G. Kresse}, + Journal = {J. Chem. Phys.}, + Pages = {094103}, + Volume = {132}, + Year = {2010}} + +@article{PaiMarHumKreGerAng-JCP-06, + Author = {J. Paier and M. Marsman and K. Hummer and G. Kresse and I. C. Gerber and J. G. \'Angy\'an}, + Journal = {J. Chem. Phys.}, + Pages = {154709}, + Volume = {124}, + Year = {2006}} + +@article{PaiMarKre-JCP-07, + Author = {Joachim Paier and Martijn Marsman and Georg Kresse}, + Doi = {10.1063/1.2747249}, + Eid = {024103}, + Journal = {The Journal of Chemical Physics}, + Keywords = {HF calculations; density functional theory; heat of formation; lattice constants; electron gas; electron correlations; elastic moduli}, + Number = {2}, + Numpages = {10}, + Pages = {024103}, + Publisher = {AIP}, + Title = {Why does the B3LYP hybrid functional fail for metals?}, + Url = {http://link.aip.org/link/?JCP/127/024103/1}, + Volume = {127}, + Year = {2007}, + Bdsk-Url-1 = {http://link.aip.org/link/?JCP/127/024103/1}, + Bdsk-Url-2 = {https://doi.org/10.1063/1.2747249}} + +@article{PaiRenRinScuGruKreSch-NJP-12, + Author = {Joachim Paier and Xinguo Ren and Patrick Rinke and Gustavo E. Scuseria and Andreas Gr\"uneis and Georg Kresse and Matthias Scheffler}, + Journal = {New J. Phys.}, + Pages = {043002}, + Volume = {14}, + Year = {2012}} + +@article{PalCiz-JCP-70, + Author = {J. Paldus and J. {\v C}i{\v z}ek}, + Journal = {J. Chem. Phys.}, + Pages = {2919}, + Volume = {52}, + Year = {1970}} + +@article{PalCis-JCP-71, + Author = {J. Paldus and J. {\v C}i{\v z}ek}, + Journal = {J. Chem. Phys.}, + Pages = {2293}, + Volume = {54}, + Year = {1971}} + +@article{PalCiz-PRA-70, + Author = {J. Paldus and J. {\v C}i{\v z}ek}, + Journal = {Phys. Rev. A}, + Pages = {2268}, + Volume = {2}, + Year = {1970}} + +@article{Pal-JCP-77, + Author = {J. Paldus}, + Journal = {J. Chem. Phys.}, + Pages = {303}, + Volume = {67}, + Year = {1977}} + +@article{PalOniDelCorRei-PRB-99, + Author = {Maurizia Palummo and Giovanni Onida and Rodolfo Del Sole and Massimiliano Corradini and Lucia Reining}, + Journal = {Phys. Rev. B}, + Pages = {11329}, + Volume = {{60}}, + Year = {1999}} + +@article{PalPavHubSch-EPJB-11, + Author = {G. Pal and Y. Pavlyukh and W. H\"ubner and H. C. Schneider}, + Journal = {Eur. Phys. J. B.}, + Pages = {327}, + Volume = {79}, + Year = {2011}} + +@article{Pan-CPL-93, + Author = {I. Panas}, + Journal = {Chem. Phys. Lett.}, + Pages = {255}, + Volume = {201}, + Year = {1993}} + +@article{pan:5642, + Author = {Xiao-Yin Pan and Viraht Sahni}, + Issue = {12}, + Journal = {The Journal of Chemical Physics}, + Keywords = {ground states; molecular configurations; density functional theory; Schrodinger equation; ionisation potential; wave functions; eigenvalues and eigenfunctions; electron correlations; hydrogen}, + Page = {5642-5649}, + Publisher = {AIP}, + Title = {Quantal density functional theory of the hydrogen molecule}, + Url = {http://link.aip.org/link/?JCP/120/5642/1}, + Volume = {120}, + Year = {2004}, + Bdsk-Url-1 = {http://link.aip.org/link/?JCP/120/5642/1}} + +@article{PaqTou-JCP-18, + Author = {Julien Paquier and Julien Toulouse}, + Doi = {10.1063/1.5049773}, + Journal = {J. Chem. Phys.}, + Pages = {174110}, + Title = {Four-component relativistic range-separated density-functional theory: Short-range exchange local-density approximation}, + Volume = {149}, + Year = {2018}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.5049773}} + +@misc{PaqTou-JJJ-XX-note, + Note = {See Supplementary Information for details on the calculation of the sums over spins in Eqs.~(\ref{epsxC}) and (\ref{epsxB}), details on the calculation of the large-$\ct$ expansions in Eqs.~(\ref{epsCsrasymp}) and~(\ref{epsBsrasymp}), and for a Mathematica notebook containing the complete expressions of the large-$\ct$ expansions in Eqs.~(\ref{epsCsrasymp}) and~(\ref{epsBsrasymp}) and of the Pad\'e approximants in Eqs.~(\ref{PadeC}) and~(\ref{PadeB}).}} + +@article{parr:3801, + Author = {Robert G. Parr and Robert A. Donnelly and Mel Levy and William E. Palke}, + Journal = {The Journal of Chemical Physics}, + Number = {8}, + Pages = {3801-3807}, + Publisher = {AIP}, + Title = {Electronegativity: The density functional viewpoint}, + Url = {http://link.aip.org/link/?JCP/68/3801/1}, + Volume = {68}, + Year = {1978}, + Bdsk-Url-1 = {http://link.aip.org/link/?JCP/68/3801/1}} + +@book{ParYan-BOOK-89, + Address = {New York}, + Author = {R. G. Parr and W. Yang}, + Publisher = {Oxford University Press}, + Title = {Density-Functional Theory of Atoms and Molecules}, + Year = {1989}} + +@article{PasGidPer-PRA-13, + Author = {E. Pastorczak and N. I. Gidopoulos and K. Pernal}, + Journal = {Phys. Rev. A}, + Pages = {062501}, + Volume = {87}, + Year = {2013}} + +@article{PatCenJezJezSza-JPCA-07, + Author = {Konrad Patkowski and Wojciech Cencek and M. Jeziorska and B. Jeziorski and Krzysztof Szalewicz}, + Journal = {J. Phys. Chem. A}, + Pages = {7611}, + Volume = {111}, + Year = {2007}} + +@article{PatVas-PRB-72, + Author = {K. N. Pathak and P. Vashishta}, + Journal = {Phys. Rev. B.}, + Pages = {3649}, + Volume = {{7}}, + Year = {1972}} + +@book{Pauncz-BOOK-79, + Address = {New York}, + Author = {R. Pauncz}, + Publisher = {Plenum Press}, + Title = {Spin Eigenfunctions}, + Year = {1979}} + +@article{PauPilTouEll-JCP-10, + Author = {Fran\c{c}oise Pauzat and Julien Pilm\'e and Julien Toulouse and Yves Ellinger}, + Journal = {J. Chem. Phys.}, + Pages = {054301}, + Title = {About the collapse of the 3.3 µm CH stretching band with ionization in polycyclic aromatic hydrocarbons: Configuration interaction and quantum Monte Carlo studies of the CH fragment}, + Volume = {133}, + Year = {2010}} + +@article{PazMorGorBac-PRB-06, + Author = {Simone Paziani and Saverio Moroni and Paola Gori-Giorgi and Giovanni B. Bachelet}, + Journal = {Phys. Rev. B}, + Pages = {155111}, + Volume = {73}, + doi = {10.1103/PhysRevB.73.155111}, + Year = {2006}} + +@article{PazMorGorBac-xxx-xx, + Author = {S. Paziani and S. Moroni and P. Gori-Giorgi and G. B. Bachelet}, + Journal = {in preparation}, + Volume = {{}}, + Year = {2005}} + +@article{PeaBenHelToz-JCP-08, + Author = {M. J. Peach and P. Benfield and T. Helgaker and D. J. Tozer}, + Journal = {J. Chem. Phys.}, + Pages = {044118}, + Volume = {128}, + Year = {2008}} + +@article{PeaCohToz-PCCP-06, + Author = {M. J. G. Peach and A. J. Cohen and D. J. Tozer}, + Journal = {Phys. Chem. Chem. Phys.}, + Pages = {4543}, + Volume = {8}, + Year = {2006}} + +@article{PeaHelSalKeaLutTozHan-PCCP-06, + Author = {M. J. G. Peach and T. Helgaker and P. Salek and T. W. Keal and O. B. Lutn{\ae}s and D. J. Tozer and N. C. Handy}, + Journal = {Phys. Chem. Chem. Phys.}, + Pages = {558}, + Volume = {8}, + Year = {2006}} + +@article{PeaTeaHelToz-JCTC-15, + Author = {M. J. G. Peach and A. M. Teale and T. Helgaker and D. J. Tozer}, + Journal = {J. Chem. Theory Comput.}, + Pages = {5262}, + Volume = {11}, + Year = {2015}} + +@article{PeaToz-JPCA-12, + Author = {M. J. G. Peach and D. J. Tozer}, + Journal = {J. Phys. Chem. A}, + Pages = {9783}, + Volume = {116}, + Year = {2012}} + +@misc{PedJen-JJJ-XX, + Author = {J. K. Pedersen and H. J. A. Jensen}, + Note = {(unpublished)}, + Title = {{A second order MCSCF-DFT hybrid algorithm}}} + +@misc{PedJen-JJJ-XX_fr, + Author = {J. K. Pedersen and H. J. A. Jensen}, + Note = {non publi{\'e}}} + +@phdthesis{Ped-THESIS-04, + Address = {Odense}, + Author = {J. K. Pedersen}, + School = {University of Southern Denmark}, + Title = {Description of correlation and relativistic effects in calculations of molecular properties}, + Type = {{PhD thesis}}, + Year = {2004}} + +@article{PeiNecWar-PRA-03, + Author = {K. Peirs and D. Van Neck and M. Waroquier}, + Eid = {012505}, + Issue = {1}, + Journal = {Physical Review A (Atomic, Molecular, and Optical Physics)}, + Keywords = {density functional theory; atomic structure; Green's function methods; ab initio calculations}, + Numpages = {12}, + Page = {012505}, + Publisher = {APS}, + Title = {Algorithm to derive exact exchange-correlation potentials from correlated densities in atoms}, + Url = {http://link.aps.org/abstract/PRA/v67/e012505}, + Volume = {67}, + Year = {2003}, + Bdsk-Url-1 = {http://link.aps.org/abstract/PRA/v67/e012505}} + +@article{PerBurErn-PRL-96, + Author = {J. P. Perdew and K. Burke and M. Ernzerhof}, + Journal = {Phys. Rev. Lett.}, + Pages = {3865}, + Title = {Generalized Gradient Approximation Made Simple}, + doi = {10.1103/PhysRevLett.77.3865}, + Volume = {77}, + Year = {1996}} + +@article{PerCheVosJacPedSinFio-PRB-92, + Author = {J. P. Perdew and J. A. Chevary and S. H. Vosko and K. A. Jackson and M. R. Pederson and D. J. Singh and C. Fiolhais}, + Journal = {Phys. Rev. B}, + Pages = {6671}, + Volume = {{46}}, + Year = {1992}} + +@book{PerEngDreGroGodNogCasMar-BOOK-03, + Address = {Berlin}, + Editor = {C. Fiolhais and F. Nogueira and M. A. L. Marques}, + Publisher = {Springer}, + Series = {Vol. 620 of Lecture Notes in Physics}, + Title = {A Primer in Density Functional Theory}, + Year = {2003}} + +@article{PerErnBur-JCP-96, + Author = {J. P. Perdew and M. Ernzerhof and K. Burke}, + Journal = {J. Chem. Phys}, + Pages = {9982}, + Volume = {105}, + Year = {1996}} + +@article{PerErnBurSav-IJQC-97, + Author = {J. P. Perdew and M. Ernzerhof and K. Burke and A. Savin}, + Journal = {Int. J. Quantum Chem.}, + Pages = {197}, + Volume = {{61}}, + Year = {1997}} + +@article{PerGieGriBae-JCP-07, + Author = {K. Pernal and K. Giesbertz and O. Gritsenko and E. J. Baerends}, + Journal = {J. Chem. Phys.}, + Pages = {214101}, + Volume = {127}, + Year = {2007}} + +@article{PerGriBae-PRA-07, + Author = {K. Pernal and O. Gritsenko and E. J. Baerends}, + Journal = {Phys. Rev. A}, + Pages = {012506}, + Volume = {75}, + Year = {2007}} + +@article{Per-IJQC-93, + Author = {J. P. Perdew}, + Journal = {Int. J. Quantum. Chem.: Quantum Chem. Symp.}, + Pages = {93}, + Volume = {27}, + Year = {1993}} + +@incollection{Per-INC-91, + Address = {berlin}, + Author = {J. P. Perdew}, + Booktitle = {Electronic Structure of Solids '91}, + Editor = {P. Ziesche and H. Eschrig}, + Publisher = {Akademie Verlag}, + Year = {1991}} + +@article{Per-JCP-12, + Author = {K. Pernal}, + Journal = {J. Chem. Phys.}, + Pages = {184105}, + Title = {{Excitation energies from range-separated time-dependent density and density matrix functional theory}}, + Volume = {136}, + Year = {2012}} + +@article{PerLanSah-PRL-77, + Author = {J. P. Perdew and D. C. Langreth and V. Sahni}, + Journal = {Phys. Rev. Lett.}, + Pages = {1030}, + Volume = {{38}}, + Year = {1977}} + +@article{PerLev-PRL-83, + Author = {J. P. Perdew and M. Levy}, + Journal = {Phys. Rev. Lett.}, + Pages = {1884}, + Volume = {51}, + Year = {1983}} + +@article{PerMcMZun-PRA-81, + Author = {J. P. Perdew and E. R. McMullen and A. Zunger}, + Journal = {Phys. Rev. A}, + Pages = {2785}, + Volume = {{23}}, + Year = {1981}} + +@article{PerParLevBal-PRL-82, + Author = {J. P. Perdew and R. G. Parr and M. Levy and J. L. Balduz}, + Journal = {Phys. Rev. Lett.}, + Pages = {1691}, + Title = {Density-functional theory for fractional particle number: Derivative discontinuity of the energy}, + Volume = {49}, + Year = {1982}} + +@article{PerPerMorIll-JCC-07, + Author = {A. J. P\'erez-Jim\'enez and J. M. P\'erez-Jord\'a and I. De P. R. Moreira and F. Illas}, + Journal = {J. Comput. Chem.}, + Pages = {2559}, + Volume = {28}, + Year = {2007}} + +@article{PerPerSan-JCP-07, + Author = {A. J. P\'erez-Jim\'enez and J. M. P\'erez-Jord\'a and J. C. Sancho-Garc\'ia}, + Journal = {J. Chem. Phys.}, + Pages = {104102}, + Title = {Combining two-body density correlation functionals with multiconfigurational wave functions using natural orbitals and occupation numbers}, + Volume = {127}, + Year = {2007}} + +@article{Per-PRA-10, + Author = {Katarzyna Pernal}, + Journal = {Phys. Rev. A}, + Pages = {052511}, + Volume = {81}, + Year = {2010}} + +@article{Per-PRB-86, + Author = {J. P. Perdew}, + Journal = {Phys. Rev. B}, + Pages = {8822}, + Volume = {{33}}, + Year = {1986}} + +@article{Per-PRL-85, + Author = {J. P. Perdew}, + Journal = {Phys. Rev. Lett.}, + Pages = {1665}, + Volume = {{55}}, + Year = {1985}} + +@article{PerSavBur-PRA-95, + Author = {J. P. Perdew and A. Savin and K. Burke}, + Journal = {Phys. Rev. A}, + doi = {10.1103/PhysRevA.51.4531}, + Pages = {4531}, + Title = {{Escaping the symmetry dilemma through a pair-density interpretation of spin-density functional theory}}, + Volume = {51}, + Year = {1995}} + +@article{PerWan2-PRB-92, + Author = {J. P. Perdew and Y. Wang}, + Journal = {Phys. Rev. B}, + Pages = {12947}, + Volume = {{46}}, + Year = {1992}} + +@article{PerWan-PRB-86, + Author = {J. P. Perdew and Y. Wang}, + Journal = {Phys. Rev. B}, + Pages = {8800}, + Volume = {{33}}, + Year = {1986}} + +@article{PerWan-PRB-92, + Author = {J. P. Perdew and Y. Wang}, + Journal = {Phys. Rev. B}, + Pages = {13244}, + Volume = {45}, + Year = {1992}} + +@article{PerZun-PRB-81, + Author = {J. P. Perdew and A. Zunger}, + Journal = {Phys. Rev. B}, + Pages = {5048}, + Volume = {23}, + Year = {1981}} + +@article{PetDun-JCP-02, + Author = {{K. A. Peterson and T. H. Dunning, Jr}}, + Journal = {J. Chem. Phys.}, + Pages = {10548}, + Volume = {117}, + Year = {2002}} + +@article{PetDun-JPC-95, + Author = {{K. A. Peterson and T. H. Dunning Jr.}}, + Journal = {J. Phys. Chem.}, + Pages = {3898}, + Volume = {99}, + Year = {1995}} + +@article{PetGosGro-PRL-96, + Author = {M. Petersilka and U. J. Gossmann and E. K. U. Gross}, + Journal = {Phys. Rev. Lett.}, + Pages = {1212}, + Volume = {76}, + Year = {1996}} + +@article{PetGro-IJQC-96, + Author = {M. Petersilka and E. K. U. Gross}, + Journal = {Int. J. Quantum Chem. Symp.}, + Pages = {1393}, + Volume = {30}, + Year = {1996}} + +@article{PetHolChaNigUmr-PRL-12, + Author = {F. R. Petruzielo and A. A. Holmes and H. J. Changlani and M. P. Nightingale and C. J. Umrigar}, + Journal = {Phys. Rev. Lett.}, + Pages = {230201}, + Title = {Semistochastic Projector Monte Carlo Method}, + Volume = {109}, + Year = {2012}} + +@article{PetKenDun-JCP-93b, + Author = {{K. A. Peterson, R. A. Kendall and T. H. Dunning Jr.}}, + Journal = {J. Chem. Phys.}, + Pages = {9790}, + Volume = {99}, + Year = {1993}} + +@article{PetTouUmr-JCP-11, + Author = {F. R. Petruzielo and J. Toulouse and C. J. Umrigar}, + Journal = {J. Chem. Phys.}, + Pages = {064104}, + Title = {Basis set construction for molecular electronic structure theory: Natural orbital and Gauss--Slater basis for smooth pseudopotentials}, + Volume = {134}, + Year = {2011}} + +@article{PetTouUmr-JCP-12, + Author = {F. R. Petruzielo and J. Toulouse and C. J. Umrigar}, + Journal = {J. Chem. Phys.}, + Pages = {124116}, + Title = {{Approaching chemical accuracy with quantum Monte Carlo}}, + Volume = {136}, + Year = {2012}} + +@article{PevHea-JCP-13, + Author = {Roberto Peverati and Martin Head-Gordon}, + Journal = {J. Chem. Phys.}, + Pages = {024110}, + Title = {Orbital optimized double-hybrid density functionals}, + Volume = {139}, + Year = {2013}} + +@article{PevTru-JPCL-11, + Author = {Roberto Peverati and Donald G. Truhlar}, + Journal = {J. Phys. Chem. Lett.}, + Pages = {2810}, + Volume = {2}, + Year = {2011}} + +@article{PevTru-JPCL-12, + Author = {Roberto Peverati and Donald G. Truhlar}, + Journal = {J. Phys. Chem. Lett.}, + Pages = {117}, + Volume = {3}, + Year = {2012}} + +@article{PevTru-PTRSA-14, + Abstract = {Kohn{\textendash}Sham density functional theory is in principle an exact formulation of quantum mechanical electronic structure theory, but in practice we have to rely on approximate exchange{\textendash}correlation (xc) functionals. The objective of our work has been to design an xc functional with broad accuracy across as wide an expanse of chemistry and physics as possible, leading{\textemdash}as a long-range goal{\textemdash}to a functional with good accuracy for all problems, i.e. a universal functional. To guide our path towards that goal and to measure our progress, we have developed{\textemdash}building on earlier work of our group{\textemdash}a set of databases of reference data for a variety of energetic and structural properties in chemistry and physics. These databases include energies of molecular processes, such as atomization, complexation, proton addition and ionization; they also include molecular geometries and solid-state lattice constants, chemical reaction barrier heights, and cohesive energies and band gaps of solids. For this paper, we gather many of these databases into four comprehensive databases, two with 384 energetic data for chemistry and solid-state physics and another two with 68 structural data for chemistry and solid-state physics, and we test two wave function methods and 77 density functionals (12 Minnesota meta functionals and 65 others) in a consistent way across this same broad set of data. We especially highlight the Minnesota density functionals, but the results have broader implications in that one may see the successes and failures of many kinds of density functionals when they are all applied to the same data. Therefore, the results provide a status report on the quest for a universal functional.}, + Author = {Peverati, Roberto and Truhlar, Donald G.}, + Doi = {10.1098/rsta.2012.0476}, + Isbn = {1471-2962}, + Issn = {1364-503X}, + Journal = {Phil. Trans. R. Soc. A}, + Note = {preprint at http://arxiv.org/abs/1212.0944}, + Number = {2011}, + Pages = {20120476}, + Publisher = {The Royal Society}, + Title = {Quest for a universal density functional: the accuracy of density functionals across a broad spectrum of databases in chemistry and physics}, + Volume = {372}, + Year = {2014}, + Bdsk-Url-1 = {https://doi.org/10.1098/rsta.2012.0476}} + +@article{PhiKanZgi-JCP-15, + Author = {J. J. Phillips and A. A. Kananenka and D. Zgid}, + Journal = {J. Chem. Phys.}, + Pages = {194108}, + Volume = {142}, + Year = {2015}} + +@article{PicGos-MP-73, + Author = {B. T. Pickup and O. Goscinski}, + Journal = {Mol. Phys.}, + Pages = {1013}, + Volume = {26}, + Year = {1973}} + +@book{PinNoz-BOOK-89, + Author = {D. Pines and P. Nozi\`eres}, + Publisher = {Addison-Wesley Publishing Company}, + Title = {The Theory of Quantum Liquids}, + Year = {1989}} + +@article{PisDov-IJQC-80, + Author = {C. Pisani and R. Dovesi}, + Journal = {Int. J. Quantum Chem.}, + Pages = {501}, + Volume = {17}, + Year = {1980}} + +@article{PisMasCasHalSchUsv-JCC-08, + Author = {C. Pisani and L. Maschio and S. Casassa and M. Halo and M. Sch\"utz and D. Usvyat}, + Journal = {J. Comput. Chem.}, + Pages = {2113}, + Volume = {29}, + Year = {2008}} + +@article{BytLaiRuedenJCP05, + Author = {Bytautas,Laimutis and Ruedenberg,Klaus}, + Date-Modified = {2019-12-13 10:34:29 +0100}, + Doi = {10.1063/1.1869493}, + Journal = {J. Chem. Phys.}, + Number = {15}, + Pages = {154110}, + Title = {Correlation energy extrapolation by intrinsic scaling. IV. Accurate binding energies of the homonuclear diatomic molecules carbon, nitrogen, oxygen, and fluorine}, + Volume = {122}, + Year = {2005}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.1869493}} + +@article{pittner:10275, + Author = {Jiri Pittner and Petr Nachtigall and Petr Carsky and Jozef Masik and Ivan Hubac}, + Journal = {The Journal of Chemical Physics}, + Keywords = {organic compounds; silicon compounds; coupled cluster calculations; configuration interactions}, + Number = {21}, + Pages = {10275-10282}, + Publisher = {AIP}, + Title = {Assessment of the single-root multireference Brillouin--Wigner coupled- cluster method: Test calculations on CH[sub 2], SiH[sub 2], and twisted ethylene}, + Url = {http://link.aip.org/link/?JCP/110/10275/1}, + Volume = {110}, + Year = {1999}, + Bdsk-Url-1 = {http://link.aip.org/link/?JCP/110/10275/1}} + +@article{PolColLeiStoWerSav-IJQC-03, + Author = {R. Pollet and F. Colonna and T. Leininger and H. Stoll and H.-J. Werner and A. Savin}, + Journal = {Int. J. Quantum. Chem.}, + Pages = {84}, + Title = {Exchange-Correlation Energies and Correlation Holes for Some Two- and Four-Electron Atoms along a Nonlinear Adiabatic Connection in Density Functional Theory}, + Volume = {{91}}, + Year = {2003}} + +@article{PolSavLeiSto-JCP-02, + Author = {R. Pollet and A. Savin and T. Leininger and H. Stoll}, + Journal = {J. Chem. Phys.}, + Pages = {1250}, + Title = {Combining multideterminantal wave functions with density functionals to handle near-degeneracy in atoms and molecules}, + Volume = {{116}}, + Year = {2002}} + +@phdthesis{Pol-THESIS-01, + Author = {R. Pollet}, + School = {Universit\'e Paris 6}, + Title = {D\'eveloppement et applications d'une m\'ethode g\'en\'eralisant la th\'eorie de la fonctionnelle de la densit\'e \`a plusieurs d\'eterminants de Slater}, + Type = {Th\`ese de doctorat}, + Year = {2001}} + +@article{Pop-RMP-99, + Author = {J. A. Pople}, + Date-Modified = {2020-01-27 08:35:34 +0100}, + Doi = {10.1103/RevModPhys.71.1267}, + Journal = {Rev. Mod. Phys.}, + Pages = {1267}, + Title = {{Nobel Lecture: Quantum chemical models}}, + Volume = {{71}}, + Year = {1999}} + +@article{PorDer-PRA-02, + Author = {S. G. Porsev and A. Derevianko}, + Journal = {Phys. Rev. A}, + Pages = {020701(R)}, + Volume = {65}, + Year = {2002}} + +@article{PraWooPetDunWil-TCA-11, + Author = {{B. P. Prascher, D. E. Woon, K. A. Peterson, T. H. Dunning, Jr., and A. K. Wilson}}, + Journal = {Theor. Chem. Acc.}, + Pages = {69}, + Volume = {128}, + Year = {2011}} + +@article{PreBevFah-PRB-02, + Author = {David Prendergast and David Bevan and Stephen Fahy}, + Journal = {Phys. Rev. B}, + Pages = {155104}, + Volume = {66}, + Year = {2002}} + +@article{PreNolFilFahGre-JCP-01, + Author = {D. Prendergast and M. Nolan and C. Filippi and S. Fahy and J. C. Greer}, + Journal = {J. Chem. Phys.}, + Pages = {1626}, + Volume = {{115}}, + Year = {2001}} + +@book{PreTeuVetFla-BOOK-92, + Address = {Cambridge}, + Author = {W. Press and S. Teukolsky and W. Vetterling and B. Flannery}, + Publisher = {Cambridge University Press}, + Title = {Numerical Recipes}, + Year = {1992}} + +@article{Pul-MP-69, + Author = {P. Pulay}, + Journal = {Mol. Phys.}, + Pages = {197}, + Volume = {17}, + Year = {1969}} + +@article{PurZhaKra-JCP-09, + Author = {W. Purwanto and S. Zhang and H. Krakauer}, + Journal = {J. Chem. Phys.}, + Pages = {094107}, + Volume = {130}, + Year = {2009}} + +@misc{Qmc-PROG-XX, + Note = {QMCMOL, a quantum Monte Carlo program written by R. Assaraf, F. Colonna, X. Krokidis, P. Reinhardt and coworkers.}, + Url = {http://www.lct.jussieu.fr/pagesequipe/qmcmol/qmcmol/}, + Bdsk-Url-1 = {http://www.lct.jussieu.fr/pagesequipe/qmcmol/qmcmol/}} + +@misc{Quantumexpressoweb-PROG-XX, + Note = {\url{http://www.quantum-espresso.org}}, + Title = {QUANTUM ESPRESSO, an integrated suite of Open-Source computer codes for electronic-structure calculations and materials modeling at the nanoscale}} + +@article{RabBaeNeu-PRB-15, + Author = {E. Rabani and R. Baer and D. Neuhauser}, + Journal = {Phys. Rev. B}, + Pages = {235302}, + Volume = {91}, + Year = {2015}} + +@article{rabinovitch:1807, + Author = {B. S. Rabinovitch and John E. Douglas and F. S. Looney}, + Journal = {The Journal of Chemical Physics}, + Number = {11}, + Pages = {1807-1808}, + Publisher = {AIP}, + Title = {The Thermal Trans-Cis Isomerization of Dideuteroethylene}, + Url = {http://link.aip.org/link/?JCP/20/1807/2}, + Volume = {20}, + Year = {1952}, + Bdsk-Url-1 = {http://link.aip.org/link/?JCP/20/1807/2}} + +@book{Rai-BOOK-72, + Address = {Amsterdam-London}, + Author = {S. Raimes}, + Publisher = {North-Holland Publishing Company}, + Title = {Many-electron theory}, + Year = {1972}} + +@article{Raj-ACP-80, + Author = {A. K. Rajagopal}, + Journal = {Adv. Chem. Phys.}, + Pages = {59}, + Volume = {41}, + Year = {1980}} + +@article{RajCal-PRB-73, + Author = {A. K. Rajagopal and J. Callaway}, + Journal = {Phys. Rev. B}, + Pages = {1912}, + Volume = {7}, + Year = {1973}} + +@article{Raj-JPC-78, + Author = {A. K. Rajagopal}, + Journal = {J. Phys. C}, + Pages = {L943}, + Volume = {11}, + Year = {1978}} + +@article{RamLeoNeu-ARPC-16, + Author = {Krupa Ramasesha and Stephen R. Leone and Daniel M. Neumark}, + Journal = {Annu. Rev. Phys. Chem.}, + Pages = {41}, + Title = {Real-Time Probing of Electron Dynamics Using Attosecond Time-Resolved Spectroscopy}, + Volume = {67}, + Year = {2016}} + +@article{RamRaj-PRA-82, + Author = {M. V. Ramana and A. K. Rajagopal}, + Journal = {Phys. Rev. A}, + Pages = {96}, + Year = {1982}} + +@article{RasChi-JCP-96a, + Author = {V. A. Rassolov and D. M. Chipman}, + Journal = {J. Chem. Phys.}, + Pages = {1470}, + Volume = {105}, + Year = {1996}} + +@article{RasChi-JCP-96b, + Author = {V. A. Rassolov and D. M. Chipman}, + Journal = {J. Chem. Phys.}, + Pages = {1479}, + Volume = {105}, + Year = {1996}} + +@misc{RebSavTou-JJJ-XX, + Author = {E. Rebolini and A. Savin and J. Toulouse}, + Note = {unpublished}} + +@article{RebSavTou-MP-13, + Author = {E. Rebolini and A. Savin and J. Toulouse}, + Journal = {Mol. Phys.}, + Pages = {1219}, + Title = {Electronic excitations from a linear-response range-separated hybrid scheme}, + Volume = {111}, + Year = {2013}} + +@article{RebTeaHelSavTou-MP-18, + Author = {E. Rebolini and A. M. Teale and T. Helgaker and A. Savin and J. Toulouse}, + Doi = {doi.org/10.1080/00268976.2017.1422811}, + Journal = {Mol. Phys.}, + Pages = {1443-1451}, + Title = {Excitation energies from G\"orling--Levy perturbation theory along the range-separated adiabatic connection}, + Volume = {116}, + Year = {2018}, + Bdsk-Url-1 = {https://doi.org/10.1080/00268976.2017.1422811}} + +@phdthesis{Reb-THESIS-14, + Author = {E. Rebolini}, + School = {Universit\'e Pierre et Marie Curie}, + Title = {Range-separated density-functional theory for molecular excitation energies}, + Type = {{Th\`ese de doctorat}}, + Url = {https://tel.archives-ouvertes.fr/tel-01027522}, + Year = {2014}, + Bdsk-Url-1 = {https://tel.archives-ouvertes.fr/tel-01027522}} + +@phdthesis{Reb-THESIS-14_eng, + Author = {E. Rebolini}, + School = {Universit\'e Pierre et Marie Curie}, + Title = {Range-separated density-functional theory for molecular excitation energies}, + Type = {{PhD thesis}}, + Url = {https://tel.archives-ouvertes.fr/tel-01027522}, + Year = {2014}, + Bdsk-Url-1 = {https://tel.archives-ouvertes.fr/tel-01027522}} + +@article{RebTou-JCP-16, + Author = {E. Rebolini and J. Toulouse}, + Doi = {http://dx.doi.org/10.1063/1.4943003}, + Journal = {J. Chem. Phys.}, + Pages = {094107}, + Title = {Range-separated time-dependent density-functional theory with a frequency-dependent second-order Bethe-Salpeter correlation kernel}, + Volume = {144}, + Year = {2016}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4943003}} + +@misc{RebTou-JJJ-XX-note1, + Note = {The last two terms of the kernel in Eq. (31) of Ref.~\onlinecite{ZhaSteYan-JCP-13} contain non-antisymmetrized two-electron integrals. However, these terms can also be written with a factor of $1/2$ and antisymmetrized two-electron integrals, leading to our Eq.~(\ref{eq:eff kernel iajb}).}} + +@incollection{RebTouSav-INC-13, + Author = {E. Rebolini and J. Toulouse and A. Savin}, + Booktitle = {Electronic Structure and Reactivity}, + Editor = {S. K. Ghosh and P. K. Chattaraj}, + Note = {preprint at http://arxiv.org/abs/1304.1314}, + Pages = {367-390}, + Publisher = {CRC Press}, + Series = {Concepts and Methods in Modern Theoretical Chemistry Vol. 1}, + Title = {Electronic excitation energies of molecular systems from the Bethe-Salpeter equation: Example of the H$_2$ molecule}, + Year = {2013}} + +@misc{RebTouSav-JJJ-XX-note, + Note = {With the notations used here, the Hubbard model is obtained for $\Delta\varepsilon = 2 t$ and $J_{11}=J_{22}=J_{12}=J_{12}=K_{12}=U/2$ where $t$ is the hopping parameter and $U$ is the on-site Coulomb interaction.}} + +@article{RebTouTeaHelSav-JCP-14, + Author = {E. Rebolini and J. Toulouse and A. M. Teale and T. Helgaker and A. Savin}, + Doi = {http://dx.doi.org/10.1063/1.4890652}, + Journal = {Journal of Chemical Physics}, + Pages = {044123}, + Title = {Excitation energies along a range-separated adiabatic connection}, + Volume = {141}, + Year = {2014}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4890652}} + +@misc{RebTouTeaHelSav-JJJ-XX-sup, + Note = {See supplementary material at http://xxxxxxxx for the fits of the total and excitation energies.}} + +@article{RebTouTeaHelSav-MP-15, + Author = {E. Rebolini and J. Toulouse and A. M. Teale and T. Helgaker and A. Savin}, + Doi = {http://dx.doi.org/10.1080/00268976.2015.1011248}, + Journal = {Mol. Phys.}, + Pages = {1740}, + Title = {Excited states from range-separated density-functional perturbation theory}, + Volume = {113}, + Year = {2015}, + Bdsk-Url-1 = {http://dx.doi.org/10.1080/00268976.2015.1011248}} + +@article{RebTouTeaHelSav-PRA-15, + Author = {E. Rebolini and J. Toulouse and A. M. Teale and T. Helgaker and A. Savin}, + Doi = {http://dx.doi.org/10.1103/PhysRevA.91.032519}, + Journal = {Phys. Rev. A}, + Pages = {032519}, + Title = {Calculating excitation energies by extrapolation along adiabatic connections}, + Volume = {91}, + Year = {2015}, + Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRevA.91.032519}} + +@article{RefJaiShaNeaKro-PRB-15, + Author = {Sivan Refaely-Abramson and Manish Jain and Sahar Sharifzadeh and Jeffrey B. Neaton and Leeor Kronik}, + Journal = {Phys. Rev. B}, + Pages = {081204(R)}, + Volume = {92}, + Year = {2015}} + +@article{RefShaGovAutNeaBaeKro-PRL-12, + Author = {S. Refaely-Abramson and Sahar Sharifzadeh and Niranjan Govind and Jochen Autschbach and Jeffrey B. Neaton and Roi Baer and Leeor Kronik}, + Journal = {Phys. Rev. Lett.}, + Pages = {226405}, + Volume = {109}, + Year = {2012}} + +@article{RegTha-JPB-84, + Author = {P. E. Regier and A. J. Thakkar}, + Journal = {J. Phys. 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Hoggan}, + Booktitle = {Advances in Quantum Monte Carlo}, + Editor = {{S. Tanaka, S. M. Rothstein, and W. A. Lester, Jr.}}, + Pages = {53-63}, + Publisher = {American Chemical Society}, + Series = {ACS Symposium Series Vol. 1094}, + Title = {Quantum Monte Carlo Facing the Hartree-Fock Symmetry Dilemma: The Case of Hydrogen Rings}, + Year = {2012}} + +@article{ReiTouSav-TCA-18, + Author = {Peter Reinhardt and Julien Toulouse and Andreas Savin}, + Doi = {10.1007/s00214-018-2370-5}, + Journal = {Theor. Chem. Acc.}, + Pages = {168}, + Volume = {137}, + Year = {2018}, + Bdsk-Url-1 = {https://doi.org/10.1007/s00214-018-2370-5}} + +@article{RenRinBluWieTkaSanReuSch-NJP-12, + Author = {Xinguo Ren and Patrick Rinke and Volker Blum and Jurgen Wieferink and Alexandre Tkatchenko and Andrea Sanfilippo and Karsten Reuter and Matthias Scheffler}, + Journal = {New J. Phys.}, + Pages = {053020}, + Volume = {14}, + Year = {2012}} + +@article{RenRinJoaSch-JMS-12, + Affiliation = {Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany}, + Author = {Ren, Xinguo and Rinke, Patrick and Joas, Christian and Scheffler, Matthias}, + Journal = {J. Mater. Sci.}, + Keyword = {Chemistry and Materials Science}, + Optnote = {10.1007/s10853-012-6570-4}, + Opturl = {http://dx.doi.org/10.1007/s10853-012-6570-4}, + Owner = {gambort}, + Pages = {7447}, + Publisher = {Springer Netherlands}, + Timestamp = {2012.07.06}, + Title = {Random-phase approximation and its applications in computational chemistry and materials science}, + Volume = {47}, + Year = {2012}} + +@article{RenRinSch-PRB-09, + Author = {Ren, Xinguo and Rinke, Patrick and Scheffler, Matthias}, + Doi = {10.1103/PhysRevB.80.045402}, + Journal = {Phys. Rev. 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Phys.}, + Pages = {164109}, + Volume = {133}, + Year = {2010}} + +@article{RoeVes-IJQC-05, + Author = {I. R{\o}eggen and L. Veseth}, + Journal = {Int. J. Quantum Chem.}, + Pages = {201}, + Volume = {101}, + Year = {2005}} + +@article{RohGriBae-CPL-06, + Author = {D. Rohr and O. Gritsenko and E. J. Baerends}, + Journal = {Chem. Phys. Lett.}, + Pages = {336}, + Volume = {432}, + Year = {2006}} + +@article{RohHer-JCP-08, + Author = {M. A. Rohrdanz and J. M. Herbert}, + Journal = {J. Chem. Phys.}, + Pages = {034107}, + Volume = {129}, + Year = {2008}} + +@article{Roh-IJQC-00, + Author = {M. Rohlfing}, + Journal = {Int. J. Quantum. Chem.}, + Pages = {807}, + Volume = {80}, + Year = {2000}} + +@article{RohLou-PRB-00, + Author = {M. Rohlfing and S. G. Louie}, + Journal = {Phys. Rev. B}, + Pages = {4927}, + Volume = {62}, + Year = {2000}} + +@article{RohLou-PRL-98, + Author = {M. Rohlfing and S. G. Louie}, + Journal = {Phys. Rev. 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Lett.}, + Pages = {997}, + Volume = {52}, + Year = {1984}} + +@article{RuzPerCso-JCP-11, + Author = {A. Ruzsinsky and J. P. Perdew and G. I. Csonka}, + Journal = {J. Chem. Phys.}, + Pages = {114110}, + Volume = {134}, + Year = {2011}} + +@article{RuzPerCso-JCTC-10, + Author = {A. Ruzsinszky and J. P. Perdew and G. I. Csonka}, + Journal = {J. Chem. Theory Comput.}, + Pages = {127}, + Volume = {6}, + Year = {2010}} + +@article{RuzPerCsoVydScu-JCP-06, + Author = {A. Ruzsinszky and J. P. Perdew and G. I. Csonka and O. A. Vydrov and G. E. Scuseria}, + Journal = {J. Chem. Phys.}, + Pages = {194112}, + Volume = {125}, + Year = {2006}} + +@article{RuzPerCsoVydScu-JCP-07, + Author = {A. Ruzsinszky and J. P. Perdew and G. I. Csonka and O. A. Vydrov and G. E. Scuseria}, + Journal = {J. Chem. Phys.}, + Pages = {104102}, + Volume = {126}, + Year = {2007}} + +@article{SagPer-PRA-08, + Author = {E. Sagvolden and J. P. Perdew}, + Journal = {Phys. Rev. A}, + Pages = {012517}, + Volume = {77}, + Year = {2008}} + +@article{SahEhaNak-JCP-06, + Author = {B. Saha and M. Ehara and H. Nakatsuji}, + Journal = {J. Chem. Phys.}, + Pages = {014316}, + Volume = {125}, + Year = {2006}} + +@article{SaiTiaCheReb-PRB-08, + Author = {N. Sai and M. L. Tiago and J. R. Chelikowsky and F. A. Reboredo}, + Journal = {Phys. Rev. B}, + Pages = {161306}, + Volume = {77}, + Year = {2008}} + +@article{salvador:5046, + Author = {P. Salvador and I. Mayer}, + Issue = {11}, + Journal = {The Journal of Chemical Physics}, + Keywords = {SCF calculations; molecular electronic states; integration; density functional theory}, + Page = {5046-5052}, + Publisher = {AIP}, + Title = {Energy partitioning for ``fuzzy'' atoms}, + Url = {http://link.aip.org/link/?JCP/120/5046/1}, + Volume = {120}, + Year = {2004}, + Bdsk-Url-1 = {http://link.aip.org/link/?JCP/120/5046/1}} + +@article{SamHar-JPB-06, + Author = {P. Samal and M. K. Harbola}, + Journal = {J. Phys. 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Chem.}, + Pages = {814}, + Volume = {23}, + Year = {2002}} + +@incollection{Sau-INC-02, + Author = {Trond Saue}, + Booktitle = {Relativistic Electronic Structure Theory}, + Doi = {10.1016/S1380-7323(02)80033-4}, + Editor = {Peter Schwerdtfeger}, + Pages = {332-400}, + Publisher = {Elsevier}, + Title = {Post Dirac-Hartree-Fock methods -- properties}, + Year = {2002}, + Bdsk-Url-1 = {https://doi.org/10.1016/S1380-7323(02)80033-4}} + +@incollection{SauVis-INC-03, + Address = {Dordrecht}, + Author = {T. Saue and L. Visscher}, + Booktitle = {Theoretical Chemistry and Physics ofHeavy and Superheavy Elements}, + Editor = {S. Wilson and U. Kaldor}, + Pages = {211-267}, + Publisher = {Kluwer}, + Title = {Four-component electronic structure methods for molecules}, + Year = {2003}} + +@article{SavColAll-JCP-01, + Author = {A. Savin and F. Colonna and M. Allavena}, + Journal = {J. Chem. Phys.}, + Pages = {6827}, + Volume = {115}, + Year = {2001}} + +@article{SavCol-JMS-00a, + Author = {A. 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G. and Galli, G. and Zhang, C. and Gygi, F. and Hirao, K. and Song, J. W. and Rahul, K. and von Lilienfeld, O. A. and Podeszwa, R. and Bulik, I. W. and Henderson, T. M. and Scuseria, G. E. and Toulouse, J. and Peverati, R. and Truhlar, D. G. and Szalewicz, K.}, + Doi = {http://dx.doi.org/10.1063/1.4961095}, + Journal = {J. Chem. Phys.}, + Pages = {124105}, + Title = {Blind test of density-functional-based methods on intermolecular interaction energies}, + Url = {http://scitation.aip.org/content/aip/journal/jcp/145/12/10.1063/1.4961095}, + Volume = {145}, + Year = {2016}, + Bdsk-Url-1 = {http://scitation.aip.org/content/aip/journal/jcp/145/12/10.1063/1.4961095}, + Bdsk-Url-2 = {http://dx.doi.org/10.1063/1.4961095}} + +@book{TDDFT-BOOK-06, + Address = {Berlin}, + Editor = {M. L. Marques and C. A. Ullrich and F. Nogueira and A. Rubio and K. Burke and E. K. U. Gross}, + Publisher = {Springer}, + Series = {Lecture Notes in Physics Vol. 706}, + Title = {Time-Dependent Density Functional Theory}, + Year = {2006}} + +@book{TDDFT-BOOK-12, + Address = {Berlin}, + Editor = {M. A. L. Marques and N. T. Maitra, F. M. S. Nogueira, E. K. U. Gross, A. Rubio}, + Publisher = {Springer}, + Series = {Lecture Notes in Physics Vol. 837}, + Title = {Fundamentals of Time-Dependent Density Functional Theory}, + Year = {2012}} + +@article{TeaCorHel-JCP-09, + Author = {A. M. Teale and S. Coriani and T. Helgaker}, + Journal = {J. Chem. Phys.}, + Pages = {104111}, + Title = {{The calculation of adiabatic-connection curves from full configuration-interaction densities: Two-electron systems}}, + Volume = {130}, + Year = {2009}} + +@article{TeaCorHel-JCP-10b, + Author = {Andrew M. Teale and S. Coriani and Trygve Helgaker}, + Journal = {J. Chem. Phys.}, + Pages = {164112}, + Volume = {133}, + Year = {2010}} + +@article{TeaCorHel-JCP-10, + Author = {Andrew M. Teale and S. Coriani and Trygve Helgaker}, + Journal = {J. Chem. Phys.}, + Pages = {164115}, + Volume = {132}, + Year = {2010}} + +@article{TeaDepToz-JCP-08, + Author = {{A. M. Teale, F. De Proft, and D. J. Tozer}}, + Journal = {J. Chem. Phys.}, + Pages = {044110}, + Volume = {129}, + Year = {2008}} + +@article{TecGomEksVis-PCCP-11, + Author = {P. Tecmer and A. S. P. Gomes and U. Ekstr\"om and L. Visscher}, + Journal = {Phys. Chem. Chem. Phys.}, + Pages = {6249}, + Volume = {13}, + Year = {2011}} + +@article{TelKvaSagEksTeaHel-PRA-12, + Author = {E. I. Tellgren and S. Kvaal and E. Sagvolden and U. Ekstr\"om and A. M. Teale and T. Helgaker}, + Journal = {Phys. Rev. A}, + Pages = {062506}, + Volume = {86}, + Year = {2012}} + +@article{TelSosRozChu-PRA-13, + Author = {D. A. Telnov and K. E. Sosnova and E. Rozenbaum and S.-I. Chu}, + Journal = {Phys. Rev. A}, + Pages = {053406}, + Volume = {87}, + Year = {2013}} + +@article{Tha-CPL-03, + Author = {A. J. Thakkar}, + Journal = {Chem. Phys. Lett.}, + Pages = {80}, + Volume = {381}, + Year = {2003}} + +@incollection{Tha-INC-87, + Address = {Dordrecht}, + Author = {A. J. Thakkar}, + Booktitle = {Density Matrices and Density Functionals}, + Editor = {R. M. Erdahl and V. H. Smith}, + Pages = {553-581}, + Publisher = {Reidel}, + Year = {1987}} + +@article{ThaSmi-CPL-76, + Author = {A. J. Thakkar and V. H. Smith}, + Journal = {Chem. Phys. Lett.}, + Pages = {476}, + Volume = {42}, + Year = {1976}} + +@article{ThaSmi-JCP-77, + Author = {A. J. Thakkar and V. H. Smith}, + Journal = {J. Chem. Phys.}, + Pages = {1191}, + Volume = {67}, + Year = {1977}} + +@article{ThaTriSmi-IJQC-84, + Author = {A. J. Thakkar and A. N. Tripathi and V. H. Smith}, + Journal = {Int. J. Quantum. Chem.}, + Pages = {157}, + Volume = {26}, + Year = {1984}} + +@article{ThaTriSmi-PRA-84, + Author = {A. J. Thakkar and A. N. Tripathi and V. H. Smith}, + Journal = {Phys. Rev. A}, + Pages = {1108}, + Volume = {29}, + Year = {1984}} + +@book{Tho-BOOK-61, + Address = {New York}, + Author = {D. J. Thouless}, + Publisher = {Academic Press}, + Title = {The Quantum Mechanics of Many Body Systems}, + Year = {1961}} + +@article{Tho-NAT-25, + Author = {W. Thomas}, + Journal = {Naturwissenschaften}, + Pages = {627}, + Volume = {13}, + Year = {1925}} + +@article{Tho-NP-61, + Author = {D. J. Thouless}, + Journal = {Nucl. Phys.}, + Pages = {78}, + Volume = {22}, + Year = {1961}} + +@article{ThyFleJen-JCP-08, + Author = {J. Thyssen and T. Fleig and H. J. Aa. Jensen}, + Journal = {J. Chem. Phys.}, + Pages = {034109}, + Volume = {129}, + Year = {2008}} + +@article{TiaChe-PRB-06, + Author = {M. L. Tiago and J. R. Chelikowsky}, + Journal = {Phys. Rev. B}, + Pages = {205334}, + Volume = {73}, + Year = {2006}} + +@article{TiaChe-SSC-05, + Author = {M. L. Tiago and J. R. Chelikowsky}, + Journal = {Solid State Commun.}, + Pages = {333}, + Volume = {136}, + Year = {2005}} + +@article{TiaKenHooReb-JCP-08, + Author = {M. L. Tiago and P. R. C. Kent and R. Q. Hood and F. A. Reboredo}, + Journal = {J. Chem. Phys.}, + Pages = {084311}, + Volume = {129}, + Year = {2008}} + +@article{TkaSch-PRL-09, + Author = {A. Tkatchenko and M. Scheffler}, + Journal = {Phys. Rev. Lett.}, + Pages = {073005}, + Title = {{Accurate molecular van der Waals interactions from ground-state electron density and free-atom reference data}}, + Volume = {102}, + Year = {2009}} + +@article{TofSteDec-PRA-06, + Author = {D. Toffoli and M. Stener and P. Decleva}, + Journal = {Phys. Rev. A}, + Pages = {042704}, + Volume = {73}, + Year = {2006}} + +@article{ToiWoo-PRB-70, + Author = {F. Toigo and T. O. Woodruff}, + Journal = {Phys. Rev. B}, + Pages = {3958}, + Volume = {{2}}, + Year = {1970}} + +@article{ToiWoo-PRB-71, + Author = {F. Toigo and T. O. Woodruff}, + Journal = {Phys. Rev. B}, + Pages = {371}, + Volume = {{4}}, + Year = {1971}} + +@incollection{Tos-INC-99, + Address = {University of Oxford}, + Author = {M. P. Tosi}, + Booktitle = {Electron Correlation in the Solid State}, + Editor = {N. H. March}, + Publisher = {World Scientific Publishing Company}, + Title = {Many-Body Effects in Jellium}, + Year = {1999}} + +@incollection{TouAssUmr-INC-16, + Author = {J. Toulouse and R. Assaraf and C. J. Umrigar}, + Booktitle = {Electron Correlation in Molecules - ab initio Beyond Gaussian Quantum Chemistry}, + Doi = {10.1016/bs.aiq.2015.07.003}, + Editor = {P. E. Hoggan and T. Ozdogan}, + Pages = {285-314}, + Publisher = {Academic Press}, + Series = {Advances in Quantum Chemistry Vol. 73}, + Title = {Introduction to the Variational and Diffusion Monte Carlo Methods}, + Year = {2016}, + Bdsk-Url-1 = {https://doi.org/10.1016/bs.aiq.2015.07.003}} + +@article{TouAssUmr-JCP-07, + Author = {J. Toulouse and R. Assaraf and C. J. Umrigar}, + Journal = {J. Chem. Phys.}, + Pages = {244112}, + Title = {{Zero-variance zero-bias quantum Monte Carlo estimators of the spherically and system-averaged pair density}}, + Volume = {126}, + Year = {2007}} + +@incollection{TouCafReiHogUmr-INC-12, + Address = {Dordrecht Heidelberg London New York}, + Author = {J. Toulouse and M. Caffarel and P. Reinhardt and P. E. Hoggan and C. J. Umrigar}, + Booktitle = {Advances in the Theory of Quantum Systems in Chemistry and Physics}, + Editor = {P. E. Hoggan and J. Maruani and P. Piecuch and G. Delgado-Barrio and E. J. Br\"andas}, + Pages = {345-353}, + Publisher = {Springer}, + Series = {Progress in Theoretical Chemistry and Physics Vol. 22}, + Title = {Quantum Monte Carlo calculations of electronic excitation energies: The case of the singlet $n\to\pi^*$ (CO) transition in acrolein}, + Year = {2012}} + +@article{TouColSav-JCP-05, + Author = {J. Toulouse and F. Colonna and A. Savin}, + Journal = {J. Chem. Phys.}, + Pages = {014110}, + Title = {Short-range exchange and correlation energy density functionals: Beyond the local-density approximation}, + Volume = {122}, + Year = {2005}} + +@misc{TouColSav-JJJ-XXa, + Author = {J. Toulouse and F. Colonna and A. Savin}, + Title = {in preparation}} + +@article{TouColSav-MP-05, + Author = {Julien Toulouse and Francois Colonna and Andreas Savin}, + Journal = {Mol. Phys.}, + Pages = {2725}, + Title = {Exchange-correlation potentials and local energies per particle along non-linear adiabatic connections}, + Volume = {103}, + Year = {2005}} + +@article{TouColSav-PRA-04, + Author = {J. Toulouse and F. Colonna and A. Savin}, + Journal = {Phys. Rev. A}, + Keywords = {density functional theory; wave functions; exchange interactions (electron); electron correlations}, + Number = {6}, + Pages = {062505}, + Publisher = {APS}, + doi = {10.1103/PhysRevA.70.062505}, + Title = {Long-range--short-range separation of the electron-electron interaction in density-functional theory}, + Volume = {70}, + Year = {2004}} + +@misc{TouGerJanSavAng-JJJ-XX-note, + Note = {For Ne$_{2}$ at equilibrium distance with aug-cc-pVQZ basis, the counterpoise correction on the interaction energy is 16 $\mu$H for RSH+RPAx and 62 $\mu$H for standard MP2.}} + +@article{TouGerJanSavAng-PRL-09, + Author = {J. Toulouse and I. C. Gerber and G. Jansen and A. Savin and J. G. \'Angy\'an}, + Journal = {Phys. Rev. Lett.}, + Pages = {096404}, + Title = {Adiabatic-connection fluctuation-dissipation density-functional theory based on range separation}, + doi = {10.1103/PhysRevLett.102.096404}, + Volume = {102}, + Year = {2009}} + +@article{TouGorSav-IJQC-06, + Author = {J. Toulouse and P. Gori-Giorgi and A. Savin}, + Journal = {Int. J. Quantum Chem.}, + Pages = {2026}, + Title = {Scaling relations, virial theorem and energy densities for long-range and short-range density functionals}, + Volume = {106}, + Year = {2006}} + +@article{TouGorSav-TCA-05, + Author = {J. Toulouse and P. Gori-Giorgi and A. Savin}, + doi = {10.1007/s00214-005-0688-2}, + Journal = {Theor. Chem. Acc.}, + Pages = {305}, + Volume = {114}, + Year = {2005}} + +@misc{Tou-JJJ-XX, + Author = {S. Dubillard and E. Eliav and L. Visscher and R. Bast and T. Saue}, + Note = {unpublished}} + +@article{Tou-PRB-05, + Author = {Julien Toulouse}, + Journal = {Phys. Rev. B}, + Pages = {035117}, + Title = {Simple model of the static exchange-correlation kernel of a uniform electron gas with long-range electron-electron interaction}, + Volume = {72}, + Year = {2005}} + +@article{TouRebGouDobSeaAng-JCP-13, + Author = {Julien Toulouse and Elisa Rebolini and Tim Gould and John F. Dobson and Prasenjit Seal and J\'anos G. \'Angy\'an}, + Journal = {J. Chem. Phys.}, + Pages = {194106}, + Title = {Assessment of range-separated time-dependent density-functional theory for calculating $C_6$ dispersion coefficients}, + Volume = {138}, + Year = {2013}} + +@article{TouSavFla-IJQC-04, + Author = {J. Toulouse and A. Savin and H.-J. Flad}, + Journal = {Int. J. Quantum Chem.}, + Pages = {1047}, + Title = {Short-range exchange-correlation energy of a uniform electron gas with modified electron-electron interaction}, + Volume = {100}, + Year = {2004}} + +@misc{TouSav-JJJ-XX, + Author = {J. Toulouse and A. Savin}, + Title = {in preparation}} + +@article{TouSav-JMS-06, + Author = {Julien Toulouse and Andreas Savin}, + Journal = {J. Mol. Struct. (Theochem)}, + Pages = {147}, + Title = {Local density approximation for long-range or for short-range energy functionals?}, + Volume = {762}, + Year = {2006}} + +@article{TouShaBreAda-JCP-11, + Author = {J. Toulouse and K. Sharkas and E. Br\'emond and C. Adamo}, + Journal = {J. Chem. Phys.}, + Pages = {101102}, + Title = {Rationale for a new class of double-hybrid approximations in density-functional theory}, + Volume = {135}, + Year = {2011}} + +@phdthesis{Tou-THESIS-05, + Author = {J. Toulouse}, + Note = {tel.archives-ouvertes.fr/tel-00550772}, + School = {Universit\'e Pierre et Marie Curie (Paris 6)}, + url = {https://tel.archives-ouvertes.fr/tel-00550772/document}, + Year = {2005}} + +@article{TouUmr-JCP-07, + Author = {J. Toulouse and C. J. Umrigar}, + Journal = {J. Chem. Phys.}, + Pages = {084102}, + Title = {{Optimization of quantum Monte Carlo wave functions by energy minimization}}, + Volume = {126}, + Year = {2007}} + +@article{TouUmr-JCP-08, + Author = {J. Toulouse and C. J. Umrigar}, + Journal = {J. Chem. Phys.}, + Pages = {174101}, + Title = {{Full optimization of Jastrow-Slater wave functions with application to the first-row atoms and homonuclear diatomic molecules}}, + Volume = {128}, + Year = {2008}} + +@misc{TouUmr-JJJ-XX, + Author = {J. Toulouse and C. J. Umrigar}, + Note = {unpublished}} + +@misc{TouZhuAngSav-JJJ-XX-note2, + Note = {In the context of density-functional theory RPA is usually derived from the Kohn-Sham reference, while in the context of many-body perturbation theory (see appendices) RPA is usually derived from the HF reference. Therefore, both PBE+RPA and HF+RPA are theoretically justified.}} + +@misc{TouZhuAngSav-JJJ-XX-note3, + Note = {The inverse of a 4-point function $\chi (1,2;1',2')$ is defined according to $\int d1'd2' \chi (1,2;1',2') \chi^{-1} (2',1';4,3) = \delta(1,3) \delta (2,4)$.}} + +@misc{TouZhuAngSav-JJJ-XX-note, + Note = {The short-range self-energy correction $\Delta \Sigma^\sr_\l$ is wrongly missing in Eq.~(11) of Ref.~\onlinecite{TouGerJanSavAng-PRL-09}. However, in practice, this term vanishes in the RPA or RPAx approximation so that the results of Ref.~\onlinecite{TouGerJanSavAng-PRL-09} are correct.}} + +@article{TouZhuAngSav-PRA-10, + Author = {Julien Toulouse and Wuming Zhu and J\'anos G. \'Angy\'an and Andreas Savin}, + Journal = {Phys. Rev. A}, + Pages = {032502}, + Title = {Range-separated density-functional theory with the random-phase approximation: Detailed formalism and illustrative applications}, + Volume = {82}, + Year = {2010}} + +@article{TouZhuSavJanAng-JCP-11, + Author = {J. Toulouse and W. Zhu and A. Savin and G. Jansen and J. G. \'Angy\'an}, + Journal = {J. Chem. Phys.}, + Pages = {084119}, + Title = {Closed-shell ring coupled cluster doubles theory with range separation applied on weak intermolecular interactions}, + Volume = {135}, + doi = {10.1063/1.3626551}, + Year = {2011}} + +@misc{TouZhuSavJanAng-JJJ-XX, + Author = {J. Toulouse and W. Zhu and A. Savin and G. Jansen and J. G. \'Angyan}, + Note = {unpublished}} + +@misc{TouZhuSavJanAng-JJJ-XX-note, + Note = {In Ref.~\onlinecite{Hes-JCP-11}, AC-RPA, NRPA1, NRPA3, and NRPA4 refer to what we call here RPAx-I, RPAx-II, RPA-SO2, and RPA-SO1, respectively. In addition, NRPA2 corresponds to the variant of Fukuda {\it et al.}, i.e. $2E_{c,\RPAxII}-E_{c,\text{MP2}}$.}} + +@article{TowZupCau-CPC-96, + Author = {M. D. Towler and A. Zupan and M. Caus\'a}, + Journal = {Comp. Phys. Comm.}, + Pages = {181}, + Volume = {98}, + Year = {1996}} + +@article{TozHan-JCP-98, + Author = {D. J. Tozer and N. C. Handy}, + Journal = {J. Chem. Phys.}, + Pages = {10180}, + Volume = {109}, + Year = {1998}} + +@article{Tri-JPB-80, + Author = {H. P. Trivedi}, + Journal = {J. Phys. B}, + Pages = {839}, + Volume = {13}, + Year = {1980}} + +@article{Tru-CPL-98, + Author = {D. G. Truhlar}, + Journal = {Chem. Phys. Lett.}, + Pages = {45}, + Volume = {294}, + Year = {1998}} + +@article{TsuScu-JCP-11, + Author = {T. Tsuchimochi and G. E. Scuseria}, + Journal = {J. Chem. Phys.}, + Pages = {064101}, + Volume = {134}, + Year = {2011}} + +@article{TsuScuSav-JCP-10, + Author = {T. Tsuchimochi and G. E. Scuseria and A. Savin}, + Journal = {J. Chem. Phys.}, + Pages = {024111}, + doi = {https://doi.org/10.1063/1.3292640}, + Volume = {132}, + Year = {2010}} + +@article{TsuSonSuzHir-JCP-10, + Author = {Takao Tsuneda and Jong-Won Song and Satoshi Suzuki and Kimihiko Hirao}, + Journal = {J. Chem. Phys.}, + Pages = {174101}, + Volume = {133}, + Year = {2010}} + +@article{TsuSuzHir-JCP-99, + Author = {Takao Tsuneda and Toshihisa Suzumura and Kimihiko Hirao}, + Journal = {J. Chem. Phys.}, + Pages = {5656}, + Volume = {111}, + Year = {1999}} + +@article{UgaBoy-IJQC-85, + Author = {J. M. Ugalde and R. J. Boyd}, + Journal = {Int. J. Quantum. Chem.}, + Pages = {439}, + Volume = {27}, + Year = {1985}} + +@book{Ull-BOOK-11, + Author = {Carsten A. Ullrich}, + Publisher = {OUP Oxford}, + Title = {Time-Dependent Density-Functional Theory: Concepts and Applications}, + Year = {2011}} + +@article{UllKoh-PRL-01, + Author = {C. A. Ullrich and W. Kohn}, + Journal = {Phys. Rev. Lett.}, + Pages = {093001}, + Volume = {87}, + Year = {2001}} + +@article{UmrFil-PRL-05, + Author = {C. J. Umrigar and C. Filippi}, + Journal = {Phys. Rev. Lett.}, + Pages = {150201}, + Title = {{Energy and variance optimization of many-body wave functions}}, + Volume = {94}, + Year = {2005}} + +@misc{UmrFil-PRL-05-note, + Note = {The convergence in 2 iterations, mentioned near the end of Ref.~\onlinecite{UmrFil-PRL-05} was obtained using the correlated sampling adjustment of $a_\diag$ and the TU Hessian.}} + +@incollection{UmrGon-INC-93, + Address = {Singapore}, + Author = {C. J. Umrigar and X. Gonze}, + Booktitle = {High Performance Computing and its Application to the Physical Sciences, Proceedings of the Mardi Gras '93 Conference}, + Editor = {D. A. Browne {\it et al.}}, + Publisher = {World Scientific}, + Year = {1993}} + +@article{UmrGon-PRA-94, + Author = {C. J. Umrigar and X. Gonze}, + Journal = {Phys. Rev. A}, + Pages = {3827}, + Volume = {50}, + Year = {1994}} + +@article{Umr-IJQC-89, + Author = {C. J. Umrigar}, + Journal = {Int. J. Quantum Chem.}, + Pages = {217}, + Volume = {{23}}, + Year = {1989}} + +@incollection{Umr-INC-99, + Address = {Dordrecht}, + Author = {C. J. Umrigar}, + Booktitle = {Quantum Monte Carlo Methods in Physics and Chemistry}, + Editor = {M. P. Nightingale and C. J. Umrigar}, + Pages = {129}, + Publisher = {Kluwer}, + Series = {NATO ASI Ser. C 525}, + Year = {1999}} + +@article{UmrNigRun-JCP-93, + Author = {C. J. Umrigar and M. P. Nightingale and K. J. Runge}, + Journal = {J. Chem. Phys.}, + Pages = {2865}, + Title = {{A diffusion Monte Carlo algorithm with very small time-step errors}}, + Volume = {99}, + Year = {1993}} + +@article{Umr-PRL-93, + Author = {C. J. Umrigar}, + Journal = {Phys. Rev. Lett.}, + Pages = {408}, + Title = {{Accelerated Metropolis method}}, + Volume = {71}, + Year = {1993}} + +@incollection{UmrSavGon-INC-98, + Author = {Umrigar, C.J. and Savin, A. and Gonze, Xavier}, + Booktitle = {Electronic Density Functional Theory}, + Doi = {10.1007/978-1-4899-0316-7_12}, + Editor = {Dobson, John F. and Vignale, Giovanni and Das, Mukunda P.}, + Pages = {167-176}, + Publisher = {Springer US}, + Title = {Are Unoccupied Kohn-Sham Eigenvalues Related to Excitation Energies?}, + Year = {1998}, + Bdsk-Url-1 = {https://doi.org/10.1007/978-1-4899-0316-7_12}} + +@article{UmrTouFilSorHen-PRL-07, + Author = {C. J. Umrigar and J. Toulouse and C. Filippi and S. Sorella and R. G. Hennig}, + Journal = {Phys. Rev. Lett.}, + Pages = {110201}, + Title = {Alleviation of the Fermion-sign problem by optimization of many-body wave functions}, + Volume = {98}, + Year = {2007}} + +@misc{Umr-UNP-XXb, + Author = {C. J. Umrigar}, + Note = {unpublished}} + +@misc{Umr-UNP-XX, + Author = {C. J. Umrigar}, + Note = {unpublished}} + +@misc{Umr-UNP-XX_fr, + Author = {C. J. Umrigar}, + Note = {non publi{\'e}}} + +@incollection{UmrWilWil-INC-88, + Address = {Berlin}, + Author = {C. J. Umrigar and K. G. Wilson and J. W. Wilkins}, + Booktitle = {Computer Simulation Studies in Condensed Matter Physics: Recent Developments}, + Editor = {D. P. Landau and K. K. Mon and H. B. Sch\"uttler}, + Publisher = {Springer}, + Year = {1988}} + +@article{UmrWilWil-PRL-88, + Author = {C. J. Umrigar and K. G. Wilson and J. W. Wilkins}, + Journal = {Phys. Rev. Lett.}, + Pages = {1719}, + Title = {{Optimized trial wave functions for quantum Monte Carlo calculations}}, + Volume = {60}, + Year = {1988}} + +@article{UtsIch-PRB-80a, + Author = {K. Utsumi and S. Ichimaru}, + Journal = {Phys. Rev. B}, + Pages = {1522}, + Volume = {{22}}, + Year = {1980}} + +@article{UtsIch-PRB-80, + Author = {K. Utsumi and S. Ichimaru}, + Journal = {Phys. Rev. B}, + Pages = {5203}, + Volume = {{22}}, + Year = {1980}} + +@article{Val-JCP-80a, + Author = {S. M. Valone}, + Journal = {J. Chem. Phys.}, + Pages = {4653}, + Volume = {73}, + Year = {1980}} + +@article{VasSin-PRB-72, + Author = {P. Vashista and K.S. Singwi}, + Journal = {Phys. Rev. B}, + Pages = {875}, + Volume = {{6}}, + Year = {1972}} + +@article{VeiCle-JCP-69, + Author = {A. Veillard and E. Clementi}, + Journal = {J. Chem. Phys.}, + Pages = {2415}, + Volume = {{49}}, + Year = {1969}} + +@article{VitDelGor-JCP-05, + Author = {{V. Vitale, F. Della Sala, and A. G\"orling}}, + Journal = {J. Chem. Phys.}, + Pages = {244102}, + Volume = {122}, + Year = {2005}} + +@phdthesis{Vog-THESIS-03, + Author = {M. Vogt}, + School = {University of Cambridge}, + Title = {Spectral Moments in the Homogeneous Electron Gas}, + Year = {2003}} + +@article{VogZimNee-PRB-04, + Author = {M. Vogt and R. Zimmermann and R. J. Needs}, + Journal = {Phys. Rev. B}, + Pages = {045113}, + Volume = {{69}}, + Year = {2004}} + +@article{VosWilNus-CJP-80, + Author = {S. J. Vosko and L. Wilk and M. Nusair}, + Journal = {Can. J. Phys.}, + Pages = {1200}, + Volume = {{58}}, + Year = {1980}} + +@article{VrbDepRot-JCP-88, + Author = {J. Vrbik and M. F. DePasquale and S. M. Rothstein}, + Journal = {J. Chem. Phys.}, + Pages = {3784}, + Volume = {88}, + Year = {1988}} + +@article{VutCamGurHutparPatPetSpaDoyGabDem-JPB-10, + Author = {A. C. Vutha and W. C. Campbell and Y. V. Gurevich and N. R. Hutzler and M. Parsons and D. Patterson and E. Petrik and B. Spaun and J. M. Doyle and G. Gabrielse and D. DeMille}, + Journal = {J. Phys. B}, + Pages = {074007}, + Volume = {43}, + Year = {2010}} + +@article{VydScu-JCP-06, + Author = {O. A. Vydrov and G. E. Scuseria}, + Journal = {J. Chem. Phys.}, + Pages = {234109}, + Volume = {125}, + Year = {2006}} + +@article{VydScuPer-JCP-07, + Author = {Vydrov, O. A. and Scuseria, G. E. and Perdew, J. P.}, + Journal = {J. Chem. Phys.}, + Pages = {154109}, + Title = {Tests of functionals for systems with fractional electron number}, + Volume = {126}, + Year = {2007}} + +@article{VydVoo-JCP-09, + Author = {O. A. Vydrov and T. van Voorhis}, + Journal = {J. Chem. Phys.}, + Pages = {104105}, + Volume = {130}, + Year = {2009}} + +@article{VydVoo-JCP-10b, + Author = {O. A. Vydrov and T. V. Voorhis}, + Journal = {J. Chem. Phys.}, + Pages = {244103}, + Title = {{Nonlocal van der Waals density functional: The simpler the better}}, + Volume = {133}, + Year = {2010}} + +@article{VydVoo-PRL-09, + Author = {O. A. Vydrov and T. van Voorhis}, + Journal = {Phys. Rev. Lett.}, + Pages = {063004}, + Volume = {103}, + Year = {2009}} + +@article{Wal-TFS-45, + Author = {A. D. Walsh}, + Journal = {Trans. Faraday Soc.}, + Pages = {498}, + Volume = {41}, + Year = {1945}} + +@article{WanGauWul-JCP-08, + Author = {{F. Wang, J. Gauss, and C. van W\"ullen}}, + Journal = {J. Chem. Phys.}, + Pages = {064113}, + Volume = {129}, + Year = {2008}} + +@article{WanPer-PRB-91, + Author = {Y. Wang and J.P. Perdew}, + Journal = {Phys. Rev. B}, + Pages = {13298}, + Volume = {{44}}, + Year = {1991}} + +@article{WanSch-JCP-96, + Author = {S. G. Wang and W. H. E. Schwarz}, + Journal = {J. Chem. Phys.}, + Pages = {4641}, + Volume = {{105}}, + Year = {1996}} + +@article{WanSchSmi-PRA-00, + Author = {J. Wang and H. L. Schmider and V. H. Smith}, + Journal = {Phys. Rev. A}, + Pages = {016501}, + Volume = {62}, + Year = {2000}} + +@article{WanSmi-IJQC-94, + Author = {J. Wang and V. H. Smith}, + Journal = {Int. J. Quantum. Chem.}, + Pages = {147}, + Volume = {49}, + Year = {1994}} + +@article{WanTriSmi-JCP-92, + Author = {J. Wang and A. N. Tripathi and V. H. Smith}, + Journal = {J. Chem. Phys.}, + Pages = {9188}, + Volume = {97}, + Year = {1992}} + +@article{WanTriSmi-JCP-94, + Author = {J. Wang and A. N. Tripathi and V. H. Smith}, + Journal = {J. Chem. Phys.}, + Pages = {4842}, + Volume = {101}, + Year = {1994}} + +@article{WanZie-JCP-04, + Author = {F. Wang and T. Ziegler}, + Journal = {J. Chem. Phys.}, + Pages = {12191}, + Volume = {121}, + Year = {2004}} + +@article{WasBur-PRL-05, + Author = {A. Wasserman and K. Burke}, + Journal = {Phys. Rev. Lett.}, + Pages = {163006}, + Volume = {95}, + Year = {2005}} + +@article{WasMaiBur-PRL-03, + Author = {A. Wasserman and N. T. Maitra and K. Burke}, + Journal = {Phys. Rev. Lett.}, + Pages = {263001}, + Volume = {91}, + Year = {2003}} + +@article{WatKamYamUdaMul-MP-04, + Author = {Noboru Watanabe and Yohei Kamata and Kota Yamauchi and Yasuo Udagawa and Thomas M\"uller}, + Journal = {Mol. Phys.}, + Pages = {649}, + Volume = {102}, + Year = {2004}} + +@article{WeiDelGor-JCP-08, + Author = {M. Weimer and F. Della Sala and A. G\"orling}, + Journal = {J. Chem. Phys.}, + Pages = {144109}, + Title = {Multiconfiguration optimized effective potential method for a density-functional treatment of static correlation}, + Volume = {128}, + Year = {2008}} + +@article{WenBer-JCTC-11, + Author = {Shuhao Wen and Gregory J. O. Beran}, + Journal = {J. Chem. Theory Comput.}, + Pages = {3733}, + Volume = {7}, + Year = {2011}} + +@article{WerKno-JCP-88, + Author = {H.-J. Werner and P. J. Knowles}, + Journal = {J. Chem. Phys.}, + Pages = {5003}, + Volume = {{89}}, + Year = {1988}} + +@article{werner:3144, + Author = {Hans-Joachim Werner and Ernst-Albrecht Reinsch}, + Issue = {6}, + Journal = {The Journal of Chemical Physics}, + Keywords = {CONFIGURATION INTERACTION; ELECTRON CORRELATION; ELECTRONIC STRUCTURE; WAVE FUNCTIONS; DIPOLE MOMENTS; POTENTIAL ENERGY; MOLECULAR MODELS}, + Page = {3144-3156}, + Publisher = {AIP}, + Title = {The self-consistent electron pairs method for multiconfiguration reference state functions}, + Url = {http://link.aip.org/link/?JCP/76/3144/1}, + Volume = {76}, + Year = {1982}, + Bdsk-Url-1 = {http://link.aip.org/link/?JCP/76/3144/1}} + +@book{Weyl-56, + Address = {New York}, + Author = {H. Weyl}, + Publisher = {Dover}, + Title = {Theory of Groups and Quantum Mechanics}, + Year = {1956}} + +@article{WhiBur-JCP-05, + Author = {Takeyce K. Whittingham and Kieron Burke}, + Journal = {J. Chem. Phys.}, + Number = {13}, + Pages = {134108}, + Title = {Relations between coordinate and potential scaling in the high-density limit}, + Volume = {122}, + Year = {2005}} + +@article{WilMouDun-JMS-96, + Author = {A. K. Wilson and T. v. Mourik and T. H. Dunning}, + Journal = {J. Mol. Struct.}, + Pages = {339}, + Volume = {388}, + Year = {1996}} + +@article{WilWooPetDun-JCP-99, + Author = {A.K. Wilson and D.E. Woon and K.A. Peterson and T.H.Dunning}, + Journal = {J. Chem. Phys.}, + Pages = {7667}, + Volume = {110}, + Year = {1999}} + +@article{WodJanRagCor-JPCA-08, + Author = {{M. D. Wodrich, D. F. Jana, P. v. R. Schleyer, and C. Corminboeuf}}, + Journal = {J. Phys. Chem. A}, + Pages = {11495}, + Volume = {112}, + Year = {2008}} + +@article{Wol-PRL-89, + Author = {Wolff, U.}, + Doi = {10.1103/PhysRevLett.62.361}, + Issue = {4}, + Journal = {Phys. Rev. Lett.}, + Numpages = {0}, + Pages = {361--364}, + Publisher = {American Physical Society}, + Title = {{Collective Monte Carlo updating for spin systems}}, + Url = {http://link.aps.org/doi/10.1103/PhysRevLett.62.361}, + Volume = {62}, + Year = {1989}, + Bdsk-Url-1 = {http://link.aps.org/doi/10.1103/PhysRevLett.62.361}, + Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.62.361}} + +@article{wolinski:3647, + Author = {Krzysztof Wolinski and Peter Pulay}, + Journal = {The Journal of Chemical Physics}, + Keywords = {PERTURBATION THEORY; WAVE FUNCTIONS; ELECTRONIC STRUCTURE; DIPOLE MOMENTS; AFFINITY; IONIZATION POTENTIAL; POTENTIAL ENERGY SURFACES; HYDROFLUORIC ACID; FLUORINE; ETHYLENE; OZONE; FORMALDEHYDE; ELECTRON CORRELATION}, + Number = {7}, + Pages = {3647-3659}, + Publisher = {AIP}, + Title = {Generalized M[o-slash]ller--Plesset perturbation theory: Second order results for two-configuration, open-shell excited singlet, and doublet wave functions}, + Url = {http://link.aip.org/link/?JCP/90/3647/1}, + Volume = {90}, + Year = {1989}, + Bdsk-Url-1 = {http://link.aip.org/link/?JCP/90/3647/1}} + +@article{WooDun-JCP-93, + Author = {D.E. Woon and T.H. Dunning}, + Journal = {J. Chem. Phys.}, + Pages = {1358}, + Volume = {98}, + Year = {1993}} + +@article{WooDun-JCP-94, + Author = {D.E. Woon and T.H. Dunning}, + Journal = {J. Chem. Phys.}, + Pages = {2975}, + Volume = {100}, + Year = {1994}} + +@article{WooDun-JCP-95, + Author = {D.E. Woon and T.H. Dunning}, + Journal = {J. Chem. Phys.}, + Pages = {4572}, + Volume = {103}, + Year = {1995}} + +@article{WouNec-EPJD-14, + Author = {{S. Wouters and D. Van Neck}}, + Journal = {Eur. Phys. J. D}, + Pages = {272}, + Title = {The density matrix renormalization group for ab initio quantum chemistry}, + Volume = {68}, + Year = {2014}} + +@article{WuAyeYan-JCP-03, + Author = {Qin Wu and Paul W. Ayers and Weitao Yang}, + Journal = {J. Chem. Phys.}, + Keywords = {density functional theory; variational techniques; electron affinity; negative ions}, + Number = {6}, + Pages = {2978-2990}, + Publisher = {AIP}, + Title = {Density-functional theory calculations with correct long-range potentials}, + Url = {http://link.aip.org/link/?JCP/119/2978/1}, + Volume = {119}, + Year = {2003}, + Bdsk-Url-1 = {http://link.aip.org/link/?JCP/119/2978/1}} + +@article{WuCheVoo-JCP-07, + Author = {{Q. Wu, C.-L. Cheng, and T. Van Voorhis}}, + Journal = {J. Chem. Phys.}, + Pages = {164119}, + Title = {Configuration interaction based on constrained density functional theory: A multireference method}, + Volume = {127}, + Year = {2007}} + +@article{WuKadVoo-JCP-09, + Author = {{Q. Wu, B. Kaduk, and T. Van Voorhis}}, + Journal = {J. Chem. Phys.}, + Pages = {034109}, + Volume = {130}, + Year = {2009}} + +@incollection{Wul-INC-10, + Address = {Netherlands}, + Author = {{C. van W\"ullen}}, + Booktitle = {Relativistic Methods for Chemists, Challenges and Advances in Computational Chemistry and Physics 10}, + Editor = {M. Barysz and Y. Ishikawa}, + Pages = {191-214}, + Publisher = {Springer}, + Title = {RELATIVISTIC DENSITY FUNCTIONAL THEORY}, + Year = {2010}} + +@article{WuSha-CPL-99, + Author = {W. Wu and S. Shaik}, + Journal = {Chem. Phys. Lett.}, + Pages = {37}, + Volume = {301}, + Year = {1999}} + +@article{WuShaHib-xx, + Author = {P. Su and J. Wu. and J. Gu and W. Wu and S. Shaik and P. C. Hiberty}, + Journal = {J. Chem. Theory Comput.}, + Pages = {121}, + Volume = {7}, + Year = {2011}} + +@article{WuSonCaoZhaSha-JPCA-02, + Author = {W. Wu and L. Song and Z. Cao and Q. Zhang and S. Shaik}, + Journal = {J. Phys. Chem. A}, + Pages = {2721}, + Volume = {106}, + Year = {2002}} + +@article{WuVarNayLotSco-JCP-01, + Author = {X. Wu and M. C. Vargas and S. Nayak and V. Lotrich and G. Scoles}, + Journal = {J. Chem. Phys.}, + Pages = {8748}, + Volume = {115}, + Year = {2001}} + +@article{WuWuMo-IJQC-98, + Author = {W. Wu and A. Wu and Y. Mo and M. Lin and Q. Zhang}, + Journal = {Int. J. Quantum Chem.}, + Pages = {287}, + Volume = {67}, + Year = {1998}} + +@article{WuYan-JCP-02, + Author = {Q. Wu and W. Yang}, + Journal = {J. Chem. Phys.}, + Pages = {515}, + Volume = {116}, + Year = {2002}} + +@article{WuYan-JCP-03, + Author = {Qin Wu and Weitao Yang}, + Journal = {J. Chem. Phys.}, + Number = {6}, + Pages = {2498-2509}, + Title = {A direct optimization method for calculating density functionals and exchange--correlation potentials from electron densities}, + Volume = {118}, + Year = {2003}} + +@article{xantheas:8054, + Author = {Sotiris S. Xantheas and Gregory J. Atchity and Stephen T. Elbert and Klaus Ruedenberg}, + Journal = {The Journal of Chemical Physics}, + Keywords = {OZONE; ELECTRONIC STRUCTURE; POTENTIAL ENERGY; SELF–CONSISTENT FIELD; CONFIGURATION INTERACTION; POTENTIAL ENERGY SURFACES; QUANTUM CHEMISTRY}, + Number = {12}, + Pages = {8054-8069}, + Publisher = {AIP}, + Title = {Potential energy surfaces of ozone. I}, + Url = {http://link.aip.org/link/?JCP/94/8054/1}, + Volume = {94}, + Year = {1991}, + Bdsk-Url-1 = {http://link.aip.org/link/?JCP/94/8054/1}} + +@article{YamKocTen-JCP-07, + Author = {D. Yamaki and H. Koch and S. Ten-no}, + Journal = {J. Chem. Phys.}, + Pages = {144104}, + Volume = {127}, + Year = {2007}} + +@article{YamNakUkaTakYam-IJQC-06, + Author = {S. Yamanaka and K. Nakata and T. Ukai and T. Takada and K. Yamaguchi}, + Journal = {Int. J. Quantum Chem.}, + Pages = {3312}, + Title = {Multireference Density Functional Theory With Orbital-Dependent Correlation Corrections}, + Volume = {106}, + Year = {2006}} + +@article{YanAyeWu-PRL-04, + Author = {Weitao Yang and Paul W. Ayers and Qin Wu}, + Journal = {Phys. Rev. Lett.}, + Pages = {146404}, + Volume = {92}, + Year = {2003}} + +@article{YanCohMor-JCP-12, + Author = {Weitao Yang and Aron J. Cohen and Paula Mori-S\'anchez}, + Journal = {J. Chem. Phys.}, + Pages = {204111}, + Volume = {136}, + Year = {2012}} + +@article{YanHarHan-MP-05, + Author = {T. Yanai and R. J. Harrison and N. C. Handy}, + Journal = {Mol. Phys.}, + Pages = {413}, + Volume = {103}, + Year = {2005}} + +@article{Yan-JCP-98, + Author = {W. Yang}, + Journal = {J. Chem. Phys.}, + Pages = {10107}, + Title = {Generalized adiabatic connection in density functional theory}, + Volume = {{109}}, + Year = {1998}} + +@article{YanMorCoh-JCP-13, + Author = {W. Yang and P. Mori-S{\'a}nchez and A. J. Cohen}, + Journal = {J. Chem. Phys.}, + Pages = {104114}, + Volume = {139}, + Year = {2013}} + +@article{YanPerKur-PRB-00, + Author = {Yan, Z. and Perdew, J. P. and Kurth, S.}, + Journal = {Phys. Rev. B.}, + Pages = {16430-16439}, + Title = {Density functional for short-range correlation: Accuracy of the random-phase approximation for isoelectronic energy changes}, + Volume = {61}, + Year = {2000}} + +@article{YanTewHan-CPL-04, + Author = {Takeshi Yanai and David P. Tew and Nicholas C. Handy}, + Journal = {Chem. Phys. Lett.}, + Pages = {51}, + Volume = {393}, + Year = {2004}} + +@article{YanWu-PRL-02, + Author = {Weitao Yang and Qin Wu}, + Journal = {Phys. Rev. Lett.}, + Pages = {143002}, + Title = {Direct Method for Optimized Effective Potentials in Density-Functional Theory}, + Volume = {89}, + Year = {2002}} + +@article{YanZhaAye-PRL-00, + Author = {Weitao Yang and Yingkai Zhang and Paul W. Ayers}, + Journal = {Phys. Rev. Lett.}, + Pages = {5172}, + Volume = {84}, + Year = {2000}} + +@article{Yar-CPL-81, + Author = {D. R. Yarkony}, + Journal = {Chem. Phys. Lett.}, + Pages = {634}, + Volume = {77}, + Year = {1981}} + +@article{Yas-PRL-02, + Author = {Koji Yasuda}, + Eid = {053001}, + Journal = {Physical Review Letters}, + Keywords = {density functional theory; electron density; electron correlations; Schrodinger equation; scaling phenomena; electron gas}, + Number = {5}, + Numpages = {4}, + Pages = {053001}, + Publisher = {APS}, + Title = {Local Approximation of the Correlation Energy Functional in the Density Matrix Functional Theory}, + Url = {http://link.aps.org/abstract/PRL/v88/e053001}, + Volume = {88}, + Year = {2002}, + Bdsk-Url-1 = {http://link.aps.org/abstract/PRL/v88/e053001}} + +@article{YeaJor-JCP-79, + Author = {D. L. Yeager and P. J{\o}rgensen}, + Journal = {J. Chem. Phys.}, + Pages = {755}, + Volume = {{71}}, + Year = {1979}} + +@article{YinSuCheShaWu-JCTC-12, + Author = {F. Ying and P. Su and Z. Chen and S. Shaik and W. Wu}, + Journal = {J. Chem. Theory Comput.}, + Pages = {1608}, + Volume = {8}, + Year = {2012}} + +@article{Yu-IJQC-13, + Author = {F. Yu}, + Journal = {Int. J. Quantum Chem.}, + Pages = {2355}, + Volume = {113}, + Year = {2013}} + +@article{ZahLeaGor-JCP-13, + Author = {F. Zahariev and S. S. Leang and M. S. Gordon}, + Journal = {J. Chem. Phys.}, + Pages = {244108}, + Volume = {138}, + Year = {2013}} + +@book{ZalPapMezLes-BOOK-11, + Address = {Netherlands}, + Editor = {R. Zalesny and M. G. Papadopoulos and P. G. Mezey and J. Leszczynski}, + Publisher = {Springer}, + Series = {Challenges and Advances in Computational Chemistry and Physics Vol. 13}, + Title = {Linear-Scaling Techniques in Computational Chemistry and Physics}, + Year = {2011}} + +@article{ZanSov-PRA-80, + Author = {A. Zangwill and P. Soven}, + Journal = {Phys. Rev. A}, + Pages = {1561}, + Volume = {21}, + Year = {1980}} + +@article{ZarKoh-PRB-76, + Author = {E. Zaremba and W. Kohn}, + Journal = {Phys. Rev. B}, + Pages = {2270}, + Volume = {13}, + Year = {1976}} + +@article{ZecGorMorBac-PRB-04, + Author = {Lorenzo Zecca and Paola Gori-Giorgi and Saverio Moroni and Giovanni B. Bachelet}, + Journal = {Phys. Rev. B}, + Keywords = {density functional theory; Monte Carlo methods; ground states; electron gas; electron density; electron correlations}, + Number = {20}, + Pages = {205127}, + Publisher = {APS}, + Title = {Local density functional for the short-range part of the electron-electron interaction}, + Volume = {70}, + Year = {2004}} + +@article{ZeiMea-MP-77, + Author = {G. D. Zeiss and W. J. Meath}, + Journal = {Mol. Phys.}, + Pages = {1155}, + Volume = {33}, + Year = {1977}} + +@article{ZhaBur-PRA-04, + Author = {F. Zhang and K. Burke}, + Journal = {Phys. Rev. A}, + Pages = {052510}, + Volume = {69}, + Year = {2004}} + +@article{ZhaLuoXu-JCP-10b, + Author = {I. Y. Zhang and Y. Luo and X. Xu}, + Journal = {J. Chem. Phys.}, + Pages = {104105}, + Volume = {133}, + Year = {2010}} + +@article{ZhaLuoXu-JCP-10, + Author = {I. Y. Zhang and Y. Luo and X. Xu}, + Journal = {J. Chem. Phys.}, + Pages = {194105}, + Volume = {132}, + Year = {2010}} + +@article{ZhaLynTru-JPCA-04, + Author = {Y. Zhao and B. J. Lynch and D. G. Truhlar}, + Journal = {J. Phys. Chem. A}, + Pages = {4786}, + Volume = {108}, + Year = {2004}} + +@article{ZhaLynTru-PCCP-05, + Author = {Y. Zhao and B. J. Lynch and D. G. Truhlar}, + Journal = {Phys. Chem. Chem. Phys.}, + Pages = {43}, + Volume = {7}, + Year = {2005}} + +@article{ZhaPar-PRA-92, + Author = {Q. Zhao and R. G. Parr}, + Journal = {Phys. Rev. A.}, + Pages = {2337}, + Volume = {{46}}, + Year = {1992}} + +@article{ZhaSchTru-JCTC-06, + Author = {Y. Zhao and N. E. Schultz and D. G. Truhlar}, + Journal = {J. Chem. Theory Comput.}, + Pages = {364}, + Volume = {2}, + Year = {2006}} + +@article{ZhaSteYan-JCP-13, + Abstract = {We present a dynamical second-order kernel for the Bethe-Salpeter equation to calculate electronic excitation energies. The derivation takes explicitly the functional derivative of the exact second-order self energy with respect to the one-particle Green's function. It includes naturally a frequency dependence, going beyond the adiabatic approximation. Perturbative calculations under the Tamm-Dancoff approximation, using the configuration interaction singles (CIS) eigenvectors, reveal an appreciable improvement over CIS, time-dependent Hartree-Fock, and adiabatic time-dependent density functional theory results. The perturbative results also compare well with equation-of-motion coupled-cluster and experimental results.}, + Author = {Zhang, Du and Steinmann, Stephan N and Yang, Weitao}, + Journal = {J. Chem. Phys.}, + Month = oct, + Number = {15}, + Pages = {154109}, + Pmid = {24160502}, + Publisher = {American Institute of Physics}, + Title = {{Dynamical second-order Bethe-Salpeter equation kernel: A method for electronic excitation beyond the adiabatic approximation.}}, + Volume = {139}, + Year = {2013}} + +@article{ZhaSuBreAdaXu-JCP-12, + Author = {I. Y. Zhang and N. Q. Su and E. A. G. Br\'emond and C. Adamo}, + Journal = {J. Chem. Phys.}, + Pages = {174103}, + Volume = {136}, + Year = {2012}} + +@article{ZhaTru-JCTC-07, + Author = {Y. Zhao and D. G. Truhlar}, + Journal = {J. Chem. Theory Comput.}, + Pages = {289}, + Title = {Density Functionals for Noncovalent Interaction Energies of Biological Importance}, + Volume = {3}, + Year = {2007}} + +@article{ZhaTru-TCA-08, + Author = {Y. Zhao and D. G. Truhlar}, + Journal = {Theor. Chem. Acc.}, + Pages = {215}, + Volume = {120}, + Year = {2008}} + +@article{ZhaWuXu-CC-10, + Author = {I. Y. Zhang and J. Wu and X. Xu}, + Journal = {Chem. Commun.}, + Pages = {3057}, + Volume = {46}, + Year = {2010}} + +@article{ZhaXuGod-PNAS-09, + Author = {{Y. Zhang, X. Xu and W. A. Goddard III}}, + Journal = {Proc. Natl. Acad. Sci. U.S.A.}, + Pages = {4963}, + Volume = {106}, + Year = {2009}} + +@article{ZhaXuGod-PNAS-09_fr, + Author = {{Y. Zhang, X. Xu et W. A. Goddard III}}, + Journal = {Proc. Natl. Acad. Sci. U.S.A.}, + Pages = {4963}, + Volume = {106}, + Year = {2009}} + +@article{ZhaXu-IRPC-11, + Author = {I. Y. Zhang and X. Xu}, + Journal = {Int. Rev. Phys. Chem.}, + Pages = {115}, + Volume = {30}, + Year = {2011}} + +@article{ZhaXu-JPCL-13, + Author = {I. Y. Zhang and X. Xu}, + Journal = {J. Phys. Chem. Lett.}, + Pages = {1669}, + Volume = {4}, + Year = {2013}} + +@article{ZhaXuJunGod-PNAS-11, + Author = {{I. Y. Zhang, X. Xu, Y. Jung, and W. A. Goddard III}}, + Journal = {Proc. Natl. Acad. Sci. U.S.A.}, + Pages = {19896}, + Volume = {108}, + Year = {2011}} + +@article{ZhaYan-JCP-98, + Author = {Y. Zhang and W. Yang}, + Journal = {J. Chem. Phys.}, + Pages = {2604}, + Volume = {109}, + Year = {1998}} + +@article{ZheZhaTru-JCTC-07, + Author = {J. Zheng and Y. Zhao and D. G. Truhlar}, + Journal = {J. Chem. Theory Comput.}, + Pages = {569}, + Volume = {3}, + Year = {2007}} + +@article{ZheZhaTru-JCTC-09, + Author = {J. Zheng and Y. Zhao and D. G. Truhlar}, + Journal = {J. Chem. Theory Comput.}, + Pages = {808}, + Volume = {5}, + Year = {2009}} + +@article{ZhoChu-PRA-09, + Author = {Z. Zhou and S.-I. Chu}, + Journal = {Phys. Rev. A}, + Pages = {053412}, + Volume = {79}, + Year = {2009}} + +@article{ZhuTouSavAng-JCP-10, + Author = {Wuming Zhu and Julien Toulouse and Andreas Savin and J\'anos G. \'Angy\'an}, + Journal = {J. Chem. Phys.}, + Pages = {244108}, + Title = {Range-separated density-functional theory with random phase approximation applied to noncovalent intermolecular interactions}, + Volume = {132}, + Year = {2010}} + +@misc{ZhuTouSavAng-JJJ-XX, + Author = {W. Zhu and J. Toulouse and A. Savin and J. G. \'Angy\'an}, + Note = {unpublished}} + +@article{ZieCio-PA-05, + Author = {P. Ziesche and J. Cioslowski}, + Journal = {Physica A}, + Year = {to appear}} + +@article{Zie-JPC-80, + Author = {P. Ziesche}, + Journal = {J. Phys. C}, + Pages = {3625}, + Volume = {13}, + Year = {1980}} + +@article{ZieSetKryAutWan-JCP-09, + Author = {T. Ziegler and M. Seth and M. Krykunov and J. Autschbach and F. Wang}, + Journal = {J. Chem. Phys.}, + Pages = {154102}, + Volume = {130}, + Year = {2009}} + +@article{ZimParKou-JCP-04, + Author = {U. Zimmerli and M. Parrinello and P. Koumoutsakos}, + Journal = {J. Chem. Phys.}, + Pages = {2693}, + Volume = {120}, + Year = {2004}} + +@article{ZimTouZhaMusUmr-JCP-09, + Author = {P. M. Zimmerman and J. Toulouse and Z. Zhang and C. B. Musgrave and C. J. Umrigar}, + Journal = {J. Chem. Phys.}, + Pages = {124103}, + Title = {Excited states of methylene from quantum Monte Carlo}, + Volume = {131}, + Year = {2009}} + +@article{ZucPodMosJezSza-JCP-08, + Author = {P. S. Zuchowski and R. Podeszwa and R. Moszynski and B. Jeziorski and K. Szalewicz}, + Journal = {J. Chem. Phys.}, + Pages = {084101}, + Title = {{Symmetry-adapted perturbation theory utilizing density functional description of monomers for high-spin open-shell complexes}}, + Volume = {129}, + Year = {2008}} + +@article{FerGinTou-JCP-18, + Author = {Fert{\'e},Anthony and Giner,Emmanuel and Toulouse,Julien}, + Date-Modified = {2019-12-13 10:32:50 +0100}, + Doi = {10.1063/1.5082638}, + Journal = {J. Chem. Phys.}, + Number = {8}, + Pages = {084103}, + Title = {Range-separated multideterminant density-functional theory with a short-range correlation functional of the on-top pair density}, + Volume = {150}, + Year = {2019}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.5082638}} + +@article{GinTewGarAla-JCTC-18, + Author = {Giner, Emmanuel and Tew, David P. and Garniron, Yann and Alavi, Ali}, + Doi = {10.1021/acs.jctc.8b00591}, + Eprint = {https://doi.org/10.1021/acs.jctc.8b00591}, + Journal = {Journal of Chemical Theory and Computation}, + Note = {PMID: 30347156}, + Number = {12}, + Pages = {6240-6252}, + Title = {Interplay between Electronic Correlation and Metal--Ligand Delocalization in the Spectroscopy of Transition Metal Compounds: Case Study on a Series of Planar Cu2+ Complexes}, + Url = {https://doi.org/10.1021/acs.jctc.8b00591}, + Volume = {14}, + Year = {2018}, + Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b00591}} + +@article{GorSav-PRA-06, + Author = {Gori-Giorgi, Paola and Savin, Andreas}, + Doi = {10.1103/PhysRevA.73.032506}, + Issue = {3}, + Journal = {Phys. Rev. A}, + Month = {Mar}, + Numpages = {9}, + Pages = {032506}, + Publisher = {American Physical Society}, + Title = {Properties of short-range and long-range correlation energy density functionals from electron-electron coalescence}, + Url = {https://link.aps.org/doi/10.1103/PhysRevA.73.032506}, + Volume = {73}, + Year = {2006}, + Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevA.73.032506}, + Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevA.73.032506}} + +@article{HalHelJorKloKocOls-CPL-98, + Abstract = {Valence and all-electron correlation energies of Ne, N2, and H2O at fixed experimental geometries are computed at the levels of second-order perturbation theory (MP2) and coupled cluster theory with singles and doubles excitations (CCSD), and singles and doubles excitations with a perturbative triples correction (CCSD(T)). Correlation-consistent polarized valence and core-valence basis sets up to sextuple zeta quality are employed. Guided by basis-set limits established by rij-dependent methods, a number of extrapolation schemes for use with the correlation-consistent basis sets are investigated. Among the schemes considered here, a linear least-squares procedure applied to the quintuple and sextuple zeta results yields the most accurate extrapolations.}, + Author = {Asger Halkier and Trygve Helgaker and Poul J{\o}rgensen and Wim Klopper and Henrik Koch and Jeppe Olsen and Angela K. Wilson}, + Date-Modified = {2020-01-27 08:50:16 +0100}, + Doi = {10.1016/S0009-2614(98)00111-0}, + Journal = {Chemical Physics Letters}, + Number = {3}, + Pages = {243 - 252}, + Title = {Basis-set convergence in correlated calculations on Ne, N2, and H2O}, + Volume = {286}, + Year = {1998}, + Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0009261498001110}, + Bdsk-Url-2 = {https://doi.org/10.1016/S0009-2614(98)00111-0}} + +@article{BenErn-PhysRev-1969, + Author = {Charles F. Bender and Ernest R. Davidson}, + Doi = {10.1103/physrev.183.23}, + Journal = {Phys. Rev.}, + Month = {jul}, + Number = {1}, + Pages = {23--30}, + Publisher = {American Physical Society ({APS})}, + Title = {Studies in Configuration Interaction: The First-Row Diatomic Hydrides}, + Url = {http://dx.doi.org/10.1103/PhysRev.183.23}, + Volume = {183}, + Year = 1969, + Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRev.183.23}, + Bdsk-Url-2 = {http://dx.doi.org/10.1103/physrev.183.23}} + +@article{WhiHac-JCP-1969, + Author = {J. L. Whitten and Melvyn Hackmeyer}, + Doi = {10.1063/1.1671985}, + Journal = {J. Chem. Phys.}, + Month = {dec}, + Number = {12}, + Pages = {5584--5596}, + Publisher = {{AIP} Publishing}, + Title = {Configuration Interaction Studies of Ground and Excited States of Polyatomic Molecules. I. The {CI} Formulation and Studies of Formaldehyde}, + Url = {https://doi.org/10.1063%2F1.1671985}, + Volume = {51}, + Year = 1969, + Bdsk-Url-1 = {https://doi.org/10.1063%2F1.1671985}, + Bdsk-Url-2 = {https://doi.org/10.1063/1.1671985}} + +@article{HurMalRan-1973, + Author = {B. Huron and J. P. Malrieu and P. Rancurel}, + Doi = {10.1063/1.1679199}, + Journal = {J. Chem. Phys.}, + Month = {jun}, + Number = {12}, + Pages = {5745--5759}, + Publisher = {{AIP} Publishing}, + Title = {Iterative perturbation calculations of ground and excited state energies from multiconfigurational zeroth-order wavefunctions}, + Url = {https://doi.org/10.1063%2F1.1679199}, + Volume = {58}, + Year = 1973, + Bdsk-Url-1 = {https://doi.org/10.1063%2F1.1679199}, + Bdsk-Url-2 = {https://doi.org/10.1063/1.1679199}} + +@article{EvaDauMal-ChemPhys-83, + Author = {Stefano Evangelisti and Jean-Pierre Daudey and Jean-Paul Malrieu}, + Doi = {10.1016/0301-0104(83)85011-3}, + Journal = {Chemical Physics}, + Month = {feb}, + Number = {1}, + Pages = {91--102}, + Publisher = {Elsevier {BV}}, + Title = {Convergence of an improved {CIPSI} algorithm}, + Url = {https://doi.org/10.1016%2F0301-0104%2883%2985011-3}, + Volume = {75}, + Year = 1983, + Bdsk-Url-1 = {https://doi.org/10.1016%2F0301-0104%2883%2985011-3}, + Bdsk-Url-2 = {https://doi.org/10.1016/0301-0104(83)85011-3}} + +@article{Cim-JCP-1985, + Author = {Cimiraglia, Renzo}, + Date-Added = {2018-12-01 13:44:03 +0100}, + Date-Modified = {2018-12-01 13:44:03 +0100}, + Doi = {10.1063/1.449362}, + File = {1%2E449362.pdf:/home/scemama/Dropbox/Zotero/storage/52SWQQR4/1%2E449362.pdf:application/pdf;1.449362.pdf:/home/scemama/Dropbox/Zotero/storage/E6WCUH8T/1.449362.pdf:application/pdf;Snapshot:/home/scemama/Dropbox/Zotero/storage/HLPRZTEI/1.html:text/html}, + Issn = {0021-9606, 1089-7690}, + Journal = {J. Chem. Phys.}, + Language = {en}, + Month = aug, + Number = {4}, + Pages = {1746--1749}, + Shorttitle = {Second order perturbation correction to {CI} energies by use of diagrammatic techniques}, + Title = {Second order perturbation correction to {CI} energies by use of diagrammatic techniques: {An} improvement to the {CIPSI} algorithm}, + Url = {http://aip.scitation.org/doi/10.1063/1.449362}, + Urldate = {2017-11-14}, + Volume = {83}, + Year = {1985}, + Bdsk-Url-1 = {http://aip.scitation.org/doi/10.1063/1.449362}, + Bdsk-Url-2 = {http://dx.doi.org/10.1063/1.449362}} + +@article{Cim-JCC-1987, + Author = {Cimiraglia, Renzo and Persico, Maurizio}, + Date-Added = {2018-12-01 13:44:08 +0100}, + Date-Modified = {2018-12-01 13:44:08 +0100}, + File = {28a1e2d5fd9eba4d0cba5227b6acd8463ff941ffaa24949d1437dfca2bc8f3c7.pdf:/home/scemama/Dropbox/Zotero/storage/4L9PPHEJ/28a1e2d5fd9eba4d0cba5227b6acd8463ff941ffaa24949d1437dfca2bc8f3c7.pdf:application/pdf;540080105_ftp.pdf:/home/scemama/Dropbox/Zotero/storage/MZIFQQ9W/540080105_ftp.pdf:application/pdf}, + Journal = {J. Comput. Chem.}, + Number = {1}, + Pages = {39--47}, + Shorttitle = {Recent advances in multireference second order perturbation {CI}}, + Title = {Recent advances in multireference second order perturbation {CI}: {The} {CIPSI} method revisited}, + Volume = {8}, + Year = {1987}} + +@article{IllRubRic-JCP-88, + Author = {Illas, F. and Rubio, J. and Ricart, J. M.}, + Date-Added = {2018-12-01 13:44:13 +0100}, + Date-Modified = {2018-12-01 13:44:13 +0100}, + Doi = {10.1063/1.455405}, + File = {1.455405.pdf:/home/scemama/Dropbox/Zotero/storage/DYD93IM9/1.455405.pdf:application/pdf}, + Issn = {0021-9606, 1089-7690}, + Journal = {J. Chem. Phys.}, + Language = {en}, + Month = nov, + Number = {10}, + Pages = {6376--6384}, + Title = {Approximate natural orbitals and the convergence of a second order multireference many-body perturbation theory ({CIPSI}) algorithm}, + Url = {http://aip.scitation.org/doi/10.1063/1.455405}, + Urldate = {2017-11-14}, + Volume = {89}, + Year = {1988}, + Bdsk-Url-1 = {http://aip.scitation.org/doi/10.1063/1.455405}, + Bdsk-Url-2 = {http://dx.doi.org/10.1063/1.455405}} + +@article{PovRubIll-TCA-92, + Author = {Povill, A. and Rubio, J. and Illas, F.}, + Date-Added = {2018-12-01 13:44:18 +0100}, + Date-Modified = {2018-12-01 13:44:18 +0100}, + Doi = {10.1007/BF01113255}, + File = {BF01113255.pdf:/home/scemama/Dropbox/Zotero/storage/TH3IVJUP/BF01113255.pdf:application/pdf;Snapshot:/home/scemama/Dropbox/Zotero/storage/M8KP3TSQ/BF01113255.html:text/html}, + Journal = {Theor. Chem. Acc.}, + Number = {3}, + Pages = {229--238}, + Title = {Treating large intermediate spaces in the {CIPSI} method through a direct selected {CI} algorithm}, + Volume = {82}, + Year = {1992}, + Bdsk-Url-1 = {https://doi.org/10.1007/BF01113255}} + +@article{BunCarRam-JCP-06, + Author = {Bunge, Carlos F. and Carb{\'o}-Dorca, Ramon}, + Date-Added = {2018-12-01 13:44:28 +0100}, + Date-Modified = {2018-12-01 13:44:28 +0100}, + Doi = {10.1063/1.2207621}, + File = {Snapshot:/home/scemama/Dropbox/Zotero/storage/SB8DJGT3/1.html:text/html}, + Issn = {0021-9606}, + Journal = {J. Chem. Phys.}, + Month = jul, + Number = {1}, + Pages = {014108}, + Title = {Select-divide-and-conquer method for large-scale configuration interaction}, + Url = {http://aip.scitation.org/doi/abs/10.1063/1.2207621}, + Urldate = {2017-11-17}, + Volume = {125}, + Year = {2006}, + Bdsk-Url-1 = {http://aip.scitation.org/doi/abs/10.1063/1.2207621}, + Bdsk-Url-2 = {http://dx.doi.org/10.1063/1.2207621}} + +@article{AbrSheDav-CPL-05, + Author = {Abrams, Micah L. and Sherrill, C. David}, + Date-Added = {2018-12-01 13:44:23 +0100}, + Date-Modified = {2018-12-01 13:44:23 +0100}, + Doi = {10.1016/j.cplett.2005.06.107}, + Issn = {0009-2614}, + Journal = {Chem. Phys. Lett.}, + Month = {Aug}, + Number = {1-3}, + Pages = {121--124}, + Publisher = {Elsevier BV}, + Title = {Important configurations in configuration interaction and coupled-cluster wave functions}, + Url = {http://dx.doi.org/10.1016/j.cplett.2005.06.107}, + Volume = {412}, + Year = {2005}, + Bdsk-Url-1 = {http://dx.doi.org/10.1016/j.cplett.2005.06.107}} + +@article{MusEngels-JCC-06, + Author = {Musch, Patrick and Engels, Bernd}, + Journal = {{Journal of Computational Chemistry}}, + Number = {{10}}, + Year = {{2006}}} + +@article{BytRue-CP-09, + Author = {Laimutis Bytautas and Klaus Ruedenberg}, + Doi = {10.1016/j.chemphys.2008.11.021}, + Journal = {Chemical Physics}, + Month = {feb}, + Number = {1-3}, + Pages = {64--75}, + Publisher = {Elsevier {BV}}, + Title = {A priori identification of configurational deadwood}, + Url = {https://doi.org/10.1016%2Fj.chemphys.2008.11.021}, + Volume = {356}, + Year = 2009, + Bdsk-Url-1 = {https://doi.org/10.1016%2Fj.chemphys.2008.11.021}, + Bdsk-Url-2 = {https://doi.org/10.1016/j.chemphys.2008.11.021}} + +@article{GinSceCaf-CJC-13, + Author = {Giner, Emmanuel and Scemama, Anthony and Caffarel, Michel}, + Date-Added = {2018-11-29 14:22:53 +0100}, + Date-Modified = {2018-11-29 14:22:53 +0100}, + Doi = {10.1139/cjc-2013-0017}, + Issn = {1480-3291}, + Journal = {Can. J. Chem.}, + Month = {Sep}, + Number = {9}, + Pages = {879--885}, + Publisher = {Canadian Science Publishing}, + Title = {Using perturbatively selected configuration interaction in quantum Monte Carlo calculations}, + Url = {http://dx.doi.org/10.1139/cjc-2013-0017}, + Volume = {91}, + Year = {2013}, + Bdsk-Url-1 = {http://dx.doi.org/10.1139/cjc-2013-0017}} + +@article{CafGinScemRam-JCTC-14, + Author = {Caffarel, Michel and Giner, Emmanuel and Scemama, Anthony and Ram{\'\i}rez-Sol{\'\i}s, Alejandro}, + Date-Added = {2018-11-29 14:22:40 +0100}, + Date-Modified = {2018-11-29 14:22:40 +0100}, + Doi = {10.1021/ct5004252}, + Issn = {1549-9626}, + Journal = {J. Chem. Theory Comput.}, + Month = {Dec}, + Number = {12}, + Pages = {5286--5296}, + Publisher = {American Chemical Society (ACS)}, + Title = {Spin Density Distribution in Open-Shell Transition Metal Systems: A Comparative Post-Hartree--Fock, Density Functional Theory, and Quantum Monte Carlo Study of the CuCl$_2$ Molecule}, + Url = {http://dx.doi.org/10.1021/ct5004252}, + Volume = {10}, + Year = {2014}, + Bdsk-Url-1 = {http://dx.doi.org/10.1021/ct5004252}} + +@article{GinSceCaf-JCP-15, + Author = {Emmanuel Giner and Anthony Scemama and Michel Caffarel}, + Date-Added = {2018-11-29 14:22:53 +0100}, + Date-Modified = {2018-11-29 14:22:53 +0100}, + Doi = {10.1063/1.4905528}, + Issn = {1089-7690}, + Journal = {J. Chem. Phys.}, + Month = {Jan}, + Number = {4}, + Pages = {044115}, + Publisher = {AIP Publishing}, + Title = {Fixed-node diffusion Monte Carlo potential energy curve of the fluorine molecule F$_2$ using selected configuration interaction trial wavefunctions}, + Url = {http://dx.doi.org/10.1063/1.4905528}, + Volume = {142}, + Year = {2015}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4905528}} + +@article{CafAplGinScem-arxiv-16, + Author = {Michel Caffarel and Thomas Applencourt and Emmanuel Giner and Anthony Scemama}, + Date-Added = {2019-02-14 10:41:06 +0100}, + Date-Modified = {2019-02-14 10:41:23 +0100}, + Doi = {10.1021/bk-2016-1234.ch002}, + Eprint = {arXiv:1607.06742}, + Title = {Using CIPSI nodes in diffusion Monte Carlo}, + Year = {2016}, + Bdsk-Url-1 = {https://dx.doi.org/10.1021/bk-2016-1234.ch002}} + +@article{CafAplGinSce-JCP-16, + Author = {Michel Caffarel and Thomas Applencourt and Emmanuel Giner and Anthony Scemama}, + Date-Modified = {2019-02-14 10:41:27 +0100}, + Doi = {10.1063/1.4947093}, + Journal = {J. Chem. Phys.}, + Month = {apr}, + Number = {15}, + Pages = {151103}, + Publisher = {{AIP} Publishing}, + Title = {Communication: Toward an improved control of the fixed-node error in quantum Monte Carlo: The case of the water molecule}, + Url = {https://doi.org/10.1063%2F1.4947093}, + Volume = {144}, + Year = 2016, + Bdsk-Url-1 = {https://doi.org/10.1063%2F1.4947093}, + Bdsk-Url-2 = {https://doi.org/10.1063/1.4947093}} + +@article{SchEva-JCP-16, + Author = {Schriber, Jeffrey B. and Evangelista, Francesco A.}, + Date-Added = {2018-12-01 13:44:38 +0100}, + Date-Modified = {2018-12-01 13:44:38 +0100}, + Doi = {10.1063/1.4948308}, + File = {Full Text PDF:/home/scemama/Dropbox/Zotero/storage/XR99ZTDH/Schriber and Evangelista - 2016 - Communication An adaptive configuration interacti.pdf:application/pdf;Snapshot:/home/scemama/Dropbox/Zotero/storage/6KITP3BL/1.html:text/html}, + Issn = {0021-9606}, + Journal = {J. Chem. Phys.}, + Month = apr, + Number = {16}, + Pages = {161106}, + Shorttitle = {Communication}, + Title = {Communication: {An} adaptive configuration interaction approach for strongly correlated electrons with tunable accuracy}, + Url = {http://aip.scitation.org/doi/abs/10.1063/1.4948308}, + Urldate = {2017-11-17}, + Volume = {144}, + Year = {2016}, + Bdsk-Url-1 = {http://aip.scitation.org/doi/abs/10.1063/1.4948308}, + Bdsk-Url-2 = {http://dx.doi.org/10.1063/1.4948308}} + +@article{LiuHofJCTC-16, + Author = {Wenjian Liu and Mark R. Hoffmann}, + Doi = {10.1021/acs.jctc.5b01099}, + Journal = {J. Chem. Theory Comput.}, + Month = {jan}, + Number = {3}, + Pages = {1169--1178}, + Publisher = {American Chemical Society ({ACS})}, + Title = {{iCI}: Iterative {CI} toward full {CI}}, + Url = {https://doi.org/10.1021%2Facs.jctc.5b01099}, + Volume = {12}, + Year = 2016, + Bdsk-Url-1 = {https://doi.org/10.1021%2Facs.jctc.5b01099}, + Bdsk-Url-2 = {https://doi.org/10.1021/acs.jctc.5b01099}} + +@article{ShaHolJeaAlaUmr-JCTC-17, + Author = {Sharma, Sandeep and Holmes, Adam A. and Jeanmairet, Guillaume and Alavi, Ali and Umrigar, C. J.}, + Date-Added = {2018-12-01 13:35:29 +0100}, + Date-Modified = {2018-12-01 13:35:29 +0100}, + Doi = {10.1021/acs.jctc.6b01028}, + Issn = {1549-9626}, + Journal = {J. Chem. Theory Comput.}, + Month = {Mar}, + Number = {4}, + Pages = {1595--1604}, + Publisher = {American Chemical Society (ACS)}, + Title = {Semistochastic Heat-Bath Configuration Interaction Method: Selected Configuration Interaction with Semistochastic Perturbation Theory}, + Url = {http://dx.doi.org/10.1021/acs.jctc.6b01028}, + Volume = {13}, + Year = {2017}, + Bdsk-Url-1 = {http://dx.doi.org/10.1021/acs.jctc.6b01028}} + +@article{HolUmrSha-JCP-17, + Author = {Holmes, Adam A. and Umrigar, C. J. and Sharma, Sandeep}, + Date-Added = {2018-12-01 13:35:29 +0100}, + Date-Modified = {2018-12-01 13:35:29 +0100}, + Doi = {10.1063/1.4998614}, + Issn = {1089-7690}, + Journal = {J. Chem. Phys.}, + Month = {Oct}, + Number = {16}, + Pages = {164111}, + Publisher = {AIP Publishing}, + Title = {Excited states using semistochastic heat-bath configuration interaction}, + Url = {http://dx.doi.org/10.1063/1.4998614}, + Volume = {147}, + Year = {2017}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4998614}} + +@article{SchEva-JCTC-17, + Author = {Schriber, Jeffrey B. and Evangelista, Francesco A.}, + Date-Modified = {2018-12-01 13:45:15 +0100}, + Doi = {10.1021/acs.jctc.7b00725}, + Journal = {J. Chem. Theory Comput.}, + Month = {Oct}, + Publisher = {American Chemical Society}, + Title = {{Adaptive Configuration Interaction for Computing Challenging Electronic Excited States with Tunable Accuracy}}, + Year = {2017}, + Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.7b00725}} + +@article{PerCle-JCP-17, + Author = {Per, Manolo C. and Cleland, Deidre M.}, + Date-Added = {2018-12-01 13:44:59 +0100}, + Date-Modified = {2018-12-01 13:44:59 +0100}, + Doi = {10.1063/1.4981527}, + Issn = {1089-7690}, + Journal = {J. Chem. Phys.}, + Month = {Apr}, + Number = {16}, + Pages = {164101}, + Publisher = {AIP Publishing}, + Title = {Energy-based truncation of multi-determinant wavefunctions in quantum Monte Carlo}, + Url = {http://dx.doi.org/10.1063/1.4981527}, + Volume = {146}, + Year = {2017}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4981527}} + +@article{OhtJun-JCP-17, + Author = {Yuhki Ohtsuka and Jun-ya Hasegawa}, + Doi = {10.1063/1.4993214}, + Journal = {J. Chem. Phys.}, + Month = {jul}, + Number = {3}, + Pages = {034102}, + Publisher = {{AIP} Publishing}, + Title = {Selected configuration interaction method using sampled first-order corrections to wave functions}, + Url = {https://doi.org/10.1063%2F1.4993214}, + Volume = {147}, + Year = 2017, + Bdsk-Url-1 = {https://doi.org/10.1063%2F1.4993214}, + Bdsk-Url-2 = {https://doi.org/10.1063/1.4993214}} + +@article{Zim-JCP-17, + Author = {Zimmerman, Paul M.}, + Date-Added = {2018-12-01 13:45:04 +0100}, + Date-Modified = {2018-12-01 13:45:04 +0100}, + Doi = {10.1063/1.4977727}, + Issn = {1089-7690}, + Journal = {J. Chem. Phys.}, + Month = {Mar}, + Number = {10}, + Pages = {104102}, + Publisher = {AIP Publishing}, + Title = {Incremental full configuration interaction}, + Url = {http://dx.doi.org/10.1063/1.4977727}, + Volume = {146}, + Year = {2017}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4977727}} + +@article{LiOttHolShaUmr-JCP-2018, + Author = {J. Li and M. Otten and A. A. Holmes and S. Sharma and C. J. Umrigar}, + Date-Added = {2019-02-13 09:25:07 +0100}, + Date-Modified = {2019-02-13 09:26:03 +0100}, + Doi = {10.1063/1.5055390}, + Journal = {J. Chem. Phys.}, + Pages = {214110}, + Title = {Fast semistochastic heat-bath configuration interaction}, + Volume = {149}, + Year = {2018}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.5055390}} + +@article{ChiHolOttUmrShaZim-JPCA-18, + Author = {Chien, Alan D. and Holmes, Adam A. and Otten, Matthew and Umrigar, C. J. and Sharma, Sandeep and Zimmerman, Paul M.}, + Date-Added = {2018-12-01 13:35:29 +0100}, + Date-Modified = {2018-12-01 13:35:29 +0100}, + Doi = {10.1021/acs.jpca.8b01554}, + File = {/Users/loos/Zotero/storage/J96RZ7JP/Chien et al. - 2018 - Excited States of Methylene, Polyenes, and Ozone f.pdf}, + Issn = {1089-5639, 1520-5215}, + Journal = {J. Phys. Chem. A}, + Language = {en}, + Month = mar, + Number = {10}, + Pages = {2714--2722}, + Title = {Excited {{States}} of {{Methylene}}, {{Polyenes}}, and {{Ozone}} from {{Heat}}-{{Bath Configuration Interaction}}}, + Volume = {122}, + Year = {2018}, + Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpca.8b01554}} + +@article{SceBenJacCafLoo-JCP-18, + Author = {Anthony Scemama and Anouar Benali and Denis Jacquemin and Michel Caffarel and Pierre-Fran{\c{c}}ois Loos}, + Date-Modified = {2018-11-29 13:30:28 +0100}, + Doi = {10.1063/1.5041327}, + Journal = {J. Chem. Phys.}, + Month = {jul}, + Number = {3}, + Pages = {034108}, + Publisher = {{AIP} Publishing}, + Title = {Excitation energies from diffusion Monte Carlo using selected configuration interaction nodes}, + Url = {https://doi.org/10.1063%2F1.5041327}, + Volume = {149}, + Year = 2018, + Bdsk-Url-1 = {https://doi.org/10.1063%2F1.5041327}, + Bdsk-Url-2 = {https://doi.org/10.1063/1.5041327}} + +@article{LooSceBloGarCafJac-JCTC-18, + Author = {Pierre-Fran{\c{c}}ois Loos and Anthony Scemama and Aymeric Blondel and Yann Garniron and Michel Caffarel and Denis Jacquemin}, + Date-Modified = {2019-02-05 11:23:26 +0100}, + Doi = {10.1021/acs.jctc.8b00406}, + Journal = {J. Chem. Theory Comput.}, + Month = {jul}, + Number = {8}, + Pages = {4360--4379}, + Publisher = {American Chemical Society ({ACS})}, + Title = {A Mountaineering Strategy to Excited States: Highly Accurate Reference Energies and Benchmarks}, + Url = {https://doi.org/10.1021%2Facs.jctc.8b00406}, + Volume = {14}, + Year = 2018, + Bdsk-Url-1 = {https://doi.org/10.1021%2Facs.jctc.8b00406}, + Bdsk-Url-2 = {https://doi.org/10.1021/acs.jctc.8b00406}} + +@article{GarSceGinCaffLoo-JCP-18, + Author = {Y. Garniron and A. Scemama and E. Giner and M. Caffarel and P. F. Loos}, + Date-Added = {2018-11-29 14:23:11 +0100}, + Date-Modified = {2018-11-29 14:23:11 +0100}, + Doi = {10.1063/1.5044503}, + Journal = {J. Chem. Phys.}, + Pages = {064103}, + Title = {Selected Configuration Interaction Dressed by Perturbation}, + Volume = {149}, + Year = {2018}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.5044503}} + +@article{SceGarCafLoo-JCTC-18, + Author = {A. Scemama and Y. Garniron and M. Caffarel and P. F. Loos}, + Date-Added = {2018-11-29 13:30:19 +0100}, + Date-Modified = {2018-11-29 13:30:23 +0100}, + Doi = {10.1021/acs.jctc.7b01250}, + Journal = {J. Chem. Theory Comput.}, + Pages = {1395}, + Title = {Deterministic construction of nodal surfaces within quantum Monte Carlo: the case of FeS}, + Volume = {14}, + Year = {2018}, + Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.7b01250}} + +@article{GarGinMalSce-JCP-16, + Author = {Yann Garniron and Emmanuel Giner and Jean-Paul Malrieu and Anthony Scemama}, + Doi = {10.1063/1.4980034}, + Journal = {J. Chem. Phys.}, + Month = {apr}, + Number = {15}, + Pages = {154107}, + Publisher = {{AIP} Publishing}, + Title = {Alternative definition of excitation amplitudes in multi-reference state-specific coupled cluster}, + Url = {https://doi.org/10.1063%2F1.4980034}, + Volume = {146}, + Year = 2017, + Bdsk-Url-1 = {https://doi.org/10.1063%2F1.4980034}, + Bdsk-Url-2 = {https://doi.org/10.1063/1.4980034}} + +@article{LooBogSceCafJac-JCTC-19, + Author = {P. F. Loos and M. Boggio-Pasqua and A. Scemama and M. Caffarel and D. Jacquemin}, + Date-Added = {2019-02-05 09:37:37 +0100}, + Date-Modified = {2019-02-05 09:37:37 +0100}, + Doi = {10.1021/acs.jctc.8b01205}, + Journal = {J. Chem. Theory Comput.}, + Pages = {in press}, + Title = {Reference energies for double excitations}, + Volume = {15}, + Year = {2019}, + Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b01205}} + +@article{Thom-PRL-10, + Author = {Thom, Alex J. W.}, + Doi = {10.1103/PhysRevLett.105.263004}, + Issue = {26}, + Journal = {Phys. Rev. Lett.}, + Month = {Dec}, + Numpages = {4}, + Pages = {263004}, + Publisher = {American Physical Society}, + Title = {Stochastic Coupled Cluster Theory}, + Url = {https://link.aps.org/doi/10.1103/PhysRevLett.105.263004}, + Volume = {105}, + Year = {2010}, + Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.105.263004}, + Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.105.263004}} + +@article{ScoTho-JCP-17, + Author = {Scott,Charles J. C. and Thom,Alex J. W.}, + Doi = {10.1063/1.4991795}, + Eprint = {https://doi.org/10.1063/1.4991795}, + Journal = {The Journal of Chemical Physics}, + Number = {12}, + Pages = {124105}, + Title = {Stochastic coupled cluster theory: Efficient sampling of the coupled cluster expansion}, + Url = {https://doi.org/10.1063/1.4991795}, + Volume = {147}, + Year = {2017}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.4991795}} + +@article{SpeNeuVigFraTho-JCP-18, + Author = {Spencer,J. S. and Neufeld,V. A. and Vigor,W. A. and Franklin,R. S. T. and Thom,A. J. W.}, + Doi = {10.1063/1.5047420}, + Eprint = {https://doi.org/10.1063/1.5047420}, + Journal = {The Journal of Chemical Physics}, + Number = {20}, + Pages = {204103}, + Title = {Large scale parallelization in stochastic coupled cluster}, + Url = {https://doi.org/10.1063/1.5047420}, + Volume = {149}, + Year = {2018}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.5047420}} + +@article{DeuEmiShePie-PRL-17, + Author = {Deustua, J. Emiliano and Shen, Jun and Piecuch, Piotr}, + Doi = {10.1103/PhysRevLett.119.223003}, + Issue = {22}, + Journal = {Phys. Rev. Lett.}, + Month = {Nov}, + Numpages = {5}, + Pages = {223003}, + Publisher = {American Physical Society}, + Title = {Converging High-Level Coupled-Cluster Energetics by Monte Carlo Sampling and Moment Expansions}, + Url = {https://link.aps.org/doi/10.1103/PhysRevLett.119.223003}, + Volume = {119}, + Year = {2017}, + Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.119.223003}, + Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.119.223003}} + +@article{DeuEmiMagShePie-JCP-18, + Author = {Deustua,J. Emiliano and Magoulas,Ilias and Shen,Jun and Piecuch,Piotr}, + Doi = {10.1063/1.5055769}, + Eprint = {https://doi.org/10.1063/1.5055769}, + Journal = {The Journal of Chemical Physics}, + Number = {15}, + Pages = {151101}, + Title = {Communication: Approaching exact quantum chemistry by cluster analysis of full configuration interaction quantum Monte Carlo wave functions}, + Url = {https://doi.org/10.1063/1.5055769}, + Volume = {149}, + Year = {2018}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.5055769}} + +@article{DeuEmiYumShePie-JCP-19, + Author = {Deustua,J. Emiliano and Yuwono,Stephen H. and Shen,Jun and Piecuch,Piotr}, + Doi = {10.1063/1.5090346}, + Eprint = {https://doi.org/10.1063/1.5090346}, + Journal = {The Journal of Chemical Physics}, + Number = {11}, + Pages = {111101}, + Title = {Accurate excited-state energetics by a combination of Monte Carlo sampling and equation-of-motion coupled-cluster computations}, + Url = {https://doi.org/10.1063/1.5090346}, + Volume = {150}, + Year = {2019}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.5090346}} + +@article{QiuHenZhaScu-JCP-17, + Author = {Qiu,Yiheng and Henderson,Thomas M. and Zhao,Jinmo and Scuseria,Gustavo E.}, + Doi = {10.1063/1.4991020}, + Eprint = {https://doi.org/10.1063/1.4991020}, + Journal = {The Journal of Chemical Physics}, + Number = {6}, + Pages = {064111}, + Title = {Projected coupled cluster theory}, + Url = {https://doi.org/10.1063/1.4991020}, + Volume = {147}, + Year = {2017}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.4991020}} + +@article{QiuHenZhaScu-JCP-18, + Author = {Qiu,Yiheng and Henderson,Thomas M. and Zhao,Jinmo and Scuseria,Gustavo E.}, + Doi = {10.1063/1.5053605}, + Eprint = {https://doi.org/10.1063/1.5053605}, + Journal = {The Journal of Chemical Physics}, + Number = {16}, + Pages = {164108}, + Title = {Projected coupled cluster theory: Optimization of cluster amplitudes in the presence of symmetry projection}, + Url = {https://doi.org/10.1063/1.5053605}, + Volume = {149}, + Year = {2018}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.5053605}} + +@article{GomHenScu-JCP-19, + Author = {Gomez,John A. and Henderson,Thomas M. and Scuseria,Gustavo E.}, + Doi = {10.1063/1.5085314}, + Eprint = {https://doi.org/10.1063/1.5085314}, + Journal = {The Journal of Chemical Physics}, + Number = {14}, + Pages = {144108}, + Title = {Polynomial-product states: A symmetry-projection-based factorization of the full coupled cluster wavefunction in terms of polynomials of double excitations}, + Url = {https://doi.org/10.1063/1.5085314}, + Volume = {150}, + Year = {2019}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.5085314}} + +@article{Hyl-ZP-29, + Author = {E. A. Hylleraas}, + Date-Added = {2019-04-07 14:28:17 +0200}, + Date-Modified = {2020-01-27 08:44:37 +0100}, + Doi = {10.1007/BF01375457}, + Journal = {Z. Phys.}, + Pages = {347}, + Title = {Neue Berechnung der Energie des Heliums im Grundzustande, sowie des tiefsten Terms von Ortho-Helium}, + Volume = {54}, + Year = {1929}} + +@article{KutKlo-JCP-91, + Author = {W. Kutzelnigg and W. Klopper}, + Date-Added = {2019-04-03 21:35:04 +0200}, + Date-Modified = {2020-01-27 08:45:50 +0100}, + Doi = {10.1063/1.459921}, + Journal = {J. Chem. Phys.}, + Pages = {1985}, + Title = {Wave functions with terms linear in the interelectronic coordinates to take care of the correlation cusp. I. General theory}, + Volume = {94}, + Year = {1991}} + +@article{IrmHulGru-PRL-19, + Author = {Andreas Irmler and Felix Hummel and Andreas Gr{\"u}neis}, + Date-Modified = {2020-01-06 14:18:23 +0100}, + Doi = {10.1103/PhysRevLett.123.156401}, + Journal = {Phys. Rev. Lett.}, + Pages = {156401}, + Title = {On the duality of ring and ladder diagrams and its importance for many-electron perturbation theories}, + Volume = {123}, + Year = {2019}, + Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.123.156401}} + +@article{GruHirOhnTen-JCP-17, + Author = {A. Gr\"uneis and S. Hirata and Y.-Y. Ohnishi and S. Ten-no}, + Date-Added = {2019-05-08 10:24:45 +0200}, + Date-Modified = {2019-05-08 10:27:42 +0200}, + Doi = {10.1063/1.4976974}, + Journal = {J. Chem. Phys.}, + Pages = {080901}, + Title = {Perspective: Explicitly correlated electronic structure theory for complex systems}, + Volume = {146}, + Year = {2017}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.4976974}} + +@article{MaWer-WIREs-18, + Author = {Q. Ma and H.-J. Werner}, + Date-Added = {2019-05-08 10:32:33 +0200}, + Date-Modified = {2019-05-08 10:33:31 +0200}, + Journal = {WIREs Comput. Mol. Sci.}, + doi = {10.1002/wcms.1371}, + Pages = {e1371}, + Title = {Explicitly correlated local coupled‐cluster methods using pair natural orbitals}, + Volume = {8}, + Url = {https://doi.org/10.1002/wcms.1371}, + Year = {2018}} + +@article{LooPraSceTouGin-JCPL-19, + Author = {Loos, Pierre-Fran{\c c}ois and Pradines, Barth{\'e}l{\'e}my and Scemama, Anthony and Toulouse, Julien and Giner, Emmanuel}, + Date-Modified = {2019-12-13 10:33:25 +0100}, + Doi = {10.1021/acs.jpclett.9b01176}, + Journal = {J. Phys. Chem. Lett.}, + Number = {11}, + Pages = {2931-2937}, + Title = {A Density-Based Basis-Set Correction for Wave Function Theory}, + Volume = {10}, + Year = {2019}, + Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpclett.9b01176}} + +@article{GoriSav-PRA-06, + Author = {Gori-Giorgi, Paola and Savin, Andreas}, + Doi = {10.1103/PhysRevA.73.032506}, + Issue = {3}, + Journal = {Phys. Rev. A}, + Month = {Mar}, + Numpages = {9}, + Pages = {032506}, + Publisher = {American Physical Society}, + Title = {Properties of short-range and long-range correlation energy density functionals from electron-electron coalescence}, + Url = {https://link.aps.org/doi/10.1103/PhysRevA.73.032506}, + Volume = {73}, + Year = {2006}, + Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevA.73.032506}, + Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevA.73.032506}} + +@article{GritMeePer-PRA-18, + Author = {Gritsenko, Oleg V. and van Meer, Robert and Pernal, Katarzyna}, + Doi = {10.1103/PhysRevA.98.062510}, + Issue = {6}, + Journal = {Phys. Rev. A}, + Month = {Dec}, + Numpages = {8}, + Pages = {062510}, + Publisher = {American Physical Society}, + Title = {Efficient evaluation of electron correlation along the bond-dissociation coordinate in the ground and excited ionic states with dynamic correlation suppression and enhancement functions of the on-top pair density}, + Url = {https://link.aps.org/doi/10.1103/PhysRevA.98.062510}, + Volume = {98}, + Year = {2018}, + Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevA.98.062510}, + Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevA.98.062510}} + +@article{GinSceTouLoo-JCP-19, + Author = {Giner,Emmanuel and Scemama,Anthony and Toulouse,Julien and Loos,Pierre-Fran{\c c}ois}, + Date-Modified = {2019-12-13 10:33:54 +0100}, + Doi = {10.1063/1.5122976}, + Journal = {J. Chem. Phys.}, + Number = {14}, + Pages = {144118}, + Title = {Chemically accurate excitation energies with small basis sets}, + Volume = {151}, + Year = {2019}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.5122976}} + +@article{QP2, + Author = {Y. Garniron and K. Gasperich and T. Applencourt and A. Benali and A. Fert{\'e} and J. Paquier and B. Pradines and R. Assaraf and P. Reinhardt and J. Toulouse and P. Barbaresco and N. Renon and G. David and J. P. Malrieu and M. V{\'e}ril and M. Caffarel and P. F. Loos and E. Giner and A. Scemama}, + Date-Added = {2019-04-07 13:54:16 +0200}, + Date-Modified = {2019-06-12 14:59:52 +0200}, + Doi = {10.1021/acs.jctc.9b00176}, + Journal = {J. Chem. Theory Comput.}, + Pages = {3591}, + Title = {Quantum Package 2.0: A Open-Source Determinant-Driven Suite Of Programs}, + Volume = {15}, + Year = {2019}, + Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.9b00176}} + +@inbook{gamess, + Address = {Amsterdam}, + Author = {Gordon, M. S. and Schmidt, M. W.}, + Booktitle = {Theory and Applications of Computational Chemistry: the first forty years}, + Editor = {Dykstra, C. E. and Frenking, G. and Kim, K. S. and Scuseria, G. E.}, + Pages = {1167--1189}, + Publisher = {Elsevier}, + Title = {{Advances in electronic structure theory: GAMESS a decade later}}, + doi = {http://dx.doi.org/10.1016/b978-044451719-7/50084-6 }, + Year = {2005}} + +@article{h10_prx, + Author = {Motta, Mario and Ceperley, David M. and Chan, Garnet Kin-Lic and Gomez, John A. and Gull, Emanuel and Guo, Sheng and Jim\'enez-Hoyos, Carlos A. and Lan, Tran Nguyen and Li, Jia and Ma, Fengjie and Millis, Andrew J. and Prokof'ev, Nikolay V. and Ray, Ushnish and Scuseria, Gustavo E. and Sorella, Sandro and Stoudenmire, Edwin M. and Sun, Qiming and Tupitsyn, Igor S. and White, Steven R. and Zgid, Dominika and Zhang, Shiwei}, + Collaboration = {Simons Collaboration on the Many-Electron Problem}, + Doi = {10.1103/PhysRevX.7.031059}, + Issue = {3}, + Journal = {Phys. Rev. X}, + Month = {Sep}, + Numpages = {28}, + Pages = {031059}, + Publisher = {American Physical Society}, + Title = {Towards the Solution of the Many-Electron Problem in Real Materials: Equation of State of the Hydrogen Chain with State-of-the-Art Many-Body Methods}, + Url = {https://link.aps.org/doi/10.1103/PhysRevX.7.031059}, + Volume = {7}, + Year = {2017}, + Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevX.7.031059}, + Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevX.7.031059}} + +@article{ShiWer-JCP-10, + Author = {Shiozaki,Toru and Werner,Hans-Joachim}, + Date-Modified = {2019-12-13 10:30:35 +0100}, + Doi = {10.1063/1.3489000}, + Journal = {J. Chem. Phys.}, + Number = {14}, + Pages = {141103}, + Title = {Communication: Second-order multireference perturbation theory with explicit correlation: CASPT2-F12}, + Volume = {133}, + Year = {2010}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.348900}, + Bdsk-Url-2 = {https://doi.org/10.1063/1.3489000}} + +@article{GuoSivValNee-JCP-17, + Author = {Guo,Yang and Sivalingam,Kantharuban and Valeev,Edward F. and Neese,Frank}, + Date-Modified = {2019-12-13 10:30:13 +0100}, + Doi = {10.1063/1.4996560}, + Journal = {J. Chem. Phys.}, + Number = {6}, + Pages = {064110}, + Title = {Explicitly correlated N-electron valence state perturbation theory (NEVPT2-F12)}, + Volume = {147}, + Year = {2017}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.4996560}} + +@article{TorVal-JCP-09, + Author = {Torheyden,Martin and Valeev,Edward F.}, + Date-Modified = {2019-12-13 10:31:19 +0100}, + Doi = {10.1063/1.3254836}, + Journal = {J. Chem. Phys.}, + Number = {17}, + Pages = {171103}, + Title = {Universal perturbative explicitly correlated basis set incompleteness correction}, + Volume = {131}, + Year = {2009}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.3254836}} + +@article{Ten-CPL-07, + Abstract = {We propose a simple F12 geminal correction in multi-reference perturbation theory. An explicitly correlated term is introduced in the external excitations of the first order wave function in an internally contracted manner. By the use of the s- and p-wave cusp conditions, the F12 correction is expressed as the expectation value of a two-body effective operator, which reduces to the MP2-F12/A∗(SP) energy in the single reference limit. The performance of the F12 multi-reference perturbation method is demonstrated for C, CH2, O2, and SiC3.}, + Author = {Seiichiro Ten-no}, + Date-Modified = {2019-12-13 10:29:25 +0100}, + Doi = {https://doi.org/10.1016/j.cplett.2007.09.006}, + Issn = {0009-2614}, + Journal = {Chem. Phys. Lett.}, + Number = {1}, + Pages = {175 - 179}, + Title = {A simple F12 geminal correction in multi-reference perturbation theory}, + Url = {http://www.sciencedirect.com/science/article/pii/S0009261407012286}, + Volume = {447}, + Year = {2007}, + Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0009261407012286}, + Bdsk-Url-2 = {https://doi.org/10.1016/j.cplett.2007.09.006}} + +@article{KedDemPitTenNog-CPL-11, + Abstract = {This Letter reports development and implementation of the explicitly correlated multireference Brillouin Wigner (MR BW-CC) coupled cluster method with Slater type geminals. The performance of the new approach is tested on the H4 model system and the dissociation curve of the fluorine molecule. Like in single reference methods, results show a dramatically improved convergence of total energies towards complete basis set limit as compared to a conventional MR BW-CC approach. In comparison with previously reported calculations with a linear correlation factor, there is a better performance for calculations in smaller basis sets.}, + Author = {Stanislav Ked{\v z}uch and Ond{\v r}ej Demel and Ji{\v r}{\'\i} Pittner and Seiichiro Ten-no and Jozef Noga}, + Doi = {https://doi.org/10.1016/j.cplett.2011.06.023}, + Issn = {0009-2614}, + Journal = {Chemical Physics Letters}, + Number = {4}, + Pages = {418 - 423}, + Title = {Multireference F12 coupled cluster theory: The Brillouin-Wigner approach with single and double excitations}, + Url = {http://www.sciencedirect.com/science/article/pii/S0009261411007160}, + Volume = {511}, + Year = {2011}, + Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0009261411007160}, + Bdsk-Url-2 = {https://doi.org/10.1016/j.cplett.2011.06.023}} + +@article{TorKniWer-JCP-11, + Author = {Shiozaki,Toru and Knizia,Gerald and Werner,Hans-Joachim}, + Date-Modified = {2019-12-13 10:30:31 +0100}, + Doi = {10.1063/1.3528720}, + Journal = {J. Chem. Phys.}, + Number = {3}, + Pages = {034113}, + Title = {Explicitly correlated multireference configuration interaction: MRCI-F12}, + Volume = {134}, + Year = {2011}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.3528720}} + +@article{DemStanMatTenPitNog-PCCP-12, + Abstract = {This paper reports development of the explicitly correlated variant of Mukherjee{'}s state specific multireference coupled cluster method (MkCC-F12). The current implementation is restricted to conventional single and double excitations and to pseudo-double excitations related to the Slater Type Geminal (STG) correlation factor using the SP ansatz. The performance of the MkCCSD-F12 was tested on calculations of singlet methylene{,} dissociation curve of the fluorine molecule{,} and the BeH2 insertion pathway. As expected{,} the results of the newly developed method reconfirm the significantly faster convergence with respect to the basis set limit compared to the traditional expansion in Slater determinants. Results prove that treating the correlation factor separately for each reference is appropriate.}, + Author = {Demel, Ond{\v r}ej and Ked{\v z}uch, Stanislav and {\v S}va{\v n}a, Matej and Ten-no, Seiichiro and Pittner, Ji{\v r}{\'\i} and Noga, Jozef}, + Doi = {10.1039/C2CP23198K}, + Issue = {14}, + Journal = {Phys. Chem. Chem. Phys.}, + Pages = {4753-4762}, + Publisher = {The Royal Society of Chemistry}, + Title = {An explicitly correlated Mukherjee{'}s state specific coupled cluster method: development and pilot applications}, + Url = {http://dx.doi.org/10.1039/C2CP23198K}, + Volume = {14}, + Year = {2012}, + Bdsk-Url-1 = {http://dx.doi.org/10.1039/C2CP23198K}} + +@article{HauMaoMukKlo-CPL-12, + Abstract = {We present a universally applicable explicit electron correlation (F12) correction and apply it to Mukherjee's multi-reference perturbation theory (Mk-MRPT2). Two different F12 corrections are proposed: one is a universal F12 correction which is added to the conventional orbital correction, which is referred to as Mk-MRPT2+F12. In the second type of F12 correction the individual F12 contributions are added to each matrix element of the effective Hamiltonian. Subsequent diagonalization yields the Mk-MRPT2-F12 correction. Thereby, we achieve for both F12 corrections the accuracy of a quadruple-ζ basis set calculation when a triple-ζ basis set is employed and the F12 correction is added. Both F12 corrections reduce to MP2-F12/1A (fixed) in the single-reference limit.}, + Author = {Robin Haunschild and Shuneng Mao and Debashis Mukherjee and Wim Klopper}, + Date-Modified = {2019-12-13 10:32:04 +0100}, + Journal = {Chem. Phys. Lett.}, + Pages = {247 - 251}, + Title = {A universal explicit electron correlation correction applied to Mukherjee's multi-reference perturbation theory}, + Volume = {531}, + Year = {2012}, + url = {http://www.sciencedirect.com/science/article/pii/S0009261412002072}, + doi = {10.1016/j.cplett.2012.02.020}} + +@article{KonVal-JCP-11, + Author = {Kong,Liguo and Valeev,Edward F.}, + Date-Modified = {2019-12-13 10:31:43 +0100}, + Doi = {10.1063/1.3664729}, + Journal = {J. Chem. Phys.}, + Number = {21}, + Pages = {214105}, + Title = {SF-[2]R12: A spin-adapted explicitly correlated method applicable to arbitrary electronic states}, + Volume = {135}, + Year = {2011}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.3664729}} + +@article{BooCleAlaTew-JCP-12, + Author = {Booth,George H. and Cleland,Deidre and Alavi,Ali and Tew,David P.}, + Date-Modified = {2019-12-13 10:32:27 +0100}, + Doi = {10.1063/1.4762445}, + Journal = {J. Chem. Phys.}, + Number = {16}, + Pages = {164112}, + Title = {An explicitly correlated approach to basis set incompleteness in full configuration interaction quantum Monte Carlo}, + Volume = {137}, + Year = {2012}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.4762445}} diff --git a/Revised_Manuscript/srDFT_SC.blg b/Revised_Manuscript/srDFT_SC.blg new file mode 100644 index 0000000..583950a --- /dev/null +++ b/Revised_Manuscript/srDFT_SC.blg @@ -0,0 +1,69 @@ +This is BibTeX, Version 0.99d (TeX Live 2017/Debian) +Capacity: max_strings=100000, hash_size=100000, hash_prime=85009 +The top-level auxiliary file: srDFT_SC.aux +The style file: aipnum4-1.bst +Reallocated singl_function (elt_size=4) to 100 items from 50. +Reallocated singl_function (elt_size=4) to 100 items from 50. +Reallocated singl_function (elt_size=4) to 100 items from 50. +Reallocated singl_function (elt_size=4) to 100 items from 50. +Reallocated singl_function (elt_size=4) to 100 items from 50. +Reallocated singl_function (elt_size=4) to 100 items from 50. +Reallocated wiz_functions (elt_size=4) to 6000 items from 3000. +Database file #1: srDFT_SCNotes.bib +Database file #2: srDFT_SC.bib +Warning--I'm ignoring Sav-INC-96a's extra "publisher" field +--line 9527 of file srDFT_SC.bib +control{REVTEX41Control}, control.key{N/A}, control.author{N/A}, control.editor{N/A}, control.title{N/A}, control.pages{N/A}, control.year{N/A}, control.eprint{N/A}, +control{aip41Control}, control.key{N/A}, control.author{N/A}, control.editor{N/A}, control.title{}, control.pages{0}, control.year{N/A}, control.eprint{N/A}, +Warning--jnrlst (dependency: not reversed) set 1 +Reallocated singl_function (elt_size=4) to 100 items from 50. +merlin.mbs aipnum4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked +Control: key (0) +Control: author (8) initials jnrlst +Control: editor formatted (1) identically to author +Control: production of article title (-1) disabled +Control: page (0) single +Control: year (1) truncated +Control: production of eprint (0) enabled +You've used 94 entries, + 5918 wiz_defined-function locations, + 2264 strings with 32904 characters, +and the built_in function-call counts, 97884 in all, are: += -- 6210 +> -- 3197 +< -- 601 ++ -- 1011 +- -- 817 +* -- 15098 +:= -- 10073 +add.period$ -- 93 +call.type$ -- 94 +change.case$ -- 371 +chr.to.int$ -- 91 +cite$ -- 94 +duplicate$ -- 8692 +empty$ -- 6973 +format.name$ -- 1644 +if$ -- 19453 +int.to.chr$ -- 4 +int.to.str$ -- 101 +missing$ -- 1163 +newline$ -- 328 +num.names$ -- 279 +pop$ -- 3609 +preamble$ -- 1 +purify$ -- 460 +quote$ -- 0 +skip$ -- 3450 +stack$ -- 0 +substring$ -- 2628 +swap$ -- 8533 +text.length$ -- 308 +text.prefix$ -- 0 +top$ -- 10 +type$ -- 1330 +warning$ -- 1 +while$ -- 376 +width$ -- 0 +write$ -- 791 +(There were 2 warnings) diff --git a/Revised_Manuscript/srDFT_SC.log b/Revised_Manuscript/srDFT_SC.log new file mode 100644 index 0000000..b74b2b5 --- /dev/null +++ b/Revised_Manuscript/srDFT_SC.log @@ -0,0 +1,2460 @@ +This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=pdflatex 2019.3.29) 29 JAN 2020 08:29 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**srDFT_SC +(./srDFT_SC.tex +LaTeX2e <2017-04-15> +Babel <3.18> and hyphenation patterns for 32 language(s) loaded. +(/usr/share/texlive/texmf-dist/tex/latex/revtex/revtex4-1.cls +Document Class: revtex4-1 2010/07/25/20:33:00 4.1r (http://publish.aps.org/revt +ex4/ for documentation) + Copyright (c) 2009 The American Physical Society. + mailto:revtex@aps.org + Licensed under the LPPL: +http://www.ctan.org/tex-archive/macros/latex/base/lppl.txt + Arthur Ogawa + Based on work by David Carlisle . +ltxutil[2010/07/25/20:33:00 4.1r utilities package (portions licensed from W. 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+\c__regex_catcode_S_int=\count163 +\c__regex_catcode_L_int=\count164 +\c__regex_catcode_O_int=\count165 +\c__regex_catcode_A_int=\count166 +\c__regex_all_catcodes_int=\count167 +\l__regex_show_lines_int=\count168 +\l__regex_min_state_int=\count169 +\l__regex_max_state_int=\count170 +\l__regex_left_state_int=\count171 +\l__regex_right_state_int=\count172 +\l__regex_capturing_group_int=\count173 +\l__regex_min_pos_int=\count174 +\l__regex_max_pos_int=\count175 +\l__regex_curr_pos_int=\count176 +\l__regex_start_pos_int=\count177 +\l__regex_success_pos_int=\count178 +\l__regex_curr_char_int=\count179 +\l__regex_curr_catcode_int=\count180 +\l__regex_last_char_int=\count181 +\l__regex_case_changed_char_int=\count182 +\l__regex_curr_state_int=\count183 +\l__regex_step_int=\count184 +\l__regex_min_active_int=\count185 +\l__regex_max_active_int=\count186 +\l__regex_replacement_csnames_int=\count187 +\l__regex_match_count_int=\count188 +\l__regex_min_submatch_int=\count189 +\l__regex_submatch_int=\count190 +\l__regex_zeroth_submatch_int=\count191 +\g__regex_trace_regex_int=\count192 +\c_empty_box=\box43 +\l_tmpa_box=\box44 +\l_tmpb_box=\box45 +\g_tmpa_box=\box46 +\g_tmpb_box=\box47 +\l__box_top_dim=\dimen123 +\l__box_bottom_dim=\dimen124 +\l__box_left_dim=\dimen125 +\l__box_right_dim=\dimen126 +\l__box_top_new_dim=\dimen127 +\l__box_bottom_new_dim=\dimen128 +\l__box_left_new_dim=\dimen129 +\l__box_right_new_dim=\dimen130 +\l__box_internal_box=\box48 +\l__coffin_internal_box=\box49 +\l__coffin_internal_dim=\dimen131 +\l__coffin_offset_x_dim=\dimen132 +\l__coffin_offset_y_dim=\dimen133 +\l__coffin_x_dim=\dimen134 +\l__coffin_y_dim=\dimen135 +\l__coffin_x_prime_dim=\dimen136 +\l__coffin_y_prime_dim=\dimen137 +\c_empty_coffin=\box50 +\l__coffin_aligned_coffin=\box51 +\l__coffin_aligned_internal_coffin=\box52 +\l_tmpa_coffin=\box53 +\l_tmpb_coffin=\box54 +\l__coffin_display_coffin=\box55 +\l__coffin_display_coord_coffin=\box56 +\l__coffin_display_pole_coffin=\box57 +\l__coffin_display_offset_dim=\dimen138 +\l__coffin_display_x_dim=\dimen139 +\l__coffin_display_y_dim=\dimen140 +\l__coffin_bounding_shift_dim=\dimen141 +\l__coffin_left_corner_dim=\dimen142 +\l__coffin_right_corner_dim=\dimen143 +\l__coffin_bottom_corner_dim=\dimen144 +\l__coffin_top_corner_dim=\dimen145 +\l__coffin_scaled_total_height_dim=\dimen146 +\l__coffin_scaled_width_dim=\dimen147 +) +(/usr/share/texlive/texmf-dist/tex/latex/l3kernel/l3pdfmode.def +File: l3pdfmode.def 2017/03/18 v L3 Experimental driver: PDF mode +\l__driver_color_stack_int=\count193 +\l__driver_tmp_box=\box58 +)) +Package: xparse 2018/02/21 L3 Experimental document command parser +\l__xparse_current_arg_int=\count194 +\g__xparse_grabber_int=\count195 +\l__xparse_m_args_int=\count196 +\l__xparse_mandatory_args_int=\count197 +\l__xparse_v_nesting_int=\count198 +) +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \vnabla with sig. '' on line 25. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \varE with sig. '' on line 29. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \ordersymbol with sig. '' on line 30. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \lparen with sig. '' on line 31. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \rparen with sig. '' on line 32. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \quantity with sig. '' on line 35. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \@quantity with sig. ' t\big t\Big t\bigg t\Bigg g o d() +. d|| ' on line 62. +................................................. + +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\big ' for the command '\@quantity' should be a single +* character. +************************************************* +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\Big ' for the command '\@quantity' should be a single +* character. +************************************************* +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\bigg ' for the command '\@quantity' should be a single +* character. +************************************************* +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\Bigg ' for the command '\@quantity' should be a single +* character. +************************************************* +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qty with sig. '' on line 63. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \pqty with sig. ' l m ' on line 64. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \bqty with sig. ' l m ' on line 65. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \Bqty with sig. ' l m ' on line 66. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \vqty with sig. ' l m ' on line 67. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \pmqty with sig. 'm' on line 70. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \Pmqty with sig. 'm' on line 71. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \bmqty with sig. 'm' on line 72. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \vmqty with sig. 'm' on line 73. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \matrixquantity with sig. '' on line 74. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \@matrixquantity with sig. ' s g o d() d|| ' on line 100. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \mqty with sig. '' on line 101. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \matrixdeterminant with sig. 'm' on line 102. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \mdet with sig. '' on line 103. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \spmqty with sig. 'm' on line 105. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \sPmqty with sig. 'm' on line 106. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \sbmqty with sig. 'm' on line 107. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \svmqty with sig. 'm' on line 108. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \smallmatrixquantity with sig. ' s g o d() d|| ' on line +. 133. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \smqty with sig. '' on line 134. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \smallmatrixdeterminant with sig. 'm' on line 135. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \smdet with sig. '' on line 136. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \argopen with sig. 's' on line 138. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \argclose with sig. 's' on line 139. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \braces with sig. '' on line 141. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \@braces with sig. ' s t\big t\Big t\bigg t\Bigg m m m ' on +. line 156. +................................................. +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\big ' for the command '\@braces' should be a single +* character. +************************************************* +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\Big ' for the command '\@braces' should be a single +* character. +************************************************* +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\bigg ' for the command '\@braces' should be a single +* character. +************************************************* +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\Bigg ' for the command '\@braces' should be a single +* character. +************************************************* +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \fbraces with sig. ' s t\big t\Big t\bigg t\Bigg m m m m ' +. on line 174. +................................................. +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\big ' for the command '\fbraces' should be a single +* character. +************************************************* +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\Big ' for the command '\fbraces' should be a single +* character. +************************************************* +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\bigg ' for the command '\fbraces' should be a single +* character. +************************************************* +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\Bigg ' for the command '\fbraces' should be a single +* character. +************************************************* +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \absolutevalue with sig. ' l m ' on line 176. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \abs with sig. '' on line 177. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \norm with sig. ' l m ' on line 178. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \order with sig. ' l m ' on line 179. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \evaluated with sig. ' s g d[| d(| ' on line 194. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \eval with sig. '' on line 195. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \poissonbracket with sig. ' l m m ' on line 197. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \pb with sig. '' on line 198. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \commutator with sig. ' l m m ' on line 201. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \comm with sig. '' on line 202. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \anticommutator with sig. ' l m m ' on line 203. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \acommutator with sig. '' on line 204. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \acomm with sig. '' on line 205. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \vectorbold with sig. ' s m ' on line 208. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \vb with sig. '' on line 209. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \vectorarrow with sig. ' s m ' on line 211. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \va with sig. '' on line 212. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \vectorunit with sig. ' s m ' on line 214. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \vu with sig. '' on line 215. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \dotproduct with sig. '' on line 217. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \vdot with sig. '' on line 218. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \crossproduct with sig. '' on line 220. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \cross with sig. '' on line 221. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \cp with sig. '' on line 222. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \gradient with sig. ' g o d() ' on line 232. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \grad with sig. '' on line 233. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \divergence with sig. ' g o d() ' on line 243. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \div with sig. '' on line 246. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \curl with sig. ' g o d() ' on line 256. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \laplacian with sig. ' g o d() ' on line 266. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \principalvalue with sig. 'g' on line 274. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \pv with sig. '' on line 275. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \PV with sig. 'g' on line 276. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \Re with sig. 'g' on line 277. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \Im with sig. 'g' on line 278. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \opbraces with sig. ' m g o d() ' on line 299. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \trigbraces with sig. ' m o d() ' on line 305. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \sin with sig. '' on line 309. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \cos with sig. '' on line 310. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \tan with sig. '' on line 311. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \csc with sig. '' on line 312. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \sec with sig. '' on line 313. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \cot with sig. '' on line 314. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \arcsin with sig. '' on line 316. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \arccos with sig. '' on line 317. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \arctan with sig. '' on line 318. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \arccsc with sig. '' on line 320. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \arcsec with sig. '' on line 322. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \arccot with sig. '' on line 324. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \asin with sig. '' on line 327. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \acos with sig. '' on line 329. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \atan with sig. '' on line 331. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \acsc with sig. '' on line 333. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \asec with sig. '' on line 335. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \acot with sig. '' on line 337. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \sinh with sig. '' on line 339. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \cosh with sig. '' on line 340. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \tanh with sig. '' on line 341. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \csch with sig. '' on line 343. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \sech with sig. '' on line 345. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \coth with sig. '' on line 346. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \exp with sig. '' on line 348. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \log with sig. '' on line 349. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \ln with sig. '' on line 350. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \det with sig. '' on line 351. +................................................. +................................................. +. LaTeX info: "xparse/redefine-command" +. +. Redefining command \Pr with sig. '' on line 352. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \tr with sig. '' on line 353. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \Tr with sig. '' on line 354. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \Res with sig. '' on line 355. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qqtext with sig. ' s m ' on line 377. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qq with sig. '' on line 378. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qcomma with sig. '' on line 380. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qc with sig. '' on line 381. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qif with sig. 's' on line 383. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qthen with sig. 's' on line 384. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qelse with sig. 's' on line 385. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qotherwise with sig. 's' on line 386. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qunless with sig. 's' on line 387. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qgiven with sig. 's' on line 388. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qusing with sig. 's' on line 389. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qassume with sig. 's' on line 390. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qsince with sig. 's' on line 391. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qlet with sig. 's' on line 392. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qfor with sig. 's' on line 393. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qall with sig. 's' on line 394. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qeven with sig. 's' on line 395. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qodd with sig. 's' on line 396. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qinteger with sig. 's' on line 397. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qand with sig. 's' on line 398. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qor with sig. 's' on line 399. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qas with sig. 's' on line 400. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qin with sig. 's' on line 401. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \qcc with sig. 's' on line 402. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \differential with sig. ' o g d() ' on line 415. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \dd with sig. '' on line 416. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \derivative with sig. ' s o m g d() ' on line 435. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \dv with sig. '' on line 436. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \partialderivative with sig. ' s o m g g d() ' on line 460. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \pderivative with sig. '' on line 461. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \pdv with sig. '' on line 462. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \variation with sig. ' o g d() ' on line 474. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \var with sig. '' on line 475. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \functionalderivative with sig. ' s o m g d() ' on line +. 494. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \fderivative with sig. '' on line 495. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \fdv with sig. '' on line 496. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \bra with sig. ' s m t\ket s g ' on line 526. +................................................. +************************************************* +* LaTeX warning: "xparse/not-single-char" +* +* Argument delimiter '\ket ' for the command '\bra' should be a single +* character. +************************************************* +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \ket with sig. ' s m ' on line 533. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \innerproduct with sig. ' s m g ' on line 548. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \braket with sig. '' on line 549. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \ip with sig. '' on line 550. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \outerproduct with sig. ' s m g ' on line 565. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \dyad with sig. '' on line 566. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \op with sig. '' on line 567. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \ketbra with sig. '' on line 568. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \expectationvalue with sig. ' s s m g ' on line 587. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \expval with sig. '' on line 588. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \ev with sig. '' on line 589. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \vev with sig. ' m ' on line 590. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \matrixelement with sig. ' s s m m m ' on line 601. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \matrixel with sig. '' on line 602. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \mel with sig. '' on line 603. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \identitymatrix with sig. 'm' on line 633. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \imat with sig. '' on line 634. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \xmatrix with sig. ' s m m m ' on line 659. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \xmat with sig. '' on line 660. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \zeromatrix with sig. ' m g ' on line 662. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \zmat with sig. '' on line 663. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \paulixmatrix with sig. '' on line 665. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \pauliymatrix with sig. '' on line 666. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \paulizmatrix with sig. '' on line 667. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \paulimatrix with sig. 'm' on line 678. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \pmat with sig. '' on line 679. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \diagonalmatrix with sig. 'O{} >{\SplitList {,}}m ' on line +. 681. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \@dmat with sig. 'mmggggggg' on line 714. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \dmat with sig. '' on line 715. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \antidiagonalmatrix with sig. 'O{} >{\SplitList {,}}m ' on +. line 717. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \@admat with sig. 'mmggggggg' on line 750. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \admat with sig. '' on line 751. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \flatfrac with sig. ' m m ' on line 754. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \homework with sig. '' on line 764. +................................................. +) (/usr/share/texlive/texmf-dist/tex/latex/mhchem/mhchem.sty +Package: mhchem 2017/07/24 v4.07 for typesetting chemical formulae + +(/usr/share/texlive/texmf-dist/tex/latex/l3packages/l3keys2e/l3keys2e.sty +Package: l3keys2e 2018/02/21 LaTeX2e option processing using LaTeX3 keys +) +(/usr/share/texlive/texmf-dist/tex/latex/tools/calc.sty +Package: calc 2014/10/28 v4.3 Infix arithmetic (KKT,FJ) +\calc@Acount=\count199 +\calc@Bcount=\count266 +\calc@Adimen=\dimen148 +\calc@Bdimen=\dimen149 +\calc@Askip=\skip60 +\calc@Bskip=\skip61 +LaTeX Info: Redefining \setlength on input line 80. +LaTeX Info: Redefining \addtolength on input line 81. +\calc@Ccount=\count267 +\calc@Cskip=\skip62 +) +(/usr/share/texlive/texmf-dist/tex/latex/chemgreek/chemgreek.sty +Package: chemgreek 2016/12/20 v1.1 interfaceforuprightgreeklettersforuseinchemi +stry (CN) +\l__chemgreek_tmpa_int=\count268 +\g__chemgreek_tmpa_int=\count269 +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \newchemgreekmapping with sig. 'O{}mm' on line 336. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \renewchemgreekmapping with sig. 'O{}mm' on line 339. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \declarechemgreekmapping with sig. 'O{}mm' on line 342. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \newchemgreekmappingalias with sig. 'mm' on line 347. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \renewchemgreekmappingalias with sig. 'mm' on line 350. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \declarechemgreekmappingalias with sig. 'mm' on line 353. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \changechemgreeksymbol with sig. 'mmmm' on line 383. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \chemgreekmappingsymbol with sig. 'mm' on line 477. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \activatechemgreekmapping with sig. 'sm' on line 486. +................................................. +................................................. +. LaTeX info: "xparse/define-command" +. +. Defining command \selectchemgreekmapping with sig. 'm' on line 491. +................................................. +) +\l__mhchem_ce_distanceFromLastComma_int=\count270 +\__mhchem_arrow_extPgfLength_skip=\skip63 +\mhchem@minispace@tmp=\skip64 +\mhchem@minibackspace@tmp=\skip65 +\mhchem@bondwidth=\skip66 +\mhchem@bondheight=\skip67 +\mhchem@smallbondwidth@tmpA=\skip68 +\mhchem@smallbondwidth@tmpB=\skip69 +\mhchem@smallbondwidth=\skip70 +\mhchem@bondtmp@minussidebearingleft=\skip71 +\mhchem@bondtmp@minussidebearingright=\skip72 +\l__mhchem_option_version_int=\count271 +\mhchem@option@minusmathsidebearingleft=\dimen150 +\mhchem@option@minusmathsidebearingright=\dimen151 +\mhchem@option@minustextsidebearingleft=\dimen152 +\mhchem@option@minustextsidebearingright=\dimen153 + + +Package mhchem Warning: You did not specify a 'version' option for the mhchem +(mhchem) package. Please write \usepackage[version=4]{mhchem} +(mhchem) in your preamble (or any lower number for +(mhchem) compatibility mode), because you might get slightly +(mhchem) different output with the same input in future versions +. + +) (/usr/share/texlive/texmf-dist/tex/latex/tools/xspace.sty +Package: xspace 2014/10/28 v1.13 Space after command names (DPC,MH) +) +(/usr/share/texlive/texmf-dist/tex/latex/subfigure/subfigure.sty +Package: subfigure 2002/03/15 v2.1.5 subfigure package +\subfigtopskip=\skip73 +\subfigcapskip=\skip74 +\subfigcaptopadj=\dimen154 +\subfigbottomskip=\skip75 +\subfigcapmargin=\dimen155 +\subfiglabelskip=\skip76 +\c@subfigure=\count272 +\c@lofdepth=\count273 +\c@subtable=\count274 +\c@lotdepth=\count275 + +**************************************** +* Local config file subfigure.cfg used * +**************************************** +(/usr/share/texlive/texmf-dist/tex/latex/subfigure/subfigure.cfg) +\subfig@top=\skip77 +\subfig@bottom=\skip78 +) +(/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty +Package: inputenc 2015/03/17 v1.2c Input encoding file +\inpenc@prehook=\toks24 +\inpenc@posthook=\toks25 + +(/usr/share/texlive/texmf-dist/tex/latex/base/utf8.def +File: utf8.def 2017/01/28 v1.1t UTF-8 support for inputenc +Now handling font encoding OML ... +... no UTF-8 mapping file for font encoding OML +Now handling font encoding T1 ... +... processing UTF-8 mapping file for font encoding T1 + +(/usr/share/texlive/texmf-dist/tex/latex/base/t1enc.dfu +File: t1enc.dfu 2017/01/28 v1.1t UTF-8 support for inputenc + defining Unicode char U+00A0 (decimal 160) + defining Unicode char U+00A1 (decimal 161) + defining Unicode char U+00A3 (decimal 163) + defining Unicode char U+00AB (decimal 171) + defining Unicode char U+00AD (decimal 173) + defining Unicode char U+00BB (decimal 187) + defining Unicode char U+00BF (decimal 191) + defining Unicode char U+00C0 (decimal 192) + defining Unicode char U+00C1 (decimal 193) + defining Unicode char U+00C2 (decimal 194) + defining Unicode char U+00C3 (decimal 195) + defining Unicode char U+00C4 (decimal 196) + defining Unicode char U+00C5 (decimal 197) + defining Unicode char U+00C6 (decimal 198) + defining Unicode char U+00C7 (decimal 199) + defining Unicode char U+00C8 (decimal 200) + defining Unicode char U+00C9 (decimal 201) + defining Unicode char U+00CA (decimal 202) + defining Unicode char U+00CB (decimal 203) + defining Unicode char U+00CC (decimal 204) + defining Unicode char U+00CD (decimal 205) + defining Unicode char U+00CE (decimal 206) + defining Unicode char U+00CF (decimal 207) + defining Unicode char U+00D0 (decimal 208) + defining Unicode char U+00D1 (decimal 209) + defining Unicode char U+00D2 (decimal 210) + defining Unicode char U+00D3 (decimal 211) + defining Unicode char U+00D4 (decimal 212) + defining Unicode char U+00D5 (decimal 213) + defining Unicode char U+00D6 (decimal 214) + defining Unicode char U+00D8 (decimal 216) + defining Unicode char U+00D9 (decimal 217) + defining Unicode char U+00DA (decimal 218) + defining Unicode char U+00DB (decimal 219) + defining Unicode char U+00DC (decimal 220) + defining Unicode char U+00DD (decimal 221) + defining Unicode char U+00DE (decimal 222) + defining Unicode char U+00DF (decimal 223) + defining Unicode char U+00E0 (decimal 224) + defining Unicode char U+00E1 (decimal 225) + defining Unicode char U+00E2 (decimal 226) + defining Unicode char U+00E3 (decimal 227) + defining Unicode char U+00E4 (decimal 228) + defining Unicode char U+00E5 (decimal 229) + defining Unicode char U+00E6 (decimal 230) + defining Unicode char U+00E7 (decimal 231) + defining Unicode char U+00E8 (decimal 232) + defining Unicode char U+00E9 (decimal 233) + defining Unicode char U+00EA (decimal 234) + defining Unicode char U+00EB (decimal 235) + defining Unicode char U+00EC (decimal 236) + defining Unicode char U+00ED (decimal 237) + defining Unicode char U+00EE (decimal 238) + defining Unicode char U+00EF (decimal 239) + defining Unicode char U+00F0 (decimal 240) + defining Unicode char U+00F1 (decimal 241) + defining Unicode char U+00F2 (decimal 242) + defining Unicode char U+00F3 (decimal 243) + defining Unicode char U+00F4 (decimal 244) + defining Unicode char U+00F5 (decimal 245) + defining Unicode char U+00F6 (decimal 246) + defining Unicode char U+00F8 (decimal 248) + defining Unicode char U+00F9 (decimal 249) + defining Unicode char U+00FA (decimal 250) + defining Unicode char U+00FB (decimal 251) + defining Unicode char U+00FC (decimal 252) + defining Unicode char U+00FD (decimal 253) + defining Unicode char U+00FE (decimal 254) + defining Unicode char U+00FF (decimal 255) + defining Unicode char U+0100 (decimal 256) + defining Unicode char U+0101 (decimal 257) + defining Unicode char U+0102 (decimal 258) + defining Unicode char U+0103 (decimal 259) + defining Unicode char U+0104 (decimal 260) + defining Unicode char U+0105 (decimal 261) + defining Unicode char U+0106 (decimal 262) + defining Unicode char U+0107 (decimal 263) + defining Unicode char U+0108 (decimal 264) + defining Unicode char U+0109 (decimal 265) + defining Unicode char U+010A (decimal 266) + defining Unicode char U+010B (decimal 267) + defining Unicode char U+010C (decimal 268) + defining Unicode char U+010D (decimal 269) + defining Unicode char U+010E (decimal 270) + defining Unicode char U+010F (decimal 271) + defining Unicode char U+0110 (decimal 272) + defining Unicode char U+0111 (decimal 273) + defining Unicode char U+0112 (decimal 274) + defining Unicode char U+0113 (decimal 275) + defining Unicode char U+0114 (decimal 276) + defining Unicode char U+0115 (decimal 277) + defining Unicode char U+0116 (decimal 278) + defining Unicode char U+0117 (decimal 279) + defining Unicode char U+0118 (decimal 280) + defining Unicode char U+0119 (decimal 281) + defining Unicode char U+011A (decimal 282) + defining Unicode char U+011B (decimal 283) + defining Unicode char U+011C (decimal 284) + defining Unicode char U+011D (decimal 285) + defining Unicode char U+011E (decimal 286) + defining Unicode char U+011F (decimal 287) + defining Unicode char U+0120 (decimal 288) + defining Unicode char U+0121 (decimal 289) + defining Unicode char U+0122 (decimal 290) + defining Unicode char U+0123 (decimal 291) + defining Unicode char U+0124 (decimal 292) + defining Unicode char U+0125 (decimal 293) + defining Unicode char U+0128 (decimal 296) + defining Unicode char U+0129 (decimal 297) + defining Unicode char U+012A (decimal 298) + defining Unicode char U+012B (decimal 299) + defining Unicode char U+012C (decimal 300) + defining Unicode char U+012D (decimal 301) + defining Unicode char U+012E (decimal 302) + defining Unicode char U+012F (decimal 303) + defining Unicode char U+0130 (decimal 304) + defining Unicode char U+0131 (decimal 305) + defining Unicode char U+0132 (decimal 306) + defining Unicode char U+0133 (decimal 307) + defining Unicode char U+0134 (decimal 308) + defining Unicode char U+0135 (decimal 309) + defining Unicode char U+0136 (decimal 310) + defining Unicode char U+0137 (decimal 311) + defining Unicode char U+0139 (decimal 313) + defining Unicode char U+013A (decimal 314) + defining Unicode char U+013B (decimal 315) + defining Unicode char U+013C (decimal 316) + defining Unicode char U+013D (decimal 317) + defining Unicode char U+013E (decimal 318) + defining Unicode char U+0141 (decimal 321) + defining Unicode char U+0142 (decimal 322) + defining Unicode char U+0143 (decimal 323) + defining Unicode char U+0144 (decimal 324) + defining Unicode char U+0145 (decimal 325) + defining Unicode char U+0146 (decimal 326) + defining Unicode char U+0147 (decimal 327) + defining Unicode char U+0148 (decimal 328) + defining Unicode char U+014A (decimal 330) + defining Unicode char U+014B (decimal 331) + defining Unicode char U+014C (decimal 332) + defining Unicode char U+014D (decimal 333) + defining Unicode char U+014E (decimal 334) + defining Unicode char U+014F (decimal 335) + defining Unicode char U+0150 (decimal 336) + defining Unicode char U+0151 (decimal 337) + defining Unicode char U+0152 (decimal 338) + defining Unicode char U+0153 (decimal 339) + defining Unicode char U+0154 (decimal 340) + defining Unicode char U+0155 (decimal 341) + defining Unicode char U+0156 (decimal 342) + defining Unicode char U+0157 (decimal 343) + defining Unicode char U+0158 (decimal 344) + defining Unicode char U+0159 (decimal 345) + defining Unicode char U+015A (decimal 346) + defining Unicode char U+015B (decimal 347) + defining Unicode char U+015C (decimal 348) + defining Unicode char U+015D (decimal 349) + defining Unicode char U+015E (decimal 350) + defining Unicode char U+015F (decimal 351) + defining Unicode char U+0160 (decimal 352) + defining Unicode char U+0161 (decimal 353) + defining Unicode char U+0162 (decimal 354) + defining Unicode char U+0163 (decimal 355) + defining Unicode char U+0164 (decimal 356) + defining Unicode char U+0165 (decimal 357) + defining Unicode char U+0168 (decimal 360) + defining Unicode char U+0169 (decimal 361) + defining Unicode char U+016A (decimal 362) + defining Unicode char U+016B (decimal 363) + defining Unicode char U+016C (decimal 364) + defining Unicode char U+016D (decimal 365) + defining Unicode char U+016E (decimal 366) + defining Unicode char U+016F (decimal 367) + defining Unicode char U+0170 (decimal 368) + defining Unicode char U+0171 (decimal 369) + defining Unicode char U+0172 (decimal 370) + defining Unicode char U+0173 (decimal 371) + defining Unicode char U+0174 (decimal 372) + defining Unicode char U+0175 (decimal 373) + defining Unicode char U+0176 (decimal 374) + defining Unicode char U+0177 (decimal 375) + defining Unicode char U+0178 (decimal 376) + defining Unicode char U+0179 (decimal 377) + defining Unicode char U+017A (decimal 378) + defining Unicode char U+017B (decimal 379) + defining Unicode char U+017C (decimal 380) + defining Unicode char U+017D (decimal 381) + defining Unicode char U+017E (decimal 382) + defining Unicode char U+01CD (decimal 461) + defining Unicode char U+01CE (decimal 462) + defining Unicode char U+01CF (decimal 463) + defining Unicode char U+01D0 (decimal 464) + defining Unicode char U+01D1 (decimal 465) + defining Unicode char U+01D2 (decimal 466) + defining Unicode char U+01D3 (decimal 467) + defining Unicode char U+01D4 (decimal 468) + defining Unicode char U+01E2 (decimal 482) + defining Unicode char U+01E3 (decimal 483) + defining Unicode char U+01E6 (decimal 486) + defining Unicode char U+01E7 (decimal 487) + defining Unicode char U+01E8 (decimal 488) + defining Unicode char U+01E9 (decimal 489) + defining Unicode char U+01EA (decimal 490) + defining Unicode char U+01EB (decimal 491) + defining Unicode char U+01F0 (decimal 496) + defining Unicode char U+01F4 (decimal 500) + defining Unicode char U+01F5 (decimal 501) + defining Unicode char U+0218 (decimal 536) + defining Unicode char U+0219 (decimal 537) + defining Unicode char U+021A (decimal 538) + defining Unicode char U+021B (decimal 539) + defining Unicode char U+0232 (decimal 562) + defining Unicode char U+0233 (decimal 563) + defining Unicode char U+1E02 (decimal 7682) + defining Unicode char U+1E03 (decimal 7683) + defining Unicode char U+200C (decimal 8204) + defining Unicode char U+2010 (decimal 8208) + defining Unicode char U+2011 (decimal 8209) + defining Unicode char U+2012 (decimal 8210) + defining Unicode char U+2013 (decimal 8211) + defining Unicode char U+2014 (decimal 8212) + defining Unicode char U+2015 (decimal 8213) + defining Unicode char U+2018 (decimal 8216) + defining Unicode char U+2019 (decimal 8217) + defining Unicode char U+201A (decimal 8218) + defining Unicode char U+201C (decimal 8220) + defining Unicode char U+201D (decimal 8221) + defining Unicode char U+201E (decimal 8222) + defining Unicode char U+2030 (decimal 8240) + defining Unicode char U+2031 (decimal 8241) + defining Unicode char 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+ +[9] [10 <./data/N2/DFT_avdzE_relat.pdf> <./data/N2/DFT_avdzE_relat_zoom.pdf + +pdfTeX warning: pdflatex (file ./data/N2/DFT_avdzE_relat_zoom.pdf): PDF inclusi +on: multiple pdfs with page group included in a single page +> <./data/N2/DFT_avtzE_relat.pdf + +pdfTeX warning: pdflatex (file ./data/N2/DFT_avtzE_relat.pdf): PDF inclusion: m +ultiple pdfs with page group included in a single page +> <./data/N2/DFT_avtzE_relat_zoom.pdf + +pdfTeX warning: pdflatex (file ./data/N2/DFT_avtzE_relat_zoom.pdf): PDF inclusi +on: multiple pdfs with page group included in a single page +>] [11 <./data/O2/DFT_avdzE_relat.pdf> <./data/O2/DFT_avdzE_relat_zoom.pdf + +pdfTeX warning: pdflatex (file ./data/O2/DFT_avdzE_relat_zoom.pdf): PDF inclusi +on: multiple pdfs with page group included in a single page +> <./data/O2/DFT_avtzE_relat.pdf + +pdfTeX warning: pdflatex (file ./data/O2/DFT_avtzE_relat.pdf): PDF inclusion: m +ultiple pdfs with page group included in a single page +> <./data/O2/DFT_avtzE_relat_zoom.pdf + +pdfTeX warning: pdflatex (file ./data/O2/DFT_avtzE_relat_zoom.pdf): PDF inclusi +on: multiple pdfs with page group included in a single page +>] [12 <./data/F2/DFT_avdzE_relat.pdf> <./data/F2/DFT_avdzE_relat_zoom.pdf + +pdfTeX warning: pdflatex (file ./data/F2/DFT_avdzE_relat_zoom.pdf): PDF inclusi +on: multiple pdfs with page group included in a single page +> <./data/F2/DFT_avtzE_relat.pdf + +pdfTeX warning: pdflatex (file ./data/F2/DFT_avtzE_relat.pdf): PDF inclusion: m +ultiple pdfs with page group included in a single page +> <./data/F2/DFT_avtzE_relat_zoom.pdf + +pdfTeX warning: pdflatex (file ./data/F2/DFT_avtzE_relat_zoom.pdf): PDF inclusi +on: multiple pdfs with page group included in a single page +>] [13] (./srDFT_SC.bbl +Underfull \hbox (badness 1270) in paragraph at lines 245--250 + []\T1/txr/m/n/8 (+20) G. M. Barca and P.-F. Loos, [][]J. Chem. Phys. \T1/txr/b +x/n/8 147\T1/txr/m/n/8 (+20) , 024103 (2017)[][], + [] + +[14]) [15] +Package atveryend Info: Empty hook `AfterLastShipout' on input line 947. + (./srDFT_SC.aux) +Package atveryend Info: Executing hook `AtVeryEndDocument' on input line 947. +Package atveryend Info: Executing hook `AtEndAfterFileList' on input line 947. +Package rerunfilecheck Info: File `srDFT_SC.out' has not changed. +(rerunfilecheck) Checksum: 4B6311C7B61AB256A1DDF014715C94D2;2220. + ) +Here is how much of TeX's memory you used: + 22569 strings out of 494310 + 382433 string characters out of 6167045 + 540167 words of memory out of 5000000 + 25019 multiletter control sequences out of 15000+600000 + 643732 words of font info for 539 fonts, out of 8000000 for 9000 + 309 hyphenation exceptions out of 8191 + 39i,16n,56p,10380b,708s stack positions out of 5000i,500n,10000p,200000b,80000s +{/usr/share/texlive/texmf-dist/fonts/enc/dvips/base/8r.enc} +Output written on srDFT_SC.pdf (15 pages, 415131 bytes). +PDF statistics: + 907 PDF objects out of 1000 (max. 8388607) + 823 compressed objects within 9 object streams + 217 named destinations out of 1000 (max. 500000) + 76595 words of extra memory for PDF output out of 89155 (max. 10000000) + diff --git a/Revised_Manuscript/srDFT_SC.out b/Revised_Manuscript/srDFT_SC.out new file mode 100644 index 0000000..b3d317c --- /dev/null +++ b/Revised_Manuscript/srDFT_SC.out @@ -0,0 +1,27 @@ +\BOOKMARK [0][-]{section*.2}{A basis-set error correction based on density-functional theory for strongly correlated molecular systems}{}% 2 +\BOOKMARK [1][-]{section*.1}{Abstract}{section*.2}% 1 +\BOOKMARK [1][-]{section*.3}{Introduction}{section*.2}% 3 +\BOOKMARK [1][-]{section*.4}{Theory}{section*.2}% 4 +\BOOKMARK [2][-]{section*.5}{Basic theory}{section*.4}% 5 +\BOOKMARK [2][-]{section*.6}{Effective interaction in a finite basis}{section*.4}% 6 +\BOOKMARK [2][-]{section*.7}{Local range-separation function}{section*.4}% 7 +\BOOKMARK [3][-]{section*.8}{General definition}{section*.7}% 8 +\BOOKMARK [3][-]{section*.9}{Frozen-core approximation}{section*.7}% 9 +\BOOKMARK [2][-]{section*.10}{General form of the complementary functional}{section*.4}% 10 +\BOOKMARK [3][-]{section*.11}{Generic approximate form}{section*.10}% 11 +\BOOKMARK [3][-]{section*.12}{Two limits where the complementary functional vanishes}{section*.10}% 12 +\BOOKMARK [2][-]{section*.13}{Requirements on the complementary functional for strong correlation}{section*.4}% 13 +\BOOKMARK [3][-]{section*.14}{Spin-multiplet degeneracy}{section*.13}% 14 +\BOOKMARK [3][-]{section*.15}{Size consistency}{section*.13}% 15 +\BOOKMARK [2][-]{section*.16}{Actual approximations used for the complementary functional}{section*.4}% 16 +\BOOKMARK [1][-]{section*.17}{Results}{section*.2}% 17 +\BOOKMARK [2][-]{section*.18}{Computational details}{section*.17}% 18 +\BOOKMARK [2][-]{section*.19}{H10 chain}{section*.17}% 19 +\BOOKMARK [2][-]{section*.20}{Dissociation of diatomics}{section*.17}% 20 +\BOOKMARK [1][-]{section*.21}{Conclusion}{section*.2}% 21 +\BOOKMARK [1][-]{section*.22}{Size consistency of the basis-set correction}{section*.2}% 22 +\BOOKMARK [2][-]{section*.23}{Sufficient condition for size consistency}{section*.22}% 23 +\BOOKMARK [2][-]{section*.24}{Intensivity of the on-top pair density and the local range-separation function}{section*.22}% 24 +\BOOKMARK [1][-]{section*.25}{Computational cost of the basis-set correction for a CASSCF wave function}{section*.2}% 25 +\BOOKMARK [2][-]{section*.26}{Computation of the on-top pair density}{section*.25}% 26 +\BOOKMARK [2][-]{section*.27}{Computation of the local range-separation function}{section*.25}% 27 diff --git a/Revised_Manuscript/srDFT_SC.tex b/Revised_Manuscript/srDFT_SC.tex new file mode 100644 index 0000000..9c7ba4c --- /dev/null +++ b/Revised_Manuscript/srDFT_SC.tex @@ -0,0 +1,947 @@ +\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1} +\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amsmath,amssymb,amsfonts,physics,mhchem,xspace,subfigure} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{txfonts} + +\usepackage[ + colorlinks=true, + citecolor=blue, + breaklinks=true + ]{hyperref} +\urlstyle{same} + +\newcommand{\alert}[1]{\textcolor{red}{#1}} +\definecolor{darkgreen}{HTML}{009900} +\usepackage[normalem]{ulem} +\newcommand{\titou}[1]{\textcolor{red}{#1}} +\newcommand{\jt}[1]{\textcolor{purple}{#1}} +\newcommand{\manu}[1]{\textcolor{darkgreen}{#1}} +\newcommand{\toto}[1]{\textcolor{brown}{#1}} +\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}} +\newcommand{\trashJT}[1]{\textcolor{purple}{\sout{#1}}} +\newcommand{\trashMG}[1]{\textcolor{darkgreen}{\sout{#1}}} +\newcommand{\trashAS}[1]{\textcolor{brown}{\sout{#1}}} +\newcommand{\MG}[1]{\manu{(\underline{\bf MG}: #1)}} +\newcommand{\JT}[1]{\juju{(\underline{\bf JT}: #1)}} +\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}} +\newcommand{\AS}[1]{\toto{(\underline{\bf AS}: #1)}} + +\newcommand{\ie}{\textit{i.e.}} +\newcommand{\eg}{\textit{e.g.}} +\newcommand{\cdash}{\multicolumn{1}{c}{---}} +\newcommand{\mc}{\multicolumn} +\newcommand{\fnm}{\footnotemark} +\newcommand{\fnt}{\footnotetext} +\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}} +\newcommand{\mr}{\multirow} +\newcommand{\SI}{\textcolor{blue}{supporting information}} + +% second quantized operators +\newcommand{\psix}[1]{\hat{\Psi}\left({\bf X}_{#1}\right)} +\newcommand{\psixc}[1]{\hat{\Psi}^{\dagger}\left({\bf X}_{#1}\right)} +\newcommand{\ai}[1]{\hat{a}_{#1}} +\newcommand{\aic}[1]{\hat{a}^{\dagger}_{#1}} +\newcommand{\vijkl}[0]{V_{ij}^{kl}} +\newcommand{\phix}[2]{\phi_{#1}(\bfr{#2})} +\newcommand{\phixprim}[2]{\phi_{#1}(\bfr{#2}')} + +\newcommand{\CBS}{\text{CBS}} + + +% energies +\newcommand{\Ec}{E_\text{c}} +\newcommand{\EPT}{E_\text{PT2}} +\newcommand{\EsCI}{E_\text{sCI}} +\newcommand{\EDMC}{E_\text{DMC}} +\newcommand{\EexFCI}{E_\text{exFCI}} +\newcommand{\EexFCIbasis}{E_\text{exFCI}^{\Bas}} +\newcommand{\EexFCIinfty}{E_\text{exFCI}^{\infty}} +\newcommand{\EexDMC}{E_\text{exDMC}} +\newcommand{\Ead}{\Delta E_\text{ad}} +\newcommand{\efci}[0]{E_{\text{FCI}}^{\Bas}} +\newcommand{\emodel}[0]{E_{\model}^{\Bas}} +\newcommand{\emodelcomplete}[0]{E_{\model}^{\infty}} +\newcommand{\efcicomplete}[0]{E_{\text{FCI}}^{\infty}} +\newcommand{\ecccomplete}[0]{E_{\text{CCSD(T)}}^{\infty}} +\newcommand{\ecc}[0]{E_{\text{CCSD(T)}}^{\Bas}} +\newcommand{\efuncbasisFCI}[0]{\bar{E}^\Bas[\denFCI]} +\newcommand{\efuncbasisfci}[0]{\bar{E}^\Bas[\denfci]} +\newcommand{\efuncbasis}[0]{\bar{E}^\Bas[\den]} +\newcommand{\efuncden}[1]{\bar{E}^\Bas[#1]} +\newcommand{\efuncdenpbe}[1]{\bar{E}_{\text{}}^\Bas[#1]} +\newcommand{\efuncdenpbeAB}[1]{\bar{E}_{\text{A}+\text{B}}^\Bas[#1]} +\newcommand{\ecompmodel}[0]{\bar{E}^\Bas[\denmodel]} +\newcommand{\ecmubis}[0]{\bar{E}_{\text{c,md}}^{\text{sr}}[\denr;\,\mu]} +\newcommand{\ecmubisldapbe}[0]{\bar{E}_{\text{c,md}\,\text{PBE}}^{\text{sr}}[\denr;\,\mu]} +\newcommand{\ecmuapprox}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mu]} +\newcommand{\ecmuapproxmur}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mur]} +\newcommand{\ecmuapproxmurfci}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denfci;\,\mur]} +\newcommand{\ecmuapproxmurmodel}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denmodel;\,\mur]} +\newcommand{\ecompmodellda}[0]{\bar{E}_{\text{LDA}}^{\Bas,\wf{}{\Bas}}[\denmodel]} +\newcommand{\ecompmodelldaval}[0]{\bar{E}_{\text{LDA, val}}^{\Bas,\wf{}{\Bas}}[\den]} +\newcommand{\ecompmodelpbe}[0]{\bar{E}_{\text{PBE}}^{\Bas,\wf{}{\Bas}}[\den]} +\newcommand{\ecompmodelpbeval}[0]{\bar{E}_{\text{PBE, val}}^{\Bas,\wf{}{\Bas}}[\den]} +\newcommand{\emulda}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denr;\mu({\bf r};\wf{}{\Bas})\right)} +\newcommand{\emuldamodel}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denmodelr;\mu({\bf r};\wf{}{\Bas})\right)} +\newcommand{\emuldaval}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denval ({\bf r});\murval;\wf{}{\Bas})\right)} +\newcommand{\ecmd}[0]{\bar{\varepsilon}_{\text{c,md}}^{\text{sr},\text{PBE}}} +\newcommand{\psibasis}[0]{\Psi^{\basis}} +\newcommand{\BasFC}{\mathcal{V}} + +%pbeuegxiHF +%\newcommand{\pbeuegxihf}{\text{PBE-UEG-}\zeta\text{-HF}^\Bas} +%\newcommand{\argpbeuegxihf}[0]{\den,\zeta,s,\ntwo^{\text{UEG}},\mu^{\text{HF},\basis}} +%\newcommand{\argrpbeuegxihf}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo^{\text{UEG}}(\br{}),\mu^{\text{HF},\basis}(\br{})} +%pbeuegxiCAS +%\newcommand{\pbeuegxi}{\text{PBE-UEG-}\zeta\text{-CAS}^\Bas} +%\newcommand{\argpbeuegxicas}[0]{\den,\zeta,s,\ntwo^{\text{UEG}},\mu_{\text{CAS}}^{\basis}} +%\newcommand{\argrpbeuegxicas}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo^{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})} +%pbeuegXiCAS +\newcommand{\pbeuegXi}{\text{PBE-UEG}} +\newcommand{\argpbeuegXi}[0]{\den,\tilde{\zeta},s,\ntwo^{\text{UEG}},\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeuegXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),\ntwo^{\text{UEG}}(\br{}),\mu_{\text{}}(\br{})} +%pbeontxiCAS +%\newcommand{\pbeontxi}{\text{SPBE-OT}} +%\newcommand{\argpbeontxi}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} +%\newcommand{\argrpbeontxi}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})} +%pbeontXiCAS +\newcommand{\pbeontXi}{\text{PBE-OT}} +\newcommand{\argpbeontXi}[0]{\den,\tilde{\zeta},s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeontXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{}}^{}(\br{})} +%pbeont0xiCAS +\newcommand{\pbeuegns}{\text{SU-PBE-UEG}} +\newcommand{\argpbeuegns}[0]{\den,0,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeuegns}[0]{\den(\br{}),0,s(\br{}),\ntwo^{\text{UEG}}(\br{}),\mu_{\text{}}^{}(\br{})} +%pbeont0xiCAS +\newcommand{\pbeontns}{\text{SU-PBE-OT}} +\newcommand{\argpbeontns}[0]{\den,0,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeontns}[0]{\den(\br{}),0,s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{}}^{}(\br{})} + +%%%%%% arguments + +\newcommand{\argepbe}[0]{\den,\zeta,s} +\newcommand{\argebasis}[0]{\den,\zeta,\ntwo,\mu} +\newcommand{\argecmd}[0]{\den,\zeta,s,\ntwo,\mu} +\newcommand{\argepbeueg}[0]{\den,\zeta,s,\ntwo^{\text{UEG}},\mu_{\Psi^{\basis}}} +\newcommand{\argepbeontxicas}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} +\newcommand{\argepbeuegXihf}[0]{\den,\tilde{\zeta},s,\ntwo^{\text{UEG}},\mu_{\Psi^{\basis}}} +\newcommand{\argrebasis}[0]{\denr,\zeta(\br{}),s(\br{}),\ntwo(\br{}),\mu(\br{})} +\newcommand{\argrebasisab}[0]{\denr,\zeta(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})} + + +% numbers +\newcommand{\rnum}[0]{{\rm I\!R}} +\newcommand{\bfr}[1]{{\bf r}_{#1}} +\newcommand{\dr}[1]{\text{d}\bfr{#1}} +\newcommand{\rr}[2]{\bfr{#1}, \bfr{#2}} +\newcommand{\rrrr}[4]{\bfr{#1}, \bfr{#2},\bfr{#3},\bfr{#4} } + + + +% effective interaction +\newcommand{\twodm}[4]{\mel{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}} +\newcommand{\murpsi}[0]{\mu_{\wf{}{\Bas}}({\bf r})} +\newcommand{\murpsibas}[0]{\mu_{\wf{}{\Bas}}({\bf r})} +\newcommand{\ntwo}[0]{n_{2}} +\newcommand{\ntwohf}[0]{n_2^{\text{HF}}} +\newcommand{\ntwophi}[0]{n_2^{{\phi}}} +\newcommand{\ntwoextrap}[0]{\mathring{n}_{2}^{\text{}}} +\newcommand{\ntwoextrapcas}[0]{\mathring{n}_2^{\text{}}} +\newcommand{\mur}[0]{\mu({\bf r})} +\newcommand{\murr}[1]{\mu({\bf r}_{#1})} +\newcommand{\murval}[0]{\mu_{\text{val}}({\bf r})} +\newcommand{\murpsival}[0]{\mu_{\wf{}{\Bas}}^{\text{val}}({\bf r})} +\newcommand{\murrval}[1]{\mu^{\text{val}}({\bf r}_{#1})} +\newcommand{\weeopmu}[0]{\hat{W}_{\text{ee}}^{\text{lr},\mu}} + + +\newcommand{\wbasis}[0]{W_{\wf{}{\Bas}}(\bfr{1},\bfr{2})} +\newcommand{\wbasiscoal}[0]{W_{\wf{}{\Bas}}(\bfr{},\bfr{})} +\newcommand{\wbasisval}[0]{W_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})} +\newcommand{\fbasis}[0]{f_{\wf{}{\Bas}}(\bfr{1},\bfr{2})} +\newcommand{\fbasisval}[0]{f_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})} + \newcommand{\ontop}[2]{ n^{(2)}_{#1}({\bf #2}_1)} + \newcommand{\twodmrpsi}[0]{ \ntwo^{\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})} + \newcommand{\twodmrdiagpsi}[0]{ n_{2,{\wf{}{\Bas}}}(\rr{1}{2})} + \newcommand{\twodmrdiagpsival}[0]{n_{2,\wf{}{\Bas}}^{\text{val}}(\rr{1}{2})} + \newcommand{\gammamnpq}[1]{\Gamma_{mn}^{pq}[#1]} + \newcommand{\gammamnkl}[0]{\Gamma_{mn}^{kl}} + \newcommand{\gammaklmn}[1]{\Gamma_{kl}^{mn}[#1]} +%\newcommand{\wbasiscoal}[1]{W_{\wf{}{\Bas}}({\bf r}_{#1})} +\newcommand{\ontoppsi}[1]{ n^{(2)}_{\wf{}{\Bas}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})} +\newcommand{\wbasiscoalval}{W_{\wf{}{\Bas}}^{\text{val}}({\bf r},{\bf r})} +\newcommand{\ontoppsival}[1]{ n^{(2)}_{\wf{}{\Bas}}^{\text{val}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})} + + + +\newcommand{\ex}[4]{$^{#1}#2_{#3}^{#4}$} +\newcommand{\ra}{\rightarrow} +\newcommand{\De}{D_\text{e}} + +% MODEL +\newcommand{\model}[0]{\mathcal{Y}} + +% densities +\newcommand{\denmodel}[0]{\den_{\model}^\Bas} +\newcommand{\denmodelr}[0]{\den_{\model}^\Bas ({\bf r})} +\newcommand{\denfci}[0]{\den_{\psifci}} +\newcommand{\denFCI}[0]{\den^{\Bas}_{\text{FCI}}} +\newcommand{\denbas}[0]{\den^\Bas} +\newcommand{\denhf}[0]{\den_{\text{HF}}^\Bas} +\newcommand{\denrfci}[0]{\denr_{\psifci}} +\newcommand{\dencipsir}[0]{{n}_{\text{CIPSI}}^\Bas({\bf r})} +\newcommand{\dencipsi}[0]{{n}_{\text{CIPSI}}^\Bas} +\newcommand{\den}[0]{{n}} +\newcommand{\denval}[0]{{n}^{\text{val}}} +\newcommand{\denr}[0]{{n}({\bf r})} +\newcommand{\onedmval}[0]{\rho_{ij,\sigma}^{\text{val}}} + +% wave functions +\newcommand{\psifci}[0]{\Psi^{\Bas}_{\text{FCI}}} +\newcommand{\psimu}[0]{\Psi^{\mu}} +% operators +\newcommand{\weeopbasis}[0]{\hat{W}_{\text{ee}}^\Bas} +\newcommand{\kinop}[0]{\hat{T}} + +\newcommand{\weeopbasisval}[0]{\hat{W}_{\text{ee}}^{\Basval}} +\newcommand{\weeop}[0]{\hat{W}_{\text{ee}}} + + +% units +\newcommand{\IneV}[1]{#1 eV} +\newcommand{\InAU}[1]{#1 a.u.} +\newcommand{\InAA}[1]{#1 \AA} + + +% methods +\newcommand{\UEG}{\text{UEG}} +\newcommand{\LDA}{\text{LDA}} +\newcommand{\PBE}{\text{PBE}} +\newcommand{\FCI}{\text{FCI}} +\newcommand{\CCSDT}{\text{CCSD(T)}} +\newcommand{\lr}{\text{lr}} +\newcommand{\sr}{\text{sr}} + +\newcommand{\Nel}{N} +\newcommand{\V}[2]{V_{#1}^{#2}} + + +\newcommand{\n}[2]{n_{#1}^{#2}} +\newcommand{\E}[2]{E_{#1}^{#2}} +\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}} +\newcommand{\bEc}[1]{\Bar{E}_\text{c}^{#1}} +\newcommand{\e}[2]{\varepsilon_{#1}^{#2}} +\newcommand{\be}[2]{\Bar{\varepsilon}_{#1}^{#2}} +\newcommand{\bec}[1]{\Bar{e}^{#1}} +\newcommand{\wf}[2]{\Psi_{#1}^{#2}} +\newcommand{\W}[2]{W_{#1}^{#2}} +\newcommand{\w}[2]{w_{#1}^{#2}} +\newcommand{\hn}[2]{\Hat{n}_{#1}^{#2}} +\newcommand{\rsmu}[2]{\mu_{#1}^{#2}} +\newcommand{\SO}[2]{\phi_{#1}(\br{#2})} + +\newcommand{\modX}{\text{X}} +\newcommand{\modY}{\text{Y}} + +% basis sets +\newcommand{\setdenbasis}{\mathcal{N}_{\Bas}} +\newcommand{\Bas}{\mathcal{B}} +\newcommand{\basis}{\mathcal{B}} +\newcommand{\Basval}{\mathcal{B}_\text{val}} +\newcommand{\Val}{\mathcal{V}} +\newcommand{\Cor}{\mathcal{C}} + +% operators +\renewcommand{\d}{\text{d}} +\newcommand{\hT}{\Hat{T}} +\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}} +\newcommand{\f}[2]{f_{#1}^{#2}} +\newcommand{\Gam}[2]{\Gamma_{#1}^{#2}} +\newcommand{\isEquivTo}[1]{\underset{#1}{\sim}} + +% coordinates +\newcommand{\br}[1]{{\mathbf{r}_{#1}}} +\newcommand{\bx}[1]{\mathbf{x}_{#1}} +\newcommand{\dbr}[1]{d\br{#1}} +\newcommand{\PBEspin}{PBEspin} +\newcommand{\PBEueg}{PBE-UEG-{$\tilde{\zeta}$}} + +\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique (UMR 7616), Sorbonne Universit\'e, CNRS, Paris, France} +\newcommand{\ISCD}{Institut des Sciences du Calcul et des Donn\'ees, Sorbonne Universit\'e, Paris, France} +\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} +\newcommand{\IUF}{Institut Universitaire de France, Paris, France} + + +\begin{document} + +\title{A basis-set error correction based on density-functional theory for strongly correlated molecular systems} + +\author{Emmanuel Giner} +\email{emmanuel.giner@lct.jussieu.fr} +\affiliation{\LCT} +\author{Anthony Scemama} +\affiliation{\LCPQ} +\author{Pierre-Fran\c{c}ois Loos} +\email{loos@irsamc.ups-tlse.fr} +\affiliation{\LCPQ} +\author{Julien Toulouse} +\email{toulouse@lct.jussieu.fr} +\affiliation{\LCT} +\affiliation{\IUF} + + +\begin{abstract} +We extend to strongly correlated molecular systems the recently introduced basis-set incompleteness correction based on density-functional theory (DFT) [E. Giner \textit{et al.}, \href{https://doi.org/10.1063/1.5052714}{J. Chem. Phys. \textbf{149}, 194301 (2018)}]. This basis-set correction relies on a mapping between wave-function calculations in a finite basis set and range-separated DFT (RSDFT) through the definition of an effective non-divergent interaction corresponding to the electron-electron Coulomb interaction projected in the finite basis set. This enables the use of RSDFT-type complementary density functionals to recover the dominant part of the short-range correlation effects missing in this finite basis set. To study both weak and strong correlation regimes we consider the potential energy curves of the \ce{H10}, \ce{N2}, \ce{O2}, and \ce{F2} molecules up to the dissociation limit, and we explore various approximations of complementary functionals fulfilling two key properties: spin-multiplet degeneracy (\ie, independence of the energy with respect to the spin projection $S_z$) and size consistency. Specifically, we investigate the dependence of the functional on different types of on-top pair densities and spin polarizations. The key result of this study is that the explicit dependence on the on-top pair density allows one to completely remove the dependence on any form of spin polarization without any significant loss of accuracy. Quantitatively, we show that the basis-set correction reaches chemical accuracy on atomization energies with triple-$\zeta$ quality basis sets for most of the systems studied here. Also, the present basis-set incompleteness correction provides smooth potential energy curves along the whole range of internuclear distances. +\end{abstract} + +\maketitle + +%%%%%%%%%%%%%%%%%%%%%%%% +\section{Introduction} +%%%%%%%%%%%%%%%%%%%%%%%% +The general goal of quantum chemistry is to provide reliable theoretical tools to explore the rich area of chemistry. More specifically, developments in quantum chemistry primarily aim at accurately computing the electronic structure of molecular systems. Despite intense developments, no definitive solution to this problem has been found. The theoretical challenge to tackle belongs to the quantum many-body problem, due to the intrinsic quantum nature of the electrons and the Coulomb repulsion between them. This so-called electronic correlation problem corresponds to finding a solution to the Schr\"odinger equation for a $N$-electron system, and two main roads have emerged to approximate this solution: wave-function theory (WFT) \cite{Pop-RMP-99} and density-functional theory (DFT). \cite{Koh-RMP-99} Although both WFT and DFT spring from the same Schr\"odinger equation, they rely on very different formalisms, as the former deals with the complicated $N$-electron wave function whereas the latter focuses on the much simpler one-electron density. In its Kohn-Sham (KS) formulation, \cite{KohSha-PR-65} the computational cost of DFT is very appealing since it is a simple mean-field procedure. Therefore, although continued efforts have been made to reduce the computational cost of WFT, DFT still remains the workhorse of quantum computational chemistry. + +The difficulty of obtaining a reliable theoretical description of a given chemical system can be roughly categorized by the strength of the electronic correlation. The so-called weakly correlated systems, such as closed-shell organic molecules near their equilibrium geometry, are typically dominated by correlation effects which do not affect the qualitative mean-field picture of the system. These weak-correlation effects can be either short range (near the electron-electron coalescence points) \cite{HatKloKohTew-CR-12} or long range (London dispersion interactions). \cite{AngDobJanGou-BOOK-20} The theoretical description of weakly correlated systems is one of the most concrete achievement of quantum chemistry, and the main remaining challenge for these systems is to push the limit of the chemical system size that can be treated. The case of the so-called strongly correlated systems, which are ubiquitous in chemistry, is more problematic as they exhibit a much more complex electronic structure. For example, transition metal complexes, low-spin open-shell systems, covalent bond breaking situations have all in common that they cannot be even qualitatively described by a single electronic configuration. It is now clear that the usual semilocal density-functional approximations of KS DFT fail to accurately describe these situations \cite{GorSeiSav-PCCP-08,GagTruLiCarHoyBa-ACR-17} and WFT is king for the treatment of strongly correlated systems. + +In practice, WFT uses a finite one-electron basis set. The exact solution of the Schr\"odinger equation within this basis set is then provided by full configuration interaction (FCI) which consists in a linear-algebra eigenvalue problem with a dimension scaling exponentially with the system size. Due to this exponential growth of the FCI computational cost, introducing approximations is necessary, with at least two difficulties for strongly correlated systems: i) the qualitative description of the wave function is determined by a primary set of electronic configurations (whose size can scale exponentially in many cases) among which near degeneracies and/or strong interactions appear in the Hamiltonian matrix; ii) the quantitative description of the system requires also to account for weak correlation effects which involve many other electronic configurations with typically much smaller weights in the wave function. Addressing simultaneously these two issues is a rather complicated task for a given approximate WFT method, especially if one adds the requirement of satisfying formal properties, such as spin-multiplet degeneracy (\ie, independence of the energy with respect to the spin projection $S_z$) and size consistency. + +Beside the difficulties of accurately describing the molecular electronic structure within a given basis set, a crucial limitation of WFT methods is the slow convergence of the energy (and related properties) with respect to the size of the one-electron basis set. As initially shown by the seminal work of Hylleraas \cite{Hyl-ZP-29} and further developed by Kutzelnigg and coworkers, \cite{Kut-TCA-85,KutKlo-JCP-91, NogKut-JCP-94} the main convergence problem originates from the divergence of the electron-electron Coulomb interaction at the coalescence point, which induces a discontinuity in the first derivative of the exact wave function (the so-called electron-electron cusp). Describing such a discontinuity with an incomplete one-electron basis set is impossible and, as a consequence, the convergence of the computed energies and properties are strongly affected. To alleviate this problem, extrapolation techniques have been developed, either based on a partial-wave expansion analysis, \cite{HelKloKocNog-JCP-97,HalHelJorKloKocOlsWil-CPL-98} or more recently based on perturbative arguments. \cite{IrmHulGru-PRL-19,IrmGru-JCP-2019} A more rigorous approach to tackle the basis-set convergence problem is provided by the so-called explicitly correlated F12 (or R12) methods \cite{Ten-TCA-12,TenNog-WIREs-12,HatKloKohTew-CR-12, KonBisVal-CR-12, GruHirOhnTen-JCP-17, MaWer-WIREs-18} which introduce a geminal function depending explicitly on the interelectronic distance. This ensures a correct representation of the Coulomb correlation hole around the electron-electron coalescence point, and leads to a much faster convergence of the energy than usual WFT methods. For instance, using the explicitly correlated version of coupled cluster with singles, doubles, and perturbative triples [CCSD(T)] in a triple-$\zeta$ basis set is equivalent to using a quintuple-$\zeta$ basis set with the usual CCSD(T) method, \cite{TewKloNeiHat-PCCP-07} although a computational overhead is introduced by the auxiliary basis set needed to compute the three-electron integrals involved in F12 theory. \cite{BarLoo-JCP-17} In addition to the computational cost, a possible drawback of F12 theory is its rather complex formalism which requires non-trivial developments for adapting it to a new method. For strongly correlated systems, several multi-reference methods have been extended to explicit correlation (see, for example, Refs.~\onlinecite{Ten-CPL-07,ShiWer-JCP-10,TorKniWer-JCP-11,DemStanMatTenPitNog-PCCP-12,GuoSivValNee-JCP-17}), including approaches based on the so-called universal F12 theory which are potentially applicable to any electronic-structure computational methods. \cite{TorVal-JCP-09,KonVal-JCP-11,HauMaoMukKlo-CPL-12,BooCleAlaTew-JCP-12} + +An alternative way to improve the convergence towards the complete-basis-set (CBS) limit is to treat the short-range correlation effects within DFT and to use WFT methods to deal only with the long-range and/or strong correlation effects. A rigorous approach achieving this mixing of DFT and WFT is range-separated DFT (RSDFT) (see Ref.~\onlinecite{TouColSav-PRA-04} and references therein) which relies on a decomposition of the electron-electron Coulomb interaction in terms of the interelectronic distance thanks to a range-separation parameter $\mu$. The advantage of this approach is at least two-fold: i) the DFT part deals primarily with the short-range part of the Coulomb interaction, and consequently the usual semilocal density-functional approximations are more accurate than for standard KS DFT; ii) the WFT part deals only with a smooth non-divergent interaction, and consequently the wave function has no electron-electron cusp \cite{GorSav-PRA-06} and the basis-set convergence is much faster. \cite{FraMusLupTou-JCP-15} A number of approximate RSDFT schemes have been developed involving single-reference \cite{AngGerSavTou-PRA-05, GolWerSto-PCCP-05, TouGerJanSavAng-PRL-09,JanHenScu-JCP-09, TouZhuSavJanAng-JCP-11, MusReiAngTou-JCP-15,KalTou-JCP-18,KalMusTou-JCP-19} and multi-reference \cite{LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, HedTouJen-JCP-18, FerGinTou-JCP-18} WFT methods. Nevertheless, there are still some open issues in RSDFT, such as remaining fractional-charge and fractional-spin errors in the short-range density functionals \cite{MusTou-MP-17} or the dependence of the quality of the results on the value of the range-separation parameter $\mu$. + + +Building on the development of RSDFT, a possible solution to the basis-set convergence problem has been recently proposed by some of the present authors~\cite{GinPraFerAssSavTou-JCP-18} in which RSDFT functionals are used to recover only the correlation effects outside a given basis set. The key point here is to realize that a wave function developed in an incomplete basis set is cuspless and could also originate from a Hamiltonian with a non-divergent long-range electron-electron interaction. Therefore, a mapping with RSDFT can be performed through the introduction of an effective non-divergent interaction representing the usual electron-electron Coulomb interaction projected in an incomplete basis set. First applications to weakly correlated molecular systems have been successfully carried out, \cite{LooPraSceTouGin-JCPL-19} together with extensions of this approach to the calculations of excitation energies \cite{GinSceTouLoo-JCP-19} and ionization potentials. \cite{LooPraSceGinTou-JCTC-20} The goal of the present work is to further develop this approach for the description of strongly correlated systems. + +The paper is organized as follows. In Sec.~\ref{sec:theory}, we recall the mathematical framework of the basis-set correction and we present its extension for strongly correlated systems. In particular, our focus is primarily set on imposing two key formal properties which are highly desirable in the context of strong correlation: spin-multiplet degeneracy and size consistency. To do this, we introduce i) new functionals using different flavors of spin polarizations and on-top pair densities, and ii) an effective electron-electron interaction based on a multiconfigurational wave function. This generalizes the method used in previous works on weakly correlated systems\cite{GinPraFerAssSavTou-JCP-18,LooPraSceTouGin-JCPL-19} for which it was sufficient to use an effective interaction based on a single-determinant wave function and a functional depending only on the usual density and spin polarization. +Then, in Sec.~\ref{sec:results}, we apply the method to the calculation of the potential energy curves of the \ce{H10}, \ce{N2}, \ce{O2}, and \ce{F2} molecules up to the dissociation limit. Finally, we conclude in Sec.~\ref{sec:conclusion}. + +%%%%%%%%%%%%%%%%%%%%%%%% +\section{Theory} +\label{sec:theory} +%%%%%%%%%%%%%%%%%%%%%%%% +As the theory behind the present basis-set correction has been exposed in details in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we only briefly recall the main equations and concepts needed for this study in Secs.~\ref{sec:basic}, \ref{sec:wee}, and \ref{sec:mur}. More specifically, in Sec.~\ref{sec:basic}, we recall the basic mathematical framework of the present theory by introducing the complementary functional to a basis set. Section \ref{sec:wee} introduces the effective non-divergent interaction in the basis set, which leads us to the definition of the effective \textit{local} range-separation function in Sec.~\ref{sec:mur}. Then, Sec.~\ref{sec:functional} exposes the new approximate RSDFT-based complementary correlation functionals. The generic form of such functionals is exposed in Sec.~\ref{sec:functional_form}, their properties in the context of the basis-set correction are discussed in Sec.~\ref{sec:functional_prop}, and the specific requirements for strong correlation are discussed in Sec.~\ref{sec:requirements}. Finally, the actual functionals used in this work are introduced in Sec.~\ref{sec:def_func}. + +\subsection{Basic theory} +\label{sec:basic} + +The exact ground-state energy $E_0$ of a $N$-electron system can, in principle, be obtained in DFT by a minimization over $N$-representable one-electron densities $\denr$ +\begin{equation} + \label{eq:levy} + E_0 = \min_{\den} \bigg\{ F[\den] + \int \d \br{} v_{\text{ne}} (\br{}) \denr \bigg\}, +\end{equation} +where $v_\text{ne}(\br{})$ is the nuclei-electron potential, and $F[\den]$ is the universal Levy-Lieb density functional written with the constrained search formalism as~\cite{Lev-PNAS-79,Lie-IJQC-83} +\begin{equation} + \label{eq:levy_func} + F[\den] = \min_{\Psi \to \den} \mel{\Psi}{\kinop +\weeop}{\Psi}, +\end{equation} +where $\kinop$ and $\weeop$ are the kinetic and electron-electron Coulomb operators, and the notation $\Psi \to \den$ means that the wave function $\Psi$ yields the density $\den$. +The minimizing density $n_0$ in Eq.~\eqref{eq:levy} is the exact ground-state density. Nevertheless, in practical calculations, the accessible densities are necessarily restricted to the set of densities ``representable in a basis set $\Bas$'', \ie, densities coming from wave functions expandable in the $N$-electron Hilbert space generated by the one-electron basis set $\Bas$. In the following, we always consider only such representable-in-$\Bas$ densities. With this restriction, Eq.~\eqref{eq:levy} then gives an upper bound $E_0^\Bas$ of the exact ground-state energy. Since the density has a faster convergence with the size of the basis set than the wave function, this restriction is a rather weak one and we can consider that $E_0^\Bas$ is an acceptable approximation to the exact ground-state energy, \ie, $E_0^\Bas \approx E_0$. + +In the present context, it is important to notice that the wave functions $\Psi$ defined in Eq.~\eqref{eq:levy_func} are not restricted to a finite basis set, \ie, they should be expanded in a complete basis set. In Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, it was then proposed to decompose $F[\den]$ as, for a representable-in-$\Bas$ density $\den$, +\begin{equation} + \label{eq:def_levy_bas} + F[\den] = \min_{\wf{}{\Bas} \to \den} \mel*{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\den}, +\end{equation} +where $\wf{}{\Bas}$ are wave functions expandable in the $N$-electron Hilbert space generated by $\basis$, and +\begin{equation} + \begin{aligned} + \efuncden{\den} = \min_{\Psi \to \den} \mel*{\Psi}{\kinop +\weeop }{\Psi} + - \min_{\Psi^{\Bas} \to \den} \mel*{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \end{aligned} +\end{equation} +is the complementary density functional to the basis set $\Bas$. +Introducing the decomposition in Eq.~\eqref{eq:def_levy_bas} back into Eq.~\eqref{eq:levy} yields +\begin{multline} + \label{eq:E0basminPsiB} + E_0^\Bas = \min_{\Psi^{\Bas}} \bigg\{ \mel*{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\den_{{\Psi^{\Bas}}}} + \\ + + \int \d \br{} v_{\text{ne}} (\br{}) \den_{\Psi^{\Bas}}(\br{}) \bigg\}, +\end{multline} +where the minimization is only over wave functions $\wf{}{\Bas}$ restricted to the basis set $\basis$ and $\den_{{\Psi^{\Bas}}}(\br{})$ refers to the density generated from $\wf{}{\Bas}$. Therefore, thanks to Eq.~\eqref{eq:E0basminPsiB}, one can properly combine a WFT calculation in a finite basis set with a density functional (hereafter referred to as complementary functional) accounting for the correlation effects that are not included in the basis set. + +As a simple non-self-consistent version of this approach, we can approximate the minimizing wave function $\Psi_0^{\Bas}$ in Eq.~\eqref{eq:E0basminPsiB} by the ground-state FCI wave function $\psifci$ within $\Bas$, and we then obtain the following approximation for the exact ground-state energy [see Eqs.~(12)--(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}] +\begin{equation} + \label{eq:e0approx} + E_0 \approx E_0^\Bas \approx \efci + \efuncbasisFCI, +\end{equation} +where $\efci$ and $n_\text{FCI}^\Bas$ are the ground-state FCI energy and density, respectively. As it was originally shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} and further emphasized in Refs.~\onlinecite{LooPraSceTouGin-JCPL-19,GinSceTouLoo-JCP-19}, the main role of $\efuncbasisFCI$ is to correct for the basis-set incompleteness error, a large part of which originating from the lack of electron-electron cusp in the wave function expanded in an incomplete basis set. The whole purpose of this work is to determine approximations for $\efuncbasisFCI$ which are suitable for strongly correlated molecular systems. Two key requirements for this purpose are i) spin-multiplet degeneracy, and ii) size consistency. + +\subsection{Effective interaction in a finite basis} +\label{sec:wee} +As originally shown by Kato, \cite{Kat-CPAM-57} the electron-electron cusp of the exact wave function originates from the divergence of the Coulomb interaction at the coalescence point. Therefore, a cuspless wave function $\wf{}{\Bas}$ could also be obtained from a Hamiltonian with a non-divergent electron-electron interaction. +In other words, the impact of the basis set incompleteness can be understood as the removal of the divergence of the usual electron-electron Coulomb interaction. + +As originally derived in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} (see Sec.~II D~and Appendices), one can obtain an effective non-divergent electron-electron interaction, here referred to as $\wbasis$, which reproduces the expectation value of the electron-electron Coulomb interaction operator over a given wave function $\wf{}{\Bas}$. As we are interested in the behavior at the coalescence point, we focus on the opposite-spin part of the electron-electron interaction. More specifically, the effective electron-electron interaction associated to a given wave function $\wf{}{\Bas}$ is defined as +\begin{equation} + \label{eq:wbasis} + \wbasis = + \begin{cases} + \fbasis /\twodmrdiagpsi, & \text{if $\twodmrdiagpsi \ne 0$,} +\\ + \infty, & \text{otherwise,} + \end{cases} +\end{equation} +where +\begin{equation} + \twodmrdiagpsi = \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}, +\end{equation} +is the opposite-spin pair density associated with $\wf{}{\Bas}$, and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$ its associated tensor in a basis of spatial orthonormal orbitals $\{\SO{p}{}\}$, +\begin{equation} + \label{eq:fbasis} + \fbasis + = \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2}, +\end{equation} +and $\V{pq}{rs}= \braket{pq}{rs}$ are the usual two-electron Coulomb integrals. +With such a definition, one can show that $\wbasis$ satisfies +\begin{multline} + \frac{1}{2}\iint \dr{1} \dr{2} \wbasis \twodmrdiagpsi = + \\ + \frac{1}{2} \iint \dr{1} \dr{2} \frac{\twodmrdiagpsi}{\abs{\br{1}-\br{2}}}. +\end{multline} +As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, the effective interaction $\wbasis$ is necessarily finite at coalescence for an incomplete basis set, and tends to the usual Coulomb interaction in the CBS limit for any choice of wave function $\psibasis$, \ie, +\begin{equation} + \label{eq:cbs_wbasis} + \lim_{\Bas \to \text{CBS}} \wbasis = \frac{1}{\abs{\br{1}-\br{2}}},\quad \forall\,\psibasis. +\end{equation} +The condition in Eq.~\eqref{eq:cbs_wbasis} is fundamental as it guarantees the correct behavior of the theory in the CBS limit. + +\subsection{Local range-separation function} +\label{sec:mur} +\subsubsection{General definition} +The effective interaction within a finite basis, $\wbasis$, is bounded and resembles the long-range interaction used in RSDFT +\begin{equation} + \label{eq:weelr} + w_\text{ee}^{\lr}(\mu;r_{12}) = \frac{\text{erf}\big(\mu \,r_{12} \big)}{r_{12}}, +\end{equation} +where $\mu$ is the range-separation parameter. As originally proposed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we make the correspondence between these two interactions by using the local range-separation function +\begin{equation} + \label{eq:def_mur} + \murpsi = \frac{\sqrt{\pi}}{2} \wbasiscoal, +\end{equation} +such that the two interactions coincide at the electron-electron coalescence point for each $\br{}$ +\begin{equation} + w_\text{ee}^{\lr}(\murpsi;0) = \wbasiscoal, \quad \forall \, \br{}. +\end{equation} +Because of the very definition of $\wbasis$, one has the following property in the CBS limit [see Eq.~\eqref{eq:cbs_wbasis}] +\begin{equation} + \label{eq:cbs_mu} + \lim_{\Bas \to \text{CBS}} \murpsi = \infty, \quad \forall \,\psibasis, +\end{equation} +which is again fundamental to guarantee the correct behavior of the theory in the CBS limit. + +\subsubsection{Frozen-core approximation} +\label{sec:FC} +As all WFT calculations in this work are performed within the frozen-core approximation, we use a ``valence-only'' (or no-core) version of the various quantities needed for the complementary functional introduced in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}. We partition the basis set as $\Bas = \Cor \bigcup \BasFC$, where $\Cor$ and $\BasFC$ are the sets of core and ``valence'' (\ie, non-core) orbitals, respectively, and define the valence-only local range-separation function as +\begin{equation} + \label{eq:def_mur_val} + \murpsival = \frac{\sqrt{\pi}}{2} \wbasiscoalval{}, +\end{equation} +where +\begin{equation} + \label{eq:wbasis_val} + \wbasisval = + \begin{cases} + \fbasisval /\twodmrdiagpsival, & \text{if $\twodmrdiagpsival \ne 0$,} +\\ + \infty, & \text{otherwise,} + \end{cases} +\end{equation} +is the valence-only effective interaction and +\begin{gather} + \label{eq:fbasis_val} + \fbasisval + = \sum_{pq\in \Bas} \sum_{rstu \in \BasFC} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2}, + \\ + \label{eq:twordm_val} + \twodmrdiagpsival + = \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}. +\end{gather} +One would note the restrictions of the sums to the set $\BasFC$ in Eqs.~\eqref{eq:fbasis_val} and \eqref{eq:twordm_val}. +It is also noteworthy that, with the present definition, $\wbasisval$ still tends to the usual Coulomb interaction as $\Bas \to \CBS$. + +\subsection{General form of the complementary functional} +\label{sec:functional} + +\subsubsection{Generic approximate form} +\label{sec:functional_form} + +As originally proposed and motivated in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the complementary functional $\efuncden{\den}$ by using the so-called correlation energy functional with multideterminant reference (ECMD) introduced by Toulouse \textit{et al.}.\cite{TouGorSav-TCA-05,Tou-THESIS-05} Following the recent work in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}, we propose to consider a Perdew-Burke-Ernzerhof (PBE)-like functional which uses the one-electron density $\denr$, the spin polarization $\zeta(\br{})=[n_\uparrow(\br{})-n_\downarrow(\br{})]/\denr$ (where $n_\uparrow(\br{})$ and $n_\downarrow(\br{})$ are the spin-up and spin-down densities), the reduced density gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$, and the on-top pair density $\ntwo(\br{})\equiv \ntwo(\br{},\br{})$. In the present work, all these quantities are computed with the same wave function $\psibasis$ used to define $\mur \equiv\murpsi$. +Therefore, $\efuncden{\den}$ has the following generic form +\begin{multline} + \label{eq:def_ecmdpbebasis} + \efuncdenpbe{\argebasis} = + \\ + \int \d\br{} \,\denr \ecmd(\argrebasis), +\end{multline} +where +\begin{equation} + \label{eq:def_ecmdpbe} + \ecmd(\argecmd) = \frac{\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)}{1+ \beta(\argepbe,\ntwo) \; \mu^3} +\end{equation} +is the correlation energy per particle, with +\begin{equation} + \label{eq:def_beta} + \beta(\argepbe,\ntwo) = \frac{3}{2\sqrt{\pi}(1 - \sqrt{2})}\frac{\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)}{\ntwo/\den}, +\end{equation} +where $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ is the usual PBE correlation energy per particle. \cite{PerBurErn-PRL-96} Before introducing the different flavors of approximate functionals that we will use here (see Sec.~\ref{sec:def_func}), we would like to give some motivations for this choice of functional form. + +The form of $\ecmd(\argecmd)$ in Eq.~\eqref{eq:def_ecmdpbe} has been originally proposed in Ref.~\onlinecite{FerGinTou-JCP-18} in the context of RSDFT. In the $\mu\to 0$ limit, it reduces to the usual PBE correlation functional, \ie, +\begin{equation} + \lim_{\mu \to 0} \ecmd(\argecmd) = \varepsilon_{\text{c}}^{\text{PBE}}(\argepbe), +\end{equation} +which is relevant in the weak-correlation (or high-density) limit. In the large-$\mu$ limit, it behaves as +\begin{equation} + \label{eq:lim_mularge} + \ecmd(\argecmd) \isEquivTo{\mu\to\infty} \frac{2\sqrt{\pi}(1 - \sqrt{2})}{3 \mu^3} \frac{\ntwo}{n}, +\end{equation} +which is the exact large-$\mu$ behavior of the exact ECMD correlation energy. \cite{PazMorGorBac-PRB-06,FerGinTou-JCP-18} Of course, for a specific system, the large-$\mu$ behavior will be exact only if one uses for $n_2$ the \textit{exact} on-top pair density of this system. This large-$\mu$ limit in Eq.~\eqref{eq:lim_mularge} is relevant in the strong-correlation (or low-density) limit. In the context of RSDFT, some of the present authors have illustrated in Ref.~\onlinecite{FerGinTou-JCP-18} that the on-top pair density involved in Eq.~\eqref{eq:def_ecmdpbe} plays indeed a crucial role when reaching the strong-correlation regime. The importance of the on-top pair density in the strong-correlation regime have been also recently acknowledged by Gagliardi and coworkers \cite{CarTruGag-JPCA-17} and Pernal and coworkers.\cite{GritMeePer-PRA-18} + +Note also that $\ecmd(\argecmd)$ vanishes when $\ntwo$ vanishes, \ie, +\begin{equation} + \label{eq:lim_n2} + \lim_{\ntwo \to 0} \ecmd(\argecmd) = 0, +\end{equation} +which is expected for systems with a vanishing on-top pair density. +Finally, the function $\ecmd(\argecmd)$ vanishes when $\mu \to \infty$ like all RSDFT short-range functionals, \ie, +\begin{equation} + \label{eq:lim_muinf} + \lim_{\mu \to \infty} \ecmd(\argecmd) = 0. +\end{equation} + +\subsubsection{Two limits where the complementary functional vanishes} +\label{sec:functional_prop} + +Within the definitions of Eqs.~\eqref{eq:def_mur} and \eqref{eq:def_ecmdpbebasis}, any approximate complementary functional $\efuncdenpbe{\argebasis}$ satisfies two important properties. + +First, thanks to the properties in Eqs.~\eqref{eq:cbs_mu} and~\eqref{eq:lim_muinf}, $\efuncdenpbe{\argebasis}$ vanishes in the CBS limit, independently of the type of wave function $\psibasis$ used to define the local range-separation function $\mu(\br{})$ in a given basis set $\Bas$, +\begin{equation} + \label{eq:lim_ebasis} + \lim_{\basis \to \text{CBS}} \efuncdenpbe{\argebasis} = 0, \quad \forall\, \psibasis. +\end{equation} + +Second, $\efuncdenpbe{\argebasis}$ correctly vanishes for systems with uniformly vanishing on-top pair density, such as one-electron systems and for the stretched H$_2$ molecule, +\begin{equation} + \lim_{n_2 \to 0} \efuncdenpbe{\argebasis} = 0. +\end{equation} +This property is doubly guaranteed by i) the choice of setting $\wbasis = \infty$ for a vanishing pair density [see Eq.~\eqref{eq:wbasis}], which leads to $\mu(\br{}) \to \infty$ and thus a vanishing $\ecmd(\argecmd)$ [see Eq.~\eqref{eq:lim_muinf}], and ii) the fact that $\ecmd(\argecmd)$ vanishes anyway when the on-top pair density vanishes [see Eq.~\eqref{eq:lim_n2}]. + +\subsection{Requirements on the complementary functional for strong correlation} +\label{sec:requirements} + +An important requirement for any electronic-structure method is size consistency, \ie, the additivity of the energies of non-interacting fragments, which is mandatory to avoid any ambiguity in computing interaction energies. When two subsystems \ce{A} and \ce{B} dissociate in closed-shell systems, as in the case of weak intermolecular interactions for instance, spin-restricted Hartree-Fock (RHF) is size-consistent. When the two subsystems dissociate in open-shell systems, such as in covalent bond breaking, it is well known that the RHF approach fails and an alternative is to use a complete-active-space self-consistent-field (CASSCF) wave function which, provided that the active space has been properly chosen, leads to additive energies. + +Another important requirement is spin-multiplet degeneracy, \ie, the independence of the energy with respect to the $S_z$ component of a given spin state, which is also a property of any exact wave function. Such a property is also important in the context of covalent bond breaking where the ground state of the supersystem $\ce{A + B}$ is generally of lower spin than the corresponding ground states of the fragments (\ce{A} and \ce{B}) which can have multiple $S_z$ components. + +\subsubsection{Spin-multiplet degeneracy} + +A sufficient condition to achieve spin-multiplet degeneracy is to eliminate all dependencies on $S_z$. In the case of the function $\ecmd(\argecmd)$, this means removing the dependence on the spin polarization $\zeta(\br{})$ originating from the PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ [see Eq.~\eqref{eq:def_ecmdpbe}]. + +To do so, it has been proposed to replace the dependence on the spin polarization by the dependence on the on-top pair density. Most often, it is done by introducing an effective spin polarization~\cite{MosSan-PRA-91,BecSavSto-TCA-95,Sav-INC-96a,Sav-INC-96,MieStoSav-MP-97,TakYamYam-CPL-02,TakYamYam-IJQC-04,GraCre-MP-05,TsuScuSav-JCP-10,LimCarLuoMaOlsTruGag-JCTC-14,GarBulHenScu-JCP-15,GarBulHenScu-PCCP-15,CarTruGag-JCTC-15,GagTruLiCarHoyBa-ACR-17} (see, also, Refs.~\onlinecite{PerSavBur-PRA-95,StaDav-CPL-01}) +\begin{equation} + \label{eq:def_effspin} + \tilde{\zeta}(n,n_{2}) = \sqrt{ 1 - 2 \; n_{2}/n^2 } +\end{equation} +expressed as a function of the density $n$ and the on-top pair density $n_2$ calculated from a given wave function. The advantage of this approach is that this effective spin polarization $\tilde{\zeta}$ is independent from $S_z$ since the on-top pair density is $S_z$-independent. Nevertheless, the use of $\tilde{\zeta}$ in Eq.~\eqref{eq:def_effspin} presents some disadvantages since this expression was derived for a single-determinant wave function. Hence, it does not appear justified to use it for a multideterminant wave function. More particularly, it may happen, in the multideterminant case, that $1 - 2 \; n_{2}/n^2 < 0 $ which results in a complex-valued effective spin polarization. \cite{BecSavSto-TCA-95} +Therefore, following other authors, \cite{MieStoSav-MP-97,LimCarLuoMaOlsTruGag-JCTC-14,GarBulHenScu-JCP-15} we use the following definition +\begin{equation} + \label{eq:def_effspin-0} + \tilde{\zeta}(n,n_{2}) = + \begin{cases} + \sqrt{ 1 - 2 \; n_{2}/n^2 }, & \text{if } n^2 \ge 2 n_{2}, + \\ + 0, & \text{otherwise.} + \end{cases} +\end{equation} + +An alternative way to eliminate the $S_z$ dependence is to simply set $\zeta=0$, \ie, to resort to the spin-unpolarized functional. This lowers the accuracy for open-shell systems at $\mu=0$, \ie, for the usual PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$. Nevertheless, we argue that, for sufficiently large $\mu$, it is a viable option. Indeed, the purpose of introducing the spin polarization in semilocal density-functional approximations is to mimic the exact on-top pair density, \cite{PerSavBur-PRA-95} but our functional $\ecmd(\argecmd)$ already explicitly depends on the on-top pair density [see Eqs.~\eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}]. The dependencies on $\zeta$ and $n_2$ can thus be expected to be largely redundant. Consequently, we propose here to test the use of $\ecmd$ with \textit{a zero spin polarization}. This ensures its $S_z$ independence and, as will be numerically demonstrated, very weakly affects the complementary functional accuracy. + + +\subsubsection{Size consistency} + +Since $\efuncdenpbe{\argebasis}$ is computed via a single integral over $\mathbb{R}^3$ [see Eq.~\eqref{eq:def_ecmdpbebasis}] which involves only local quantities [$n(\br{})$, $\zeta(\br{})$, $s(\br{})$, $n_2(\br{})$, and $\mu(\br{})$], in the case of non-overlapping fragments $\text{A}+\text{B}$, it can be written as the sum of two local contributions: one coming from the integration over the region of subsystem \ce{A} and the other one from the region of subsystem \ce{B}. Therefore, a sufficient condition for size consistency is that these quantities locally coincide in the isolated fragments and in the supersystem $\text{A}+\text{B}$. Since these local quantities are calculated from the wave function $\psibasis$, a sufficient condition is that the wave function is multiplicatively separable in the limit of non-interacting fragments, \ie, $\ket*{\Psi_{\text{A}+\text{B}}^{\basis}} = \ket*{\Psi_{\ce{A}}^{\basis}} \otimes \ket*{\Psi_{\ce{B}}^{\basis}}$. We refer the interested reader to Appendix~\ref{app:sizeconsistency} for a detailed proof and discussion of the latter statement. +In the case where the two subsystems \ce{A} and \ce{B} dissociate in closed-shell systems, a simple RHF wave function ensures this property, but when one or several covalent bonds are broken, a properly chosen CASSCF wave function can be used to recover this property. The underlying active space must however be chosen in such a way that it leads to size-consistent energies in the limit of dissociated fragments. + + +\subsection{Actual approximations used for the complementary functional} +\label{sec:def_func} + +As the present work focuses on the strong-correlation regime, we propose here to investigate only approximate functionals which are $S_z$ independent and size-consistent in the case of covalent bond breaking. Therefore, the wave functions $\psibasis$ used throughout this paper are CASSCF wave functions in order to ensure size consistency of all local quantities. The difference between the different flavors of functionals are only due to the types of spin polarization and on-top pair density used. + +Regarding the spin polarization that enters into the function $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$, two different types of $S_z$-independent formulations are considered: i) the \textit{effective} spin polarization $\tilde{\zeta}$ defined in Eq.~\eqref{eq:def_effspin-0} and calculated from the CASSCF wave function, and ii) a \textit{zero} spin polarization. In the latter case, the functional is referred as to ``SU'' which stands for ``spin unpolarized''. + +Regarding the on-top pair density entering in Eq.~\eqref{eq:def_beta}, we use two different approximations. The first one is based on the uniform electron gas (UEG) and reads +\begin{equation} + \label{eq:def_n2ueg} + \ntwo^{\text{UEG}}(n,\zeta) \approx n^2\big(1-\zeta^2\big)g_0(n), +\end{equation} +where the pair-distribution function $g_0(n)$ is taken from Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. As the spin polarization appears in Eq.~\eqref{eq:def_n2ueg}, we use the effective spin polarization $\tilde{\zeta}$ of Eq.~\eqref{eq:def_effspin-0} in order to ensure $S_z$ independence. Thus, $\ntwo^{\text{UEG}}$ will depend indirectly on the on-top pair density of the CASSCF wave function through $\tilde{\zeta}$. When using $\ntwo^{\text{UEG}}(\br{}) \equiv \ntwo^{\text{UEG}}(n(\br{}),\tilde{\zeta}(\br{}))$ in a functional, we will refer to it as ``UEG''. + +The second approach to approximate the exact on-top pair density consists in using directly the on-top pair density of the CASSCF wave function. Following the work of some of the present authors, \cite{FerGinTou-JCP-18,GinSceTouLoo-JCP-19} we introduce the extrapolated on-top pair density +\begin{equation} + \label{eq:def_n2extrap} + \ntwoextrap(\ntwo,\mu) = \qty( 1 + \frac{2}{\sqrt{\pi}\mu} )^{-1} \; \ntwo, +\end{equation} +which directly follows from the large-$\mu$ extrapolation of the exact on-top pair density derived by Gori-Giorgi and Savin\cite{GorSav-PRA-06} in the context of RSDFT. Thus, the extrapolated on-top pair density $\ntwoextrap$ depends on the local range-separation function $\mu$. When using $\ntwoextrap(\br{}) \equiv \ntwoextrap(\ntwo(\br{}),\mu(\br{}))$ in a functional, we will simply refer it as ``OT'', which stands for "on-top". + +We then define three complementary functionals: +\begin{itemize} + +\item[i)] $\pbeuegXi$ which combines the effective spin polarization of Eq.~\eqref{eq:def_effspin-0} and the UEG on-top pair density defined in Eq.~\eqref{eq:def_n2ueg}: +\begin{multline} + \label{eq:def_pbeueg_i} + \bar{E}^\Bas_{\pbeuegXi} + \\ + = \int \d\br{} \,\denr \ecmd(\argrpbeuegXi), +\end{multline} +\item[ii)] $\pbeontXi$ which combines the effective spin polarization of Eq.~\eqref{eq:def_effspin-0} and the on-top pair density of Eq.~\eqref{eq:def_n2extrap}: +\begin{equation} + \label{eq:def_pbeueg_ii} + \bar{E}^\Bas_{\pbeontXi} = \int \d\br{} \,\denr \ecmd(\argrpbeontXi), +\end{equation} +\item[iii)] $\pbeontns$ which combines a zero spin polarization and the on-top pair density of Eq.~\eqref{eq:def_n2extrap}: +\begin{equation} + \label{eq:def_pbeueg_iv} + \bar{E}^\Bas_{\pbeontns} = \int \d\br{} \,\denr \ecmd(\argrpbeontns). +\end{equation} +\end{itemize} +The performance of each of these functionals is tested in the following. Note that we did not define a spin-unpolarized version of the PBE-UEG functional because it would have been significantly inferior (in terms of performance) compared to the three other functionals. Indeed, because to the lack of knowledge on the spin polarization or on the accurate on-top pair density, such a functional would be inaccurate in particular for open-shell systems. This assumption has been numerically confirmed by calculations. + +%%%%%%%%%%%%%%%%%%%%%%%% +\section{Results} +\label{sec:results} +\begin{figure*} + \subfigure[cc-pVDZ]{ + \includegraphics[width=0.45\linewidth]{data/H10/DFT_vdzE_relat.pdf} + \includegraphics[width=0.45\linewidth]{data/H10/DFT_vdzE_relat_zoom.pdf} + } + \subfigure[cc-pVTZ]{ + \includegraphics[width=0.45\linewidth]{data/H10/DFT_vtzE_relat.pdf} + \includegraphics[width=0.45\linewidth]{data/H10/DFT_vtzE_relat_zoom.pdf} + } + \subfigure[cc-pVQZ]{ + \includegraphics[width=0.45\linewidth]{data/H10/DFT_vqzE_relat.pdf} + \includegraphics[width=0.45\linewidth]{data/H10/DFT_vqzE_relat_zoom.pdf} + } + \caption{ + Potential energy curves of the H$_{10}$ chain with equally-spaced atoms calculated with MRCI+Q and basis-set corrected MRCI+Q using the cc-pVDZ (top), cc-pVTZ (center), and cc-pVQZ (bottom) basis sets. + The MRCI+Q energies and the estimated exact energies have been extracted from Ref.~\onlinecite{h10_prx}. + \label{fig:H10}} +\end{figure*} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{table*} +\caption{Atomization energies (in mHa) and associated errors (in square brackets) with respect to the estimated exact values computed at different levels of theory with various basis sets.} +\begin{ruledtabular} +\begin{tabular}{lrdddd} + + System & \tabc{Basis set} & \tabc{MRCI+Q\fnm[1]} & \tabc{(MRCI+Q)+$\pbeuegXi$} & \tabc{(MRCI+Q)+$\pbeontXi$} & \tabc{(MRCI+Q)+$\pbeontns$} \\ +\hline +\ce{H10} & cc-pVDZ & 622.1 [43.3] & 642.6 [22.8] & 649.2 [16.2] & 649.5 [15.9] \\ + & cc-pVTZ & 655.2 [10.2] & 661.9 [3.5] & 666.0 [-0.6] & 666.0 [-0.6] \\ + & cc-pVQZ & 661.2 [4.2] & 664.1 [1.3] & 666.4 [-1.0] & 666.5 [-1.1] \\[0.1cm] + & &\multicolumn{4}{c}{Estimated exact:\fnm[1] 665.4} \\[0.2cm] +\hline + & & \tabc{exFCI} & \tabc{exFCI+$\pbeuegXi$} & \tabc{exFCI+$\pbeontXi$} & \tabc{exFCI+$\pbeontns$}\\ +\hline +\ce{N2} & aug-cc-pVDZ & 321.9 [40.8] & 356.2 [6.5] & 355.5 [7.2] & 354.6 [8.1] \\ + & aug-cc-pVTZ & 348.5 [14.2] & 361.5 [1.2] & 363.5 [-0.5] & 363.2 [-0.3] \\ + & aug-cc-pVQZ & 356.6 [6.1 ] & 362.8 [-0.1] & 364.2 [-1.5] & 364.3 [-1.6] \\[0.1cm] + & & \multicolumn{4}{c}{Estimated exact:\fnm[2] 362.7} \\[0.2cm] +\hline + & & \tabc{exFCI} & \tabc{exFCI+$\pbeuegXi$} & \tabc{exFCI+$\pbeontXi$} & \tabc{exFCI+$\pbeontns$}\\ +\hline +\ce{O2} & aug-cc-pVDZ & 171.4 [20.5] & 187.6 [4.3] & 187.6 [4.3] & 187.1 [4.8] \\ + & aug-cc-pVTZ & 184.5 [7.4] & 190.3 [1.6] & 191.2 [0.7] & 191.0 [0.9] \\ + & aug-cc-pVQZ & 188.3 [3.6] & 190.3 [1.6] & 191.0 [0.9] & 190.9 [1.0] \\[0.1cm] + & & \multicolumn{4}{c}{Estimated exact:\fnm[2] 191.9} \\[0.2cm] +\hline + & & \tabc{exFCI} & \tabc{exFCI+$\pbeuegXi$} & \tabc{exFCI+$\pbeontXi$} & \tabc{exFCI+$\pbeontns$}\\ +\hline +\ce{F2} & aug-cc-pVDZ & 49.6 [12.6] & 54.8 [7.4] & 54.9 [7.3] & 54.8 [7.4] \\ + & aug-cc-pVTZ & 59.3 [2.9] & 61.2 [1.0] & 61.5 [0.7] & 61.5 [0.7] \\ + & aug-cc-pVQZ & 60.1 [2.1] & 61.0 [1.2] & 61.3 [0.9] & 61.3 [0.9] \\[0.1cm] + \cline{2-6}\\[-0.3cm] + & & \tabc{CEEIS\fnm[3]} & \tabc{CEEIS\fnm[3]+$\pbeuegXi$} & \tabc{CEEIS\fnm[3]+$\pbeontXi$} & \tabc{CEEIS\fnm[3]+$\pbeontns$}\\ + \cline{2-6}\\[-0.3cm] + & cc-pVDZ & 43.7 [18.5] & 51.0 [11.2] & 51.0 [11.2] & 50.7 [11.5] \\ + & cc-pVTZ & 56.3 [5.9] & 59.2 [3.0] & 59.6 [2.6] & 59.5 [2.7] \\ + & cc-pVQZ & 59.9 [2.3] & 61.3 [0.9] & 61.6 [0.6] & 61.6 [0.6] \\[0.1cm] + & & \multicolumn{4}{c}{Estimated exact:\fnm[2] 62.2} \\ +\end{tabular} +\end{ruledtabular} +\fnt[1]{From Ref.~\onlinecite{h10_prx}.} +\fnt[2]{From the CEEIS valence-only non-relativistic calculations of Ref.~\onlinecite{BytLaiRuedenJCP05}.} +\fnt[3]{From the CEEIS valence-only non-relativistic calculations of Ref.~\onlinecite{BytNagGorRue-JCP-07}.} +\label{tab:d0} +\end{table*} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\subsection{Computational details} + +We present potential energy curves of small molecules up to the dissociation limit +to investigate the performance of the basis-set correction in regimes of both weak and strong correlation. +The considered systems are the \ce{H10} linear chain with equally-spaced atoms, and the \ce{N2}, \ce{O2}, and \ce{F2} diatomics. + +The computation of the ground-state energy in Eq.~\eqref{eq:e0approx} in a given basis set requires approximations to the FCI energy $\efci$ and to the basis-set correction $\efuncbasisFCI$. +For diatomics with the aug-cc-pVDZ and aug-cc-pVTZ basis sets,~\cite{KenDunHar-JCP-92} energies are obtained using frozen-core selected-CI calculations (using the CIPSI algorithm) followed by the extrapolation scheme proposed by Holmes \textit{et al.} (see Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJac-JCTC-19, QP2} for more detail). All these calculations are performed with the latest version of \textsc{Quantum Package}, \cite{QP2} and will be labelled as exFCI in the following. In the case of \ce{F2}, we also use the correlation energy extrapolated by intrinsic scaling (CEEIS) \cite{BytNagGorRue-JCP-07} as an estimate of the FCI correlation energy with the cc-pVXZ (X $=$ D, T, and Q) basis sets.~\cite{Dun-JCP-89} The estimated exact potential energy curves are obtained from experimental data \cite{LieCle-JCP-74a} for the \ce{N2} and \ce{O2} molecules, and from CEEIS calculations in the case of \ce{F2}. For all geometries and basis sets, the error with respect to the exact FCI energies are estimated to be of the order of $0.5$~mHa. +For the three diatomics, we performed an additional exFCI calculation with the aug-cc-pVQZ basis set at the equilibrium geometry to obtain reliable estimates of the FCI/CBS dissociation energy. +In the case of the \ce{H10} chain, the approximation to the FCI energies together with the estimated exact potential energy curves are obtained from the data of Ref.~\onlinecite{h10_prx} where the authors performed MRCI+Q calculations with a minimal valence active space as reference (see below for the description of the active space). + +Regarding the complementary functional, we first perform full-valence CASSCF calculations with the GAMESS-US software~\cite{gamess} to obtain the wave function $\psibasis$. Then, all density-related quantities involved in the functional [density $n(\br{})$, effective spin polarization $\tilde{\zeta}(\br{})$, reduced density gradient $s(\br{})$, and on-top pair density $n_2(\br{})$] together with the local range-separation function $\mu(\br{})$ are calculated with this full-valence CASSCF wave function. The CASSCF calculations are performed with the following active spaces: (10e,10o) for \ce{H10}, (10e,8o) for \ce{N2}, (12e,8o) for \ce{O2}, and (14e,8o) for \ce{F2}. We note that, instead of using CASSCF wave functions for $\psibasis$, one could of course use the same selected-CI wave functions used for calculating the energy but the calculations of $n_2(\br{})$ and $\mu(\br{})$ would then be more costly. + +Also, as the frozen-core approximation is used in all our selected-CI calculations, we use the corresponding valence-only complementary functionals (see Subsec.~\ref{sec:FC}). Therefore, all density-related quantities exclude any contribution from the 1s core orbitals, and the range-separation function follows the definition given in Eq.~\eqref{eq:def_mur_val}. + +It should be stressed that the computational cost of the basis-set correction (see Appendix~\ref{app:computational}) is negligible compared to the cost the selected-CI calculations. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{figure*} + \subfigure[aug-cc-pVDZ]{ + \includegraphics[width=0.45\linewidth]{data/N2/DFT_avdzE_relat.pdf} + \includegraphics[width=0.45\linewidth]{data/N2/DFT_avdzE_relat_zoom.pdf} + } + \subfigure[aug-cc-pVTZ]{ + \includegraphics[width=0.45\linewidth]{data/N2/DFT_avtzE_relat.pdf} + \includegraphics[width=0.45\linewidth]{data/N2/DFT_avtzE_relat_zoom.pdf} + } + \caption{ + Potential energy curves of the \ce{N2} molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets. The estimated exact energies are based on a fit of experimental data and obtained from Ref.~\onlinecite{LieCle-JCP-74a}. + \label{fig:N2}} +\end{figure*} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\subsection{H$_{10}$ chain} + +The \ce{H10} chain with equally-spaced atoms is a prototype of strongly correlated systems as it consists in the simultaneous breaking of 10 interacting covalent $\sigma$ bonds. +As it is a relatively small system, benchmark calculations at near-CBS values are available (see Ref.~\onlinecite{h10_prx} for a detailed study of this system). + +We report in Fig.~\ref{fig:H10} the potential energy curves computed using the cc-pVXZ (X $=$ D, T, and Q) basis sets for different levels of approximation, and the corresponding atomization energies are reported in Table \ref{tab:d0}. +As a general trend, the addition of the basis-set correction globally improves +the quality of the potential energy curves, independently of the approximation level of $\efuncbasis$. Also, no erratic behavior is found when stretching the bonds, which shows that the present procedure (\ie, the determination of the range-separation function and the definition of the functionals) is robust when reaching the strong-correlation regime. +In other words, smooth potential energy curves are obtained with the present basis-set correction. +More quantitatively, the values of the atomization energies are within chemical accuracy (\ie, an error below $1.4$ mHa) with the cc-pVTZ basis set when using the $\pbeontXi$ and $\pbeontns$ functionals, whereas such an accuracy is not yet reached at the standard MRCI+Q/cc-pVQZ level of theory. + +Analyzing more carefully the performance of the different types of approximate functionals, the results show that $\pbeontXi$ and $\pbeontns$ are very similar (the maximal difference on the atomization energy being 0.3 mHa), and that they give slightly more accurate results than $\pbeuegXi$. These findings provide two important clues on the role of the different physical ingredients included in these functionals: i) the explicit use of the on-top pair density originating from the CASSCF wave function [see Eq.~\eqref{eq:def_n2extrap}] is preferable over the use of the UEG on-top pair density [see Eq.~\eqref{eq:def_n2ueg}] which is somewhat understandable, and ii) removing the dependence on any kind of spin polarization does not lead to a significant loss of accuracy providing that one employs a qualitatively correct on-top pair density. The latter point is crucial as it confirms that the spin polarization in density-functional approximations essentially plays the same role as the on-top pair density. This could have significant implications for the construction of more robust families of density-functional approximations within DFT. + + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{figure*} + \subfigure[aug-cc-pVDZ]{ + \includegraphics[width=0.45\linewidth]{data/O2/DFT_avdzE_relat.pdf} + \includegraphics[width=0.45\linewidth]{data/O2/DFT_avdzE_relat_zoom.pdf} + } + \subfigure[aug-cc-pVTZ]{ + \includegraphics[width=0.45\linewidth]{data/O2/DFT_avtzE_relat.pdf} + \includegraphics[width=0.45\linewidth]{data/O2/DFT_avtzE_relat_zoom.pdf} + } + \caption{ + Potential energy curves of the \ce{O2} molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets. +The estimated exact energies are based on a fit of experimental data and obtained from Ref.~\onlinecite{LieCle-JCP-74a}. + \label{fig:O2}} +\end{figure*} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{figure*} + \subfigure[aug-cc-pVDZ]{ + \includegraphics[width=0.45\linewidth]{data/F2/DFT_avdzE_relat.pdf} + \includegraphics[width=0.45\linewidth]{data/F2/DFT_avdzE_relat_zoom.pdf} + } + \subfigure[aug-cc-pVTZ]{ + \includegraphics[width=0.45\linewidth]{data/F2/DFT_avtzE_relat.pdf} + \includegraphics[width=0.45\linewidth]{data/F2/DFT_avtzE_relat_zoom.pdf} + } + \caption{ + Potential energy curves of the \ce{F2} molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets. +The estimated exact energies are based on a fit of experimental data and obtained from Ref.~\onlinecite{LieCle-JCP-74a}. + The estimated exact energies are based on a fit of the non-relativistic valence-only CEEIS data extracted from Ref.~\onlinecite{BytNagGorRue-JCP-07}. + \label{fig:F2}} +\end{figure*} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\subsection{Dissociation of diatomics} + +The \ce{N2}, \ce{O2} and \ce{F2} molecules are complementary to the \ce{H10} system for the present study. The level of strong correlation in these diatomics also increases while stretching the bonds, similarly to the case of \ce{H10}, but in addition these molecules exhibit more important and versatile types of weak correlations due to the larger number of electrons. Indeed, the short-range correlation effects are known to play a strong differential effect on the computation of the atomization energy at equilibrium, while the shape of the curve far from the equilibrium geometry is governed by dispersion interactions which are medium to long-range weak-correlation effects. \cite{AngDobJanGou-BOOK-20} The dispersion interactions in \ce{H10} play a minor role on the potential energy curve due to the much smaller number of near-neighbor electron pairs compared to \ce{N2}, \ce{O2} or \ce{F2}. Also, \ce{O2} has a triplet ground state and is therefore a good candidate for checking the spin-polarization dependence of the various functionals proposed here. + +We report in Figs.~\ref{fig:N2}, \ref{fig:O2}, and \ref{fig:F2} the potential energy curves of \ce{N2}, \ce{O2}, and \ce{F2} computed at various approximation levels using the aug-cc-pVDZ and aug-cc-pVTZ basis sets. The atomization energies for each level of theory with different basis sets are reported in Table \ref{tab:d0}. + +Just as in \ce{H10}, the accuracy of the atomization energies is globally improved by adding the basis-set correction and it is remarkable that $\pbeontXi$ and $\pbeontns$ provide again very similar results. The latter observation confirms that the dependence on the on-top pair density allows one to remove the dependence of any kind of spin polarization for a quite wide range of covalent bonds and also for an open-shell system like \ce{O2}. More quantitatively, an error below 1.0 mHa compared to the estimated exact valence-only atomization energy is found for \ce{N2}, \ce{O2}, and \ce{F2} with the aug-cc-pVTZ basis set using the $\pbeontns$ functional, whereas such a feat is far from being reached within the same basis set at the near-FCI level. In the case of \ce{F2} it is clear that the addition of diffuse functions in the double- and triple-$\zeta$ basis sets strongly improves the accuracy of the results, which could have be anticipated due to the strong breathing-orbital effect induced by the ionic valence-bond forms in this molecule. \cite{HibHumByrLen-JCP-94} +It should be also noticed that when reaching the aug-cc-pVQZ basis set for \ce{N2}, the accuracy of the atomization energy slightly deteriorates for the $\pbeontXi$ and $\pbeontns$ functionals, but it remains nevertheless more accurate than the estimated FCI atomization energy and very close to chemical accuracy. + +Regarding now the performance of the basis-set correction along the whole potential energy curve, it is interesting to notice that it fails to provide a noticeable improvement far from the equilibrium geometry. Acknowledging that the weak-correlation effects in these regions are dominated by dispersion interactions which are long-range effects, the failure of the present approximations for the complementary functional can be understood easily. Indeed, the whole scheme designed here is based on the physics of correlation near the electron-electron coalescence point: the local range-separation function $\mu(\br{})$ is based on the value of the effective electron-electron interaction at coalescence and the ECMD functionals are suited for short-range correlation effects. Therefore, the failure of the present basis-set correction to describe dispersion interactions is theoretically expected. +We hope to report further on this in the near future. + +\section{Conclusion} +\label{sec:conclusion} + +In the present paper we have extended the recently proposed DFT-based basis-set correction to strongly correlated systems. We have applied the method to the \ce{H10}, \ce{N2}, \ce{O2}, and \ce{F2} molecules up to the dissociation limit at near-FCI level in increasingly large basis sets, and investigated how the basis-set correction affects the convergence toward the CBS limit of the potential energy curves of these molecular systems. + +The density-based basis-set correction relies on three aspects: i) the definition of an effective non-divergent electron-electron interaction obtained from the expectation value over a wave function $\psibasis$ of the Coulomb electron-electron interaction projected into an incomplete basis set $\basis$; ii) the fit of this effective interaction with the long-range interaction used in RSDFT; and iii) the use of a short-range, complementary functional borrowed from RSDFT. In the present paper, we investigated i) and iii) in the context of strong correlation and focused on potential energy curves and atomization energies. More precisely, we proposed a new scheme to design functionals fulfilling spin-multiplet degeneracy and size consistency. To fulfill such requirements we proposed to use CASSCF wave functions leading to size-consistent energies, and we developed functionals using only $S_z$-independent density-like quantities. + +The development of new $S_z$-independent and size-consistent functionals has lead us to investigate the role of two related quantities: the spin polarization and the on-top pair density. One important result of the present study is that by using functionals \textit{explicitly} depending on the on-top pair density, one can eschew its spin-polarization dependence without loss of accuracy. This avoids the commonly used effective spin polarization originally proposed in Ref.~\onlinecite{BecSavSto-TCA-95} which has the disadvantage of possibly becoming complex-valued in the multideterminant case. From a more fundamental aspect, this confirms that, in a DFT framework, the spin polarization mimics the role of the on-top pair density. +Consequently, we believe that one could potentially develop new families of density-functional approximations where the spin polarization is abandoned and replaced by the on-top pair density. + +Regarding the results of the present approach, the basis-set correction systematically improves the near-FCI calculations in a given basis set. More quantitatively, it is shown that with only triple-$\zeta$ quality basis sets chemically accurate atomization energies are obtained for all systems whereas the uncorrected near-FCI results are far from this accuracy within the same basis set. + +Also, it is shown that the basis-set correction gives substantial differential contribution to potential energy curves close to the equilibrium geometries, but at long internuclear distances it cannot recover the dispersion interaction energy missing because of the basis-set incompleteness. This behavior is actually expected as dispersion interactions are of long-range nature and the present approach is designed to recover only short-range correlation effects. + +\appendix + +\section{Size consistency of the basis-set correction} +\label{app:sizeconsistency} + +\subsection{Sufficient condition for size consistency} + +The basis-set correction is expressed as an integral in real space +\begin{multline} + \label{eq:def_ecmdpbebasisAnnex} + \efuncdenpbe{\argebasis} = \\ \int \text{d}\br{} \,\denr \ecmd(\argrebasis), +\end{multline} +where all the local quantities $\argrebasis$ are obtained from the same wave function $\Psi$. In the limit of two non-overlapping and non-interacting dissociated fragments $\text{A}+\text{B}$, this integral can be rewritten as the sum of the integral over the region $\Omega_\text{A}$ and the integral over the region $\Omega_\text{B}$ +\begin{multline} + \label{eq:def_ecmdpbebasisAB} + \efuncdenpbeAB{\argebasis} = + \\ + \int_{\Omega_\text{A}} \text{d}\br{} \,\denr \ecmd(\argrebasis) + \\ + + \int_{\Omega_\text{B}} \text{d}\br{} \,\denr \ecmd(\argrebasis). +\end{multline} +Therefore, a sufficient condition to obtain size consistency is that all the local quantities $\argrebasis$ are \textit{intensive}, \ie, they \textit{locally} coincide in the supersystem $\text{A}+\text{B}$ and in each isolated fragment $\text{X}=\text{A}$ or $\text{B}$. Hence, we must have, for $\br{} \in \Omega_\text{X}$, +\begin{subequations} +\begin{gather} +n_\text{A+B}(\br{}) = n_\text{X}(\br{}), +\label{nAB} +\\ +\zeta_\text{A+B}(\br{}) = \zeta_\text{X}(\br{}), +\label{zAB} +\\ +s_\text{A+B}(\br{}) = s_\text{X}(\br{}), +\label{sAB} +\\ +n_{2,\text{A+B}}(\br{}) = n_{2,\text{X}}(\br{}), +\label{n2AB} +\\ +\mu_{\text{A+B}}(\br{}) = \mu_{\text{X}}(\br{}), +\label{muAB} +\end{gather} +\end{subequations} +where the left-hand-side quantities are for the supersystem and the right-hand-side quantities for an isolated fragment. Such conditions can be difficult to fulfill in the presence of degeneracies since the system X can be in a different mixed state (\ie, ensemble) in the supersystem $\text{A}+\text{B}$ and in the isolated fragment. \cite{Sav-CP-09} Here, we will consider the simple case where the wave function of the supersystem is multiplicatively separable, \ie, +\begin{equation} + \ket*{\wf{\text{A}+\text{B}}{}} = \ket*{\wf{\text{A}}{}} \otimes \ket*{\wf{\text{B}}{}}, +\label{PsiAB} +\end{equation} +where $\otimes$ is the antisymmetric tensor product. In this case, it is easy to shown that Eqs.~(\ref{nAB})-(\ref{sAB}) are valid, as well known, and it remains to show that Eqs.~(\ref{n2AB}) and~(\ref{muAB}) are also valid. Before showing this, we note that even though we do not explicitly consider the case of degeneracies, the lack of size consistency which could arise from spin-multiplet degeneracies can be avoided by the same strategy used for imposing the energy independence on $S_z$, \ie, by using the effective spin polarization $\tilde{\zeta}(n(\br{}),n_{2}(\br{}))$ or a zero spin polarization $\zeta(\br{}) = 0$. Moreover, the lack of size consistency which could arise from spatial degeneracies (\eg, coming from atomic $p$ states) can also be avoided by selecting the same member of the ensemble in the supersystem and in the isolated fragment. This applies to the systems treated in this work. + +\subsection{Intensivity of the on-top pair density and the local range-separation function} + +The on-top pair density can be written in an orthonormal spatial orbital basis set $\{\SO{p}{}\}$ as +\begin{equation} +\label{eq:def_n2} + n_{2{}}(\br{}) = \sum_{pqrs \in \Bas} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{}, +\end{equation} +with $\Gam{pq}{rs} = 2 \mel*{\wf{}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{}}$. As the summations run over all orbitals in the basis set $\Bas$, $n_{2{}}(\br{})$ is invariant to orbital rotations and can thus be expressed in terms of localized orbitals. For two non-overlapping fragments $\text{A}+\text{B}$, the basis set can then be partitioned into orbitals localized on the fragment A and orbitals localized on B, \ie, $\Bas=\Bas_\text{A}\cup \Bas_\text{B}$. Therefore, we see that the on-top pair density of the supersystem $\text{A}+\text{B}$ is additively separable +\begin{equation} + n_{2,\text{A}+\text{B}}(\br{}) = n_{2,\text{A}}(\br{}) + n_{2,\text{B}}(\br{}), +\end{equation} +where $n_{2,\text{X}}(\br{})$ is the on-top pair density of the fragment X +\begin{equation} + n_{2,\text{X}}(\br{}) = \sum_{pqrs \in \Bas_\text{X}} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{}, +\end{equation} +in which the elements $\Gam{pq}{rs}$ with orbital indices restricted to the fragment X are $\Gam{pq}{rs} = 2 \mel*{\wf{\text{A}+\text{B}}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{\text{A}+\text{B}}{}} = 2 \mel*{\wf{\text{X}}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{\text{X}}{}}$, owing to the multiplicative structure of the wave function [see Eq.~\eqref{PsiAB}]. This shows that the on-top pair density is a local intensive quantity. + +The local range-separation function is defined as, for $n_{2}(\br{}) \not=0$, +\begin{equation} + \label{eq:def_murAnnex} + \mur = \frac{\sqrt{\pi}}{2} \frac{f(\bfr{},\bfr{})}{n_{2}(\br{})}, +\end{equation} +where +\begin{equation} + \label{eq:def_f} + f(\bfr{},\bfr{}) = \sum_{pqrstu\in \Bas} \SO{p}{ } \SO{q}{ } \V{pq}{rs} \Gam{rs}{tu} \SO{t}{ } \SO{u}{ }. +\end{equation} +Again, $f(\bfr{},\bfr{})$ is invariant to orbital rotations and can be expressed in terms of orbitals localized on the fragments A and B. In the limit of infinitely separated fragments, the Coulomb interaction vanishes between A and B and therefore any two-electron integral $\V{pq}{rs}$ involving orbitals on both A and B vanishes. We thus see that the quantity $f(\bfr{},\bfr{})$ of the supersystem $\text{A}+\text{B}$ is additively separable +\begin{equation} + \label{eq:def_fa+b} + f_{\text{A}+\text{B}}(\bfr{},\bfr{}) = f_{\text{A}}(\bfr{},\bfr{}) + f_{\text{B}}(\bfr{},\bfr{}), +\end{equation} +with +\begin{equation} + \label{eq:def_fX} + f_\text{X}(\bfr{},\bfr{}) = \sum_{pqrstu\in \Bas_\text{X}} \SO{p}{ } \SO{q}{ } \V{pq}{rs} \Gam{rs}{tu} \SO{t}{ } \SO{u}{ }. +\end{equation} +So, $f(\bfr{},\bfr{})$ is a local intensive quantity. +As a consequence, the local range-separation function of the supersystem $\text{A}+\text{B}$ is +\begin{equation} + \label{eq:def_murAB} + \mu_{\text{A}+\text{B}}(\bfr{}) = \frac{\sqrt{\pi}}{2} \frac{f_{\text{A}}(\bfr{},\bfr{}) + f_{\text{B}}(\bfr{},\bfr{})}{n_{2,\text{A}}(\br{}) + n_{2,\text{B}}(\br{})}, +\end{equation} +which implies +\begin{equation} + \label{eq:def_murABsum} + \mu_{\text{A}+\text{B}}(\bfr{}) = \mu_{\text{X}}(\bfr{}) \;\; \text{if} \;\; \bfr{} \in \Omega_\text{X}, +\end{equation} +where $\mu_{\text{X}}(\bfr{}) = (\sqrt{\pi}/2) f_{\text{X}}(\bfr{},\bfr{})/n_{2,\text{X}}(\br{})$. +The local range-separation function is thus a local intensive quantity. + +We can therefore conclude that, if the wave function of the supersystem $\text{A}+\text{B}$ is multiplicative separable, all local quantities used in the basis-set correction functional are intensive and therefore the basis-set correction is size consistent. + +\section{Computational cost of the basis-set correction for a CASSCF wave function} +\label{app:computational} + +The computational cost of the basis-set correction is determined by the calculation of the on-top pair density $n_{2}(\br{})$ and the local range-separation function $\mur$ on the real-space grid. For a general multideterminant wave function, the computational cost is of order $O(N_\text{grid}N_{\Bas}^4)$ where $N_\text{grid}$ is the number of grid points and $N_{\Bas}$ is the number of basis functions.\cite{LooPraSceTouGin-JCPL-19} For a CASSCF wave function, a significant reduction of the scaling of the computational cost can be achieved. + +\subsection{Computation of the on-top pair density} + +For a CASSCF wave function $\Psi$, the occupied orbitals can be partitioned into a set of active orbitals $\mathcal{A}$ and a set of inactive (doubly occupied) orbitals $\mathcal{I}$. The CASSCF on-top pair density can then be written as +\begin{equation} + \label{def_n2_good} + n_{2}(\br{}) = n_{2,\mathcal{A}}(\br{}) + n_{\mathcal{A}}(\br{}) n_{\mathcal{I}}(\br{}) + \frac{n_{\mathcal{I}}(\br{})^2}{2}, +\end{equation} +where +\begin{subequations} +\begin{align} + \label{def_n2_act} + n_{2,\mathcal{A}}(\br{}) & = \sum_{pqrs \in \mathcal{A}} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{}, + \\ + n_{\mathcal{A}}(\br{}) & = \sum_{pq \in\mathcal{A}} \phi_p (\br{}) \phi_q (\br{}) + \mel*{\wf{}{}}{ \aic{p_\uparrow}\ai{q_\uparrow} + \aic{p_\downarrow}\ai{q_\downarrow} }{\wf{}{}}, + \\ + n_{\mathcal{I}}(\br{}) & = 2 \sum_{p \in \mathcal{I}} \phi_p (\br{})^2 +\end{align} +\end{subequations} +are the purely active part of the on-top pair density, the active part of the density, and the inactive part of the density, respectively. +The leading computational cost is the evaluation of $n_{2,\mathcal{A}}(\br{})$ on the grid which, according to Eq.~\eqref{def_n2_act}, scales as $O(N_\text{grid} N_\mathcal{A}^4)$ where $N_{\mathcal{A}}$ is the number of active orbitals which is much smaller than the number of basis functions $N_{\Bas}$. + +\subsection{Computation of the local range-separation function} + +In addition to the on-top pair density, the computation of $\mur$ needs the computation of $f(\bfr{},\bfr{})$ [see Eq.~\eqref{eq:def_f}] at each grid point. It can be factorized as +\begin{equation} + \label{eq:f_good} + f(\bfr{},\bfr{}) = \sum_{rs \in \Bas} V^{rs}(\bfr{}) \, \Gamma_{rs}(\bfr{}), +\end{equation} +where +\begin{subequations} +\begin{align} + V^{rs}(\bfr{}) & = \sum_{pq \in \Bas} V_{pq}^{rs} \phi_p(\br{}) \phi_q(\br{}), + \\ + \Gamma_{rs}(\bfr{}) & = \sum_{pq \in \Bas} \Gam{rs}{pq} \phi_p(\br{}) \phi_q(\br{}) . +\end{align} +\end{subequations} +For a general multideterminant wave function, the computational cost of $f(\bfr{},\bfr{})$ thus scales as $O(N_\text{grid}N_{\Bas}^4)$. + +In the case of a CASSCF wave function, $\Gam{rs}{pq}$ vanishes if one index $p,q,r,s$ does not belong to the set of inactive or active occupied orbitals $\mathcal{I}\cup \mathcal{A}$. Therefore, at a given grid point, the number of non-zero elements $\Gamma_{rs}(\bfr{})$ is only at most $(N_{\mathcal{I}}+N_{\mathcal{A}})^2$, which is usually much smaller than $N_{\Bas}^2$. As a consequence, one can also restrict the sum in the calculation of +\begin{equation} + f(\bfr{},\bfr{}) = \sum_{rs \in \mathcal{I}\cup\mathcal{A}} V^{rs}(\bfr{}) \, \Gamma_{rs}(\bfr{}). +\end{equation} +The overall computational cost is dominated by that of $V^{rs}(\bfr{})$, which scales as $O(N_\text{grid}(N_{\mathcal{I}}+N_{\mathcal{A}})^2 N_{\Bas}^2)$, which is much smaller than $O(N_\text{grid}N_{\Bas}^4)$. + +\bibliography{srDFT_SC} + +\end{document} diff --git a/Revised_Manuscript/srDFT_SC.tex.bak b/Revised_Manuscript/srDFT_SC.tex.bak new file mode 100644 index 0000000..b18d819 --- /dev/null +++ b/Revised_Manuscript/srDFT_SC.tex.bak @@ -0,0 +1,779 @@ +\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1} +\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amsmath,amssymb,amsfonts,physics,mhchem,xspace} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{txfonts} + +\usepackage[ + colorlinks=true, + citecolor=blue, + breaklinks=true + ]{hyperref} +\urlstyle{same} + +\newcommand{\cdash}{\multicolumn{1}{c}{---}} +\newcommand{\mc}{\multicolumn} +\newcommand{\fnm}{\footnotemark} +\newcommand{\fnt}{\footnotetext} +\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}} +\newcommand{\mr}{\multirow} +\newcommand{\SI}{\textcolor{blue}{supporting information}} + +% second quantized operators +\newcommand{\psix}[1]{\hat{\Psi}\left({\bf X}_{#1}\right)} +\newcommand{\psixc}[1]{\hat{\Psi}^{\dagger}\left({\bf X}_{#1}\right)} +\newcommand{\ai}[1]{\hat{a}_{#1}} +\newcommand{\aic}[1]{\hat{a}^{\dagger}_{#1}} +\newcommand{\vijkl}[0]{V_{ij}^{kl}} +\newcommand{\phix}[2]{\phi_{#1}(\bfr{#2})} +\newcommand{\phixprim}[2]{\phi_{#1}(\bfr{#2}')} + +\newcommand{\CBS}{\text{CBS}} + + +%operators +\newcommand{\elemm}[3]{{\ensuremath{\bra{#1}{#2}\ket{#3}}\xspace}} +\newcommand{\ovrlp}[2]{{\ensuremath{\langle #1|#2\rangle}\xspace}} + +%\newcommand{\ket}[1]{{\ensuremath{|#1\rangle}\xspace}} +%\newcommand{\bra}[1]{{\ensuremath{\langle #1|}\xspace}} + +% + + +% energies +\newcommand{\Ec}{E_\text{c}} +\newcommand{\EPT}{E_\text{PT2}} +\newcommand{\EsCI}{E_\text{sCI}} +\newcommand{\EDMC}{E_\text{DMC}} +\newcommand{\EexFCI}{E_\text{exFCI}} +\newcommand{\EexFCIbasis}{E_\text{exFCI}^{\Bas}} +\newcommand{\EexFCIinfty}{E_\text{exFCI}^{\infty}} +\newcommand{\EexDMC}{E_\text{exDMC}} +\newcommand{\Ead}{\Delta E_\text{ad}} +\newcommand{\efci}[0]{E_{\text{FCI}}^{\Bas}} +\newcommand{\emodel}[0]{E_{\model}^{\Bas}} +\newcommand{\emodelcomplete}[0]{E_{\model}^{\infty}} +\newcommand{\efcicomplete}[0]{E_{\text{FCI}}^{\infty}} +\newcommand{\ecccomplete}[0]{E_{\text{CCSD(T)}}^{\infty}} +\newcommand{\ecc}[0]{E_{\text{CCSD(T)}}^{\Bas}} +\newcommand{\efuncbasisFCI}[0]{\bar{E}^\Bas[\denFCI]} +\newcommand{\efuncbasisfci}[0]{\bar{E}^\Bas[\denfci]} +\newcommand{\efuncbasis}[0]{\bar{E}^\Bas[\den]} +\newcommand{\efuncden}[1]{\bar{E}^\Bas[#1]} +\newcommand{\efuncdenpbe}[1]{\bar{E}_{\text{}}^\Bas[#1]} +\newcommand{\ecompmodel}[0]{\bar{E}^\Bas[\denmodel]} +\newcommand{\ecmubis}[0]{\bar{E}_{\text{c,md}}^{\text{sr}}[\denr;\,\mu]} +\newcommand{\ecmubisldapbe}[0]{\bar{E}_{\text{c,md}\,\text{PBE}}^{\text{sr}}[\denr;\,\mu]} +\newcommand{\ecmuapprox}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mu]} +\newcommand{\ecmuapproxmur}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mur]} +\newcommand{\ecmuapproxmurfci}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denfci;\,\mur]} +\newcommand{\ecmuapproxmurmodel}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denmodel;\,\mur]} +\newcommand{\ecompmodellda}[0]{\bar{E}_{\text{LDA}}^{\Bas,\wf{}{\Bas}}[\denmodel]} +\newcommand{\ecompmodelldaval}[0]{\bar{E}_{\text{LDA, val}}^{\Bas,\wf{}{\Bas}}[\den]} +\newcommand{\ecompmodelpbe}[0]{\bar{E}_{\text{PBE}}^{\Bas,\wf{}{\Bas}}[\den]} +\newcommand{\ecompmodelpbeval}[0]{\bar{E}_{\text{PBE, val}}^{\Bas,\wf{}{\Bas}}[\den]} +\newcommand{\emulda}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denr;\mu({\bf r};\wf{}{\Bas})\right)} +\newcommand{\emuldamodel}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denmodelr;\mu({\bf r};\wf{}{\Bas})\right)} +\newcommand{\emuldaval}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denval ({\bf r});\murval;\wf{}{\Bas})\right)} +\newcommand{\ecmd}[0]{\bar{\varepsilon}_{\text{c,md}}^{\text{sr},\text{PBE}}} +\newcommand{\psibasis}[0]{\Psi^{\basis}} +\newcommand{\BasFC}{\mathcal{A}} + +%pbeuegxiHF +\newcommand{\pbeuegxihf}{\text{PBE-UEG-}\zeta\text{-HF}^\Bas} +\newcommand{\argpbeuegxihf}[0]{\den,\zeta,s,\ntwo^{\text{UEG}},\mu^{\text{HF},\basis}} +\newcommand{\argrpbeuegxihf}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo^{\text{UEG}}(\br{}),\mu^{\text{HF},\basis}(\br{})} +%pbeuegxiCAS +\newcommand{\pbeuegxi}{\text{PBE-UEG-}\zeta\text{-CAS}^\Bas} +\newcommand{\argpbeuegxicas}[0]{\den,\zeta,s,\ntwo^{\text{UEG}},\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeuegxicas}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo^{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})} +%pbeuegXiCAS +\newcommand{\pbeuegXi}{\text{PBE-UEG-}\tilde{\zeta}} +\newcommand{\argpbeuegXi}[0]{\den,\tilde{\zeta},s,\ntwo^{\text{UEG}},\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeuegXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),\ntwo^{\text{UEG}}(\br{}),\mu_{\text{}}(\br{})} +%pbeontxiCAS +\newcommand{\pbeontxi}{\text{PBE-ot-}\zeta} +\newcommand{\argpbeontxi}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeontxi}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})} +%pbeontXiCAS +\newcommand{\pbeontXi}{\text{PBE-ot-}\tilde{\zeta}} +\newcommand{\argpbeontXi}[0]{\den,\tilde{\zeta},s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeontXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{}}^{}(\br{})} +%pbeont0xiCAS +\newcommand{\pbeontns}{\text{PBE-ot-}0\zeta} +\newcommand{\argpbeontns}[0]{\den,0,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeontns}[0]{\den(\br{}),0,s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{}}^{}(\br{})} + +%%%%%% arguments + +\newcommand{\argepbe}[0]{\den,\zeta,s} +\newcommand{\argebasis}[0]{\den,\zeta,\ntwo,\mu} +\newcommand{\argecmd}[0]{\den,\zeta,s,\ntwo,\mu} +\newcommand{\argepbeueg}[0]{\den,\zeta,s,\ntwo^{\text{UEG}},\mu_{\Psi^{\basis}}} +\newcommand{\argepbeontxicas}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} +\newcommand{\argepbeuegXihf}[0]{\den,\tilde{\zeta},s,\ntwo^{\text{UEG}},\mu_{\Psi^{\basis}}} +\newcommand{\argrebasis}[0]{\denr,\zeta(\br{}),s(\br{}),\ntwo(\br{}),\mu(\br{})} +\newcommand{\argrebasisab}[0]{\denr,\zeta(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})} + + +% numbers +\newcommand{\rnum}[0]{{\rm I\!R}} +\newcommand{\bfr}[1]{{\bf r}_{#1}} +\newcommand{\dr}[1]{\text{d}\bfr{#1}} +\newcommand{\rr}[2]{\bfr{#1}, \bfr{#2}} +\newcommand{\rrrr}[4]{\bfr{#1}, \bfr{#2},\bfr{#3},\bfr{#4} } + + + +% effective interaction +\newcommand{\twodm}[4]{\elemm{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}} +\newcommand{\murpsi}[0]{\mu_{\wf{}{\Bas}}({\bf r})} +\newcommand{\ntwo}[0]{n_{2}} +\newcommand{\ntwohf}[0]{n_2^{\text{HF}}} +\newcommand{\ntwophi}[0]{n_2^{{\phi}}} +\newcommand{\ntwoextrap}[0]{\mathring{n}_{2}^{\text{}}} +\newcommand{\ntwoextrapcas}[0]{\mathring{n}_2^{\text{}}} +\newcommand{\mur}[0]{\mu({\bf r})} +\newcommand{\murr}[1]{\mu({\bf r}_{#1})} +\newcommand{\murval}[0]{\mu_{\text{val}}({\bf r})} +\newcommand{\murpsival}[0]{\mu_{\wf{}{\Bas}}^{\text{val}}({\bf r})} +\newcommand{\murrval}[1]{\mu^{\text{val}}({\bf r}_{#1})} +\newcommand{\weeopmu}[0]{\hat{W}_{\text{ee}}^{\text{lr},\mu}} + + +\newcommand{\wbasis}[0]{W_{\wf{}{\Bas}}(\bfr{1},\bfr{2})} +\newcommand{\wbasiscoal}[0]{W_{\wf{}{\Bas}}(\bfr{},\bfr{})} +\newcommand{\wbasisval}[0]{W_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})} +\newcommand{\fbasis}[0]{f_{\wf{}{\Bas}}(\bfr{1},\bfr{2})} +\newcommand{\fbasisval}[0]{f_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})} + \newcommand{\ontop}[2]{ n^{(2)}_{#1}({\bf #2}_1)} + \newcommand{\twodmrpsi}[0]{ \ntwo^{\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})} + \newcommand{\twodmrdiagpsi}[0]{ n_{2,{\wf{}{\Bas}}}(\rr{1}{2})} + \newcommand{\twodmrdiagpsival}[0]{n_{2,\wf{}{\Bas}}^{\text{val}}(\rr{1}{2})} + \newcommand{\gammamnpq}[1]{\Gamma_{mn}^{pq}[#1]} + \newcommand{\gammamnkl}[0]{\Gamma_{mn}^{kl}} + \newcommand{\gammaklmn}[1]{\Gamma_{kl}^{mn}[#1]} +%\newcommand{\wbasiscoal}[1]{W_{\wf{}{\Bas}}({\bf r}_{#1})} +\newcommand{\ontoppsi}[1]{ n^{(2)}_{\wf{}{\Bas}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})} +\newcommand{\wbasiscoalval}{W_{\wf{}{\Bas}}^{\text{val}}({\bf r},{\bf r})} +\newcommand{\ontoppsival}[1]{ n^{(2)}_{\wf{}{\Bas}}^{\text{val}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})} + + + +\newcommand{\ex}[4]{$^{#1}#2_{#3}^{#4}$} +\newcommand{\ra}{\rightarrow} +\newcommand{\De}{D_\text{e}} + +% MODEL +\newcommand{\model}[0]{\mathcal{Y}} + +% densities +\newcommand{\denmodel}[0]{\den_{\model}^\Bas} +\newcommand{\denmodelr}[0]{\den_{\model}^\Bas ({\bf r})} +\newcommand{\denfci}[0]{\den_{\psifci}} +\newcommand{\denFCI}[0]{\den^{\Bas}_{\text{FCI}}} +\newcommand{\denhf}[0]{\den_{\text{HF}}^\Bas} +\newcommand{\denrfci}[0]{\denr_{\psifci}} +\newcommand{\dencipsir}[0]{{n}_{\text{CIPSI}}^\Bas({\bf r})} +\newcommand{\dencipsi}[0]{{n}_{\text{CIPSI}}^\Bas} +\newcommand{\den}[0]{{n}} +\newcommand{\denval}[0]{{n}^{\text{val}}} +\newcommand{\denr}[0]{{n}({\bf r})} +\newcommand{\onedmval}[0]{\rho_{ij,\sigma}^{\text{val}}} + +% wave functions +\newcommand{\psifci}[0]{\Psi^{\Bas}_{\text{FCI}}} +\newcommand{\psimu}[0]{\Psi^{\mu}} +% operators +\newcommand{\weeopbasis}[0]{\hat{W}_{\text{ee}}^\Bas} +\newcommand{\kinop}[0]{\hat{T}} + +\newcommand{\weeopbasisval}[0]{\hat{W}_{\text{ee}}^{\Basval}} +\newcommand{\weeop}[0]{\hat{W}_{\text{ee}}} + + +% units +\newcommand{\IneV}[1]{#1 eV} +\newcommand{\InAU}[1]{#1 a.u.} +\newcommand{\InAA}[1]{#1 \AA} + + +% methods +\newcommand{\UEG}{\text{UEG}} +\newcommand{\LDA}{\text{LDA}} +\newcommand{\PBE}{\text{PBE}} +\newcommand{\FCI}{\text{FCI}} +\newcommand{\CCSDT}{\text{CCSD(T)}} +\newcommand{\lr}{\text{lr}} +\newcommand{\sr}{\text{sr}} + +\newcommand{\Nel}{N} +\newcommand{\V}[2]{V_{#1}^{#2}} + + +\newcommand{\n}[2]{n_{#1}^{#2}} +\newcommand{\E}[2]{E_{#1}^{#2}} +\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}} +\newcommand{\bEc}[1]{\Bar{E}_\text{c}^{#1}} +\newcommand{\e}[2]{\varepsilon_{#1}^{#2}} +\newcommand{\be}[2]{\Bar{\varepsilon}_{#1}^{#2}} +\newcommand{\bec}[1]{\Bar{e}^{#1}} +\newcommand{\wf}[2]{\Psi_{#1}^{#2}} +\newcommand{\W}[2]{W_{#1}^{#2}} +\newcommand{\w}[2]{w_{#1}^{#2}} +\newcommand{\hn}[2]{\Hat{n}_{#1}^{#2}} +\newcommand{\rsmu}[2]{\mu_{#1}^{#2}} +\newcommand{\SO}[2]{\phi_{#1}(\br{#2})} + +\newcommand{\modX}{\text{X}} +\newcommand{\modY}{\text{Y}} + +% basis sets +\newcommand{\setdenbasis}{\mathcal{N}_{\Bas}} +\newcommand{\Bas}{\mathcal{B}} +\newcommand{\basis}{\mathcal{B}} +\newcommand{\Basval}{\mathcal{B}_\text{val}} +\newcommand{\Val}{\mathcal{V}} +\newcommand{\Cor}{\mathcal{C}} + +% operators +\renewcommand{\d}{\text{d}} +\newcommand{\hT}{\Hat{T}} +\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}} +\newcommand{\f}[2]{f_{#1}^{#2}} +\newcommand{\Gam}[2]{\Gamma_{#1}^{#2}} +\newcommand{\isEquivTo}[1]{\underset{#1}{\sim}} + +% coordinates +\newcommand{\br}[1]{{\mathbf{r}_{#1}}} +\newcommand{\bx}[1]{\mathbf{x}_{#1}} +\newcommand{\dbr}[1]{d\br{#1}} +\newcommand{\PBEspin}{PBEspin} +\newcommand{\PBEueg}{PBE-UEG-{$\tilde{\zeta}$}} + +\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique (UMR 7616), Sorbonne Universit\'e, CNRS, Paris, France} +\newcommand{\ISCD}{Institut des Sciences du Calcul et des Donn\'ees, Sorbonne Universit\'e, Paris, France} +\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} +\newcommand{\IUF}{Institut Universitaire de France, Paris, France} + + +\begin{document} + +\title{A density-based basis-set correction for strong correlation} + +\author{Emmanuel Giner} +\email{emmanuel.giner@lct.jussieu.fr} +\affiliation{\LCT} +\author{Barth\'el\'emy Pradines} +\affiliation{\LCT} +\affiliation{\ISCD} +\author{Anthony Scemama} +\affiliation{\LCPQ} +\author{Pierre-Fran\c{c}ois Loos} +\email{loos@irsamc.ups-tlse.fr} +\affiliation{\LCPQ} +\author{Julien Toulouse} +\email{toulouse@lct.jussieu.fr} +\affiliation{\LCT} +\affiliation{\IUF} + + +\begin{abstract} +We extend to strongly correlated systems the recently introduced basis-set correction based on density-functional theory (DFT) [E. Giner \textit{et al.}, J. Chem. Phys. \textbf{149}, 194301 (2018)]. This basis-set correction relies on a mapping between wave-function calculations in a finite basis set and range-separated DFT (RSDFT) through the definition of an effective non-divergent interaction corresponding to the Coulomb electron-electron interaction projected in the finite basis set, allowing one to use RSDFT-type complementary functionals to recover the dominant part of the short-range correlation effects missing in a finite basis set. Using as test cases the potential energy curves of the H$_{10}$, C$_2$, N$_2$, O$_2$, and F$_2$ molecules up to the dissociation limit, we systematically explore different approximations for the complementary functionals which are suited to describe strong-correlation regimes and which fulfill two very desirable properties: $S_z$ invariance and size consistency. Specifically, we investigate the dependence of the functionals on different flavours of on-top pair densities and spin polarizations. An important result is that the explicit dependence on the on-top pair density allows one to completely remove the dependence on any form of spin polarization without any significant loss of accuracy. +In the general context of multiconfigurational DFT, this finding shows that one can avoid the effective spin polarization whose mathematical definition is rather \textit{ad hoc} and which can become complex valued. Quantitatively, we show that the basis-set correction reaches chemical accuracy on atomization energies with triple-zeta quality basis sets for most of the systems studied. Also, the present basis-set correction provides smooth curves along the whole potential energy curves. +%We study the potential energy surfaces (PES) of the H$_{10}$, C$_2$, N$_2$, O$_2$, and F$_2$ molecules up to the dissociation limit using increasing basis sets at near full configuration interaction (FCI) level with and without the present basis-set correction. +\end{abstract} + +\maketitle + +%%%%%%%%%%%%%%%%%%%%%%%% +\section{Introduction} +%%%%%%%%%%%%%%%%%%%%%%%% +The general goal of quantum chemistry is to provide reliable theoretical tools to explore the rich area of chemistry. More specifically, developments in quantum chemistry primarily aim at accurately computing the electronic structure of molecular systems, but despite intense developments, no definitive solution to this problem has been found. The theoretical challenge to tackle belongs to the quantum many-body problem, due the intrinsic quantum nature of the electrons and the Coulomb repulsion between them. This so-called electronic correlation problem corresponds to finding a solution to the Schr\"odinger equation for a $N$-electron system, and two main roads have emerged to approximate this solution: wave-function theory (WFT)~\cite{Pop-RMP-99} and density-functional theory (DFT)~\cite{Koh-RMP-99}. Although both WFT and DFT spring from the same Schr\"odinger equation, they rely on very different formalisms, as the former deals with the complicated $N$-electron wave function whereas the latter focuses on the much simpler one-electron density. In its Kohn-Sham (KS) formulation~\cite{KohSha-PR-65}, the computational cost of DFT is very appealing since it is a simple mean-field procedure. Therefore, although continued efforts have been done to reduce the computational cost of WFT, DFT still remains the workhorse of quantum chemistry. + +The difficulty of obtaining a reliable theoretical description of a given chemical system can be roughly categorized by the strength of the electronic correlation appearing in its electronic structure. The so-called weakly correlated systems, such as closed-shell organic molecules near their equilibrium geometries, are typically dominated by correlation effects which do not affect the qualitative mean-field picture of the system. These weak correlation effects can be either short range(near the electron-electron coalescence point) or long range (London dispersion interactions). The theoretical description of weakly correlated systems is one of the most concrete achievement of quantum chemistry, and the main remaining issue for these systems is to push the limit of the size of the chemical systems that can be treated. The case of the so-called strongly correlated systems, which are ubiquitous in chemistry, is much more problematic as they exhibit a much more complex electronic structure. For example, transition metal complexes, low-spin open-shell systems, covalent bond breaking situations have all in common that they cannot be even qualitatively described by a single electronic configuration. It is now clear that the usual semilocal density-functional approximations of KS DFT fail to accurately describe these situations and WFT is king for the treatment of strongly correlated systems. + +In practice WFT uses a finite one-particle basis set (here denoted as $\basis$) to project the Schr\"odinger equation. The exact solution within the basis set is then provided by full configuration interaction (FCI) which consists in a linear-algebra problem with a dimension scaling exponentially with the system size. Due to this exponential growth of the FCI computational cost, introducing approximations is necessary, with at least two difficulties for strongly correlated systems: i) the qualitative description of the wave function is determined by a primary set of electronic configurations (whose size can scale exponentially in many cases) among which near degeneracies and/or strong interactions appear in the Hamiltonian matrix; ii) the quantitative description of the system requires to also account for weak correlation effects which involve many other electronic configurations with typically much smaller weights in the wave function. Addressing these two objectives is a rather complicated task for a given approximate WFT method, especially if one adds the requirement of satisfying formal properties, such as spin-multiplet degeneracy and size consistency. + +%To tackle this complicated problem, many methods have been proposed and an exhaustive review of the zoology of methods for strong correlation goes beyond the scope and purpose of this article. + +%To tackle this problem, many WFT methods have emerged which can be categorized in two branches: the single-reference (SR) +%and multi-reference (MR) methods. +%The SR methods rely on a single electronic configuration as a zeroth-order wave function, typically Hartree-Fock (HF). +%Then the electron correlation is introduced by increasing the rank of multiple hole-particle excitations, +%preferably treated in a coupled-cluster (CC) fashion for the sake of compactness of the wave function and extensivity of the computed energies. +%The advantage of these approaches rely on the rather straightforward way to improve the level of accuracy, +%which consists in increasing the rank of the excitation operators used to generate the CC wave function. +%Despite its appealing elegant simplicity, the computational cost of the CC methods increase drastically with the rank of the excitation +%operators, even if promising alternative approaches have been proposed using stochastic techniques\cite{Thom-PRL-10,ScoTho-JCP-17,SpeNeuVigFraTho-JCP-18,DeuEmiShePie-PRL-17,DeuEmiMagShePie-JCP-18,DeuEmiYumShePie-JCP-19} or symmetry-broken approaches\cite{QiuHenZhaScu-JCP-17,QiuHenZhaScu-JCP-18,GomHenScu-JCP-19}. +%In the MR approaches, the zeroth order wave function consists in a linear combination of Slater determinants which are supposed to concentrate most of strong interactions and near degeneracies inherent in the structure of the Hamiltonian for a strongly correlated system. The usual approach to build such a zeroth-order wave function is to perform a complete active space self consistent field (CASSCF) whose variational property prevent any divergence, and which can provide extensive energies. Of course, the choice of the active space is rather a subtle art and the CASSCF results might strongly depend on the level of chemical/physical knowledge of the user. +%On top of this zeroth-order wave function, weak correlation is introduced by the addition of other configurations through either configuration interaction\cite{WerKno-JCP-88,KnoWer-CPL-88} (MRCI) or perturbation theory (MRPT) and even coupled cluster (MRCC), which have their strengths and weaknesses, +%The advantage of MRCI approaches rely essentially in their simple linear parametrisation for the wave function together with the variational property of their energies, whose inherent drawback is the lack of size extensivity of their energies unless reaching the FCI limit. On the other hand, MRPT and MRCC can provide extensive energies but to the price of rather complicated formalisms, and these approaches might be subject to divergences and/or convergence problems due to the non linearity of the parametrisation for MRCC or a too poor choice of the zeroth-order Hamiltonian. +%A natural alternative is to combine MRCI and MRPT, which falls in the category of selected CI (SCI) which goes back to the late 60's and who has received a revival of interest and applications during the last decade \cite{BenErn-PhysRev-1969,WhiHac-JCP-1969,HurMalRan-1973,EvaDauMal-ChemPhys-83,Cim-JCP-1985,Cim-JCC-1987,IllRubRic-JCP-88,PovRubIll-TCA-92,BunCarRam-JCP-06,AbrSheDav-CPL-05,MusEngels-JCC-06,BytRue-CP-09,GinSceCaf-CJC-13,CafGinScemRam-JCTC-14,GinSceCaf-JCP-15,CafAplGinScem-arxiv-16,CafAplGinSce-JCP-16,SchEva-JCP-16,LiuHofJCTC-16,HolUmrSha-JCP-17,ShaHolJeaAlaUmr-JCTC-17,HolUmrSha-JCP-17,SchEva-JCTC-17,PerCle-JCP-17,OhtJun-JCP-17,Zim-JCP-17,LiOttHolShaUmr-JCP-2018,ChiHolOttUmrShaZim-JPCA-18,SceBenJacCafLoo-JCP-18,LooSceBloGarCafJac-JCTC-18,GarSceGinCaffLoo-JCP-18,SceGarCafLoo-JCTC-18,GarGinMalSce-JCP-16,LooBogSceCafJac-JCTC-19}. +%Among the SCI algorithms, the CI perturbatively selected iteratively (CIPSI) can be considered as a pioneer. The main idea of the CIPSI and other related SCI algorithms is to iteratively select the most important Slater determinants thanks to perturbation theory in order to build a MRCI zeroth-order wave function which automatically concentrate the strongly interacting part of the wave function. On top of this MRCI zeroth-order wave function, a rather simple MRPT approach is used to recover the missing weak correlation and the process is iterated until reaching a given convergence criterion. It is important to notice that in the SCI algorithms, neither the SCI or the MRPT are size extensive \textit{per se}, but the extensivity property is almost recovered by approaching the FCI limit. +%When the SCI are affordable, their clear advantage are that they provide near FCI wave functions and energies, whatever the level of knowledge of the user on the specific physical/chemical problem considered. The drawback of SCI is certainly their \textit{intrinsic} exponential scaling due to their linear parametrisation. Nevertheless, such an exponential scaling is lowered by the smart selection of the zeroth-order wave function together with the MRPT calculation. + +Beside the difficulties of accurately describing the molecular electronic structure within a given basis set, a crucial limitation of WFT methods is the slow convergence of the energies and properties with respect to the size of the basis set. As initially shown by the seminal work of Hylleraas\cite{Hyl-ZP-29} and further developed by Kutzelnigg and coworkers~\textit{et al.}\cite{Kut-TCA-85,KutKlo-JCP-91, NogKut-JCP-94}, the main convergence problem originates from the divergence of the Coulomb electron-electron interaction at the coalescence point, which induces a discontinuity in the first derivative of the exact wave function (the so-called electron-electron cusp). Describing such a discontinuity with an incomplete one-electron basis set is impossible and, as a consequence, the convergence of the computed energies and properties can be strongly affected. To attenuate this problem, extrapolation techniques have been developed, either based on a partial-wave expansion analysis~\cite{HelKloKocNog-JCP-97,HalHelJorKloKocOlsWil-CPL-98}, or more recently based on perturbative arguments\cite{IrmHulGru-arxiv-19}. A more rigorous approach to tackle the basis-set convergence problem is provided by the so-called R12 and F12 explicitly correlated methods\cite{Ten-TCA-12,TenNog-WIREs-12,HatKloKohTew-CR-12, KonBisVal-CR-12, GruHirOhnTen-JCP-17, MaWer-WIREs-18} which introduce a geminal function explicitly depending on the interelectronic distances ensuring the correct cusp condition in the wave function, and lead to a much faster convergence of the correlation energies than usual WFT methods. For instance, using the explicitly correlated version of coupled cluster with singles, doubles, and perturbative triples (CCSD(T)) in a triple-$\zeta$ quality basis set is equivalent to using a quintuple-$\zeta$ quality basis set with the usual CCSD(T) method\cite{TewKloNeiHat-PCCP-07}, although a computational overhead is introduced by the auxiliary basis set needed to compute the three- and four-electron integrals involved in F12 theory. In addition to the computational cost, a possible drawback of F12 theory is its rather complex formalism which requires non-trivial developments for adapting it to a new method. For strongly correlated systems, several multi-reference methods have been extended to explicitly correlation (see for instance Ref.~\onlinecite{Ten-CPL-07,ShiWer-JCP-10,TorKniWer-JCP-11,DemStanMatTenPitNog-PCCP-12,GuoSivValNee-JCP-17}), including approaches based on the so-called universal F12 theory which are potentially applicable to any electronic-structure computational methods~\cite{TorVal-JCP-09,KonVal-JCP-11,HauMaoMukKlo-CPL-12,BooCleAlaTew-JCP-12}. + +An alternative way to improve the convergence towards the complete-basis-set (CBS) limit is to treat the short-range correlation effects within DFT and to use WFT methods to deal only with the long-range and/or strong-correlation effects. A rigorous approach achieving this mixing of DFT and WFT is range-separated DFT (RSDFT) (see Ref.~\onlinecite{TouColSav-PRA-04} and references therein) which relies on a splitting of the Coulomb electron-electron interaction in terms of the interelectronic distance thanks to a range-separation parameter $\mu$. The advantage of this approach is at least two-fold: i) the DFT part deals primarily with the short-range part of the Coulomb interaction, and consequently the usual semilocal density-functional approximations are more accurate than for standard KS DFT; ii) the WFT part deals only with a smooth non-divergent interaction, and consequently the wave function has no electron-electron cusp\cite{GorSav-PRA-06} and the basis-set convergence is much faster\cite{FraMusLupTou-JCP-15}. A number of approximate RSDFT schemes have been developed involving single-reference \cite{AngGerSavTou-PRA-05, GolWerSto-PCCP-05, TouGerJanSavAng-PRL-09,JanHenScu-JCP-09, TouZhuSavJanAng-JCP-11, MusReiAngTou-JCP-15,KalTou-JCP-18,KalMusTou-JCP-19} or multi-reference \cite{LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, HedTouJen-JCP-18, FerGinTou-JCP-18} WFT methods. Nevertheless, there are still some open issues in RSDFT, such as remaining fractional-charge and fractional-spin errors in the short-range density functionals~\cite{MusTou-MP-17} or the dependence of the quality of the results on the value of the range-separation parameter $\mu$. +% which can be seen as an empirical parameter. + +Building on the development of RSDFT, a possible solution to the basis-set convergence problem has been recently proposed by some of the present authors~\cite{GinPraFerAssSavTou-JCP-18} where RSDFT functionals are used to recover only the correlation effects outside a given basis set. The key point here is to realize that a wave function developed in an incomplete basis set is cuspless and could also come from a Hamiltonian with a non divergent electron-electron interaction. Therefore, a mapping with RSDFT can be introduced through the introduction of an effective non-divergent interaction representing the usual Coulomb electron-electron interaction projected in an incomplete basis set. First applications to weakly correlated molecular systems have been successfully carried out~\cite{LooPraSceTouGin-JCPL-19}, together with extensions of this approach to the calculations of excitation energies~\cite{GinSceTouLoo-JCP-19} and ionization potentials~\cite{LooPraSceGinTou-ARX-19}. The goal of the present work is to further develop this approach for the description of strongly correlated systems. The paper is organized as follows. In Section \ref{sec:theory} we recall the mathematical framework of the basis-set correction and we present the extension for strongly correlated systems. In particular, we focus on imposition of two important formal properties: size-consistency and spin-multiplet degeneracy. +Then, in Section \ref{sec:results} we apply the method to the calculation of the potential energy curves of the C$_2$, N$_2$, O$_2$, F$_2$, and H$_{10}$ molecules up to the dissociation limit, representing prototypes of strongly correlated systems. Finally, we conclude in Section \ref{sec:conclusion}. + +%%%%%%%%%%%%%%%%%%%%%%%% +\section{Theory} +\label{sec:theory} +%%%%%%%%%%%%%%%%%%%%%%%% +As the theory of the basis-set correction has been exposed in details in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we only briefly recall the main equations and concepts needed for this study in Section \ref{sec:basic}, \ref{sec:wee}, and \ref{sec:mur}. More specifically, in Section \ref{sec:basic} we recall the basic mathematical framework of the present theory by introducing the density functional complementary to a basis set $\Bas$. In Section \ref{sec:wee} we introduce the effective non-divergent interaction in the basis set $\Bas$, which leads us to the definition of the effective local range-separation parameter in Section \ref{sec:mur}. Then, in Section \ref{sec:functional} we expose the new approximate complementary functionals based on RSDFT. The generic form of such functionals is exposed in Section \ref{sec:functional_form}, their properties in the context of the basis-set correction are discussed in Section \ref{sec:functional_prop}, and the specific requirements for strong correlation are discussed in Section \ref{sec:requirements}. Finally, the actual form of the functionals used in this work are introduced in Section \ref{sec:final_def_func}. + +\subsection{Basic formal equations} +\label{sec:basic} + +The exact ground-state energy $E_0$ of a $N$-electron system can in principle be obtained in DFT by a minimization over $N$-electron density $\denr$ +\begin{equation} + \label{eq:levy} + E_0 = \min_{\den} \bigg\{ F[\den] + \int \d \br{} v_{\text{ne}} (\br{}) \denr \bigg\}, +\end{equation} +where $v_{ne}(\br{})$ is the nuclei-electron potential, and $F[\den]$ is the universal Levy-Lieb density functional written with the constrained search formalism as~\cite{Lev-PNAS-79,Lie-IJQC-83} +\begin{equation} + \label{eq:levy_func} + F[\den] = \min_{\Psi \rightarrow \den} \elemm{\Psi}{\kinop +\weeop }{\Psi}, +\end{equation} +where the notation $\Psi \rightarrow \den$ means that the wave function $\Psi$ yields the density $n$. The minimizing density $n_0$ in Eq.~\eqref{eq:levy} is the exact ground-state density. Nevertheless, in practical calculations, the accessible densities are necessarily restricted to the set of densities ``representable in a basis set $\Bas$'', i.e. densities coming from wave functions expandable in the Hilbert space generated by the basis set $\Bas$. In the following, we always implicitly consider only such densities representable in $\Bas$. With this restriction, Eq.~\eqref{eq:levy} gives then an upper bound $E_0^\Bas$ of the exact ground-state energy. Since the density has a fast convergence with the size of the basis set, this restriction is a rather weak one and we can consider that $E_0^\Bas$ is a very good approximation to the exact ground-state energy: $E_0^\Bas \approx E_0$. + +In the present context, it is important to notice that in the definition of Eq.~\eqref{eq:levy_func} the wave functions $\Psi$ involved have no restriction to a finite basis set, i.e. they should be expanded in a complete basis set. In Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, it was then propose to decompose $F[\den]$ as +\begin{equation} + \label{eq:def_levy_bas} + F[\den] = \min_{\wf{}{\Bas} \rightarrow \den} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\den}, +\end{equation} +where $\wf{}{\Bas}$ are wave functions expandable in the Hilbert space generated by $\basis$, and $\efuncden{\den}$ is the density functional complementary to the basis set $\Bas$ defined as +\begin{equation} + \begin{aligned} + \efuncden{\den} = \min_{\Psi \rightarrow \den} \elemm{\Psi}{\kinop +\weeop }{\Psi}   + - \min_{\Psi^{\Bas} \rightarrow \den} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}}. + \end{aligned} +\end{equation} +Introducing the decomposition in Eq. \eqref{eq:def_levy_bas} back into Eq.~\eqref{eq:levy}, we arrive at the following expression for $E_0^\Bas$ +\begin{eqnarray} + \label{eq:E0basminPsiB} + E_0^\Bas &=& \min_{\Psi^{\Bas}} \bigg\{ \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\den_{{\Psi^{\Bas}}}} +\nonumber\\ +&&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + \int \d \br{} v_{\text{ne}} (\br{}) \den_{\Psi^{\Bas}}(\br{}) \bigg\}, +\end{eqnarray} +where the minimization is only over wave functions $\wf{}{\Bas}$ restricted to the basis set $\basis$ and $\den_{{\Psi^{\Bas}}}(\br{})$ refers to the density extracted from $\wf{}{\Bas}$. Therefore, with Eq.~\eqref{eq:E0basminPsiB}, one can properly combine a WFT calculation in a finite basis set with a density functional accounting for the correlation effects not included in the basis set. + +As a simple non-self-consistent version of this approach, we can approximate the minimizing wave function $\Psi^{\Bas}$ in Eq.~\eqref{eq:E0basminPsiB} by the ground-state FCI wave function $\psifci$ within $\Bas$, and we then obtain the following approximation for the exact ground-state energy (see Eqs. (12)-(15) of Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}) +\begin{equation} + \label{eq:e0approx} + E_0 \approx E_0^\Bas \approx \efci + \efuncbasisFCI, +\end{equation} +where $\efci$ and $n_\text{FCI}^\Bas$ are the ground-state FCI energy and density, respectively. As it was originally shown in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18} and further emphasized in Ref. \onlinecite{LooPraSceTouGin-JCPL-19,GinSceTouLoo-JCP-19}, the main role of $\efuncbasisFCI$ is to correct for the basis-set incompleteness error, a large part of which originating from the lack of electron-electron cusp in the wave function expanded in an incomplete basis set. The whole purpose of this work is to determine approximations for $\efuncbasisFCI$ which are suitable for treating strong correlation regimes. Two requirements on the approximations for this purpose are i) size consistency and ii) spin-multiplet degeneracy. + +\subsection{Definition of an effective interaction within $\Bas$} +\label{sec:wee} +As originally shown by Kato\cite{Kat-CPAM-57}, the cusp in the exact wave function originates from the divergence of the Coulomb interaction at the coalescence point. Therefore, a cuspless wave function $\wf{}{\Bas}$ could also be obtained from a Hamiltonian with a non-divergent electron-electron interaction. In other words, the impact of the incompleteness of a finite basis set can be understood as the removal of the divergence of the usual Coulomb electron-electron interaction at the coalescence point. + +As originally derived in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18} (Section D and Appendices), one can obtain an effective non-divergent electron-electron interaction, here referred to as $\wbasis$, which reproduces the expectation value of the Coulomb electron-electron interaction operator over a given wave function $\wf{}{\Bas}$. As we are interested in the behaviour at the coalescence point, we focus on the opposite-spin part of the electron-electron interaction. More specifically, the effective electron-electron interaction associated to a given wave function $\wf{}{\Bas}$ is defined as +\begin{equation} + \label{eq:wbasis} + \wbasis = + \begin{cases} + \fbasis /\twodmrdiagpsi, & \text{if $\twodmrdiagpsi \ne 0$,} +\\ + \infty, & \text{otherwise,} + \end{cases} +\end{equation} +where $\twodmrdiagpsi$ is the opposite-spin pair density associated with $\wf{}{\Bas}$ +\begin{equation} + \twodmrdiagpsi = \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}, +\end{equation} +and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$ its associated tensor in a basis of spatial orthonormal orbitals $\{\SO{p}{}\}$, and $\fbasis$ is +\begin{equation} + \label{eq:fbasis} + \fbasis + = \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2}, +\end{equation} +with the usual two-electron Coulomb integrals $\V{pq}{rs}=\langle pq | rs \rangle$. +With such a definition, one can show that $\wbasis$ satisfies +\begin{eqnarray} + \frac{1}{2}\iint \dr{1} \dr{2} \wbasis \twodmrdiagpsi = +\nonumber\\ +\frac{1}{2} \iint \dr{1} \dr{2} \frac{\twodmrdiagpsi}{|\br{1}-\br{2}|}. +\end{eqnarray} +As shown in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, the effective interaction $\wbasis$ is necessarily finite at coalescence for an incomplete basis set, and tends to the usual Coulomb interaction in the CBS limit for any choice of wave function $\psibasis$, i.e. +\begin{equation} + \label{eq:cbs_wbasis} + \lim_{\Bas \rightarrow \text{CBS}} \wbasis = \frac{1}{|\br{1}-\br{2}|},\quad \forall\,\psibasis. +\end{equation} +The condition in Eq.~\eqref{eq:cbs_wbasis} is fundamental as it guarantees the correct behavior of the theory in the CBS limit. + +\subsection{Definition of a local range-separation parameter} +\label{sec:mur} +\subsubsection{General definition} +As the effective interaction within a basis set, $\wbasis$, is non divergent, it ressembles the long-range interaction used in RSDFT +\begin{equation} + \label{eq:weelr} + w_\text{ee}^{\lr}(\mu;r_{12}) = \frac{\text{erf}\big(\mu \,r_{12} \big)}{r_{12}}, +\end{equation} +where $\mu$ is the range-separation parameter. As originally proposed in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we make the correspondance between these two interactions by using the local range-separation parameter $\murpsi$ +\begin{equation} + \label{eq:def_mur} + \murpsi = \frac{\sqrt{\pi}}{2} \wbasiscoal, +\end{equation} +such that the interactions coincide at the electron-electron colescence point for each $\br{}$ +\begin{equation} + w_\text{ee}^{\lr}(\murpsi;0) = \wbasiscoal, \quad \forall \, \br{}. +\end{equation} +Because of the very definition of $\wbasis$, one has the following property in the CBS limit (see Eq.~\eqref{eq:cbs_wbasis}) +\begin{equation} + \label{eq:cbs_mu} + \lim_{\Bas \rightarrow \text{CBS}} \murpsi = \infty, \quad \forall \,\psibasis, +\end{equation} +which is again fundamental to guarantee the correct behavior of the theory in the CBS limit. + +\subsubsection{Frozen-core approximation} +As all WFT calculations in this work are performed within the frozen-core approximation, we use the valence-only version of the various quantities needed for the complementary basis functional introduced in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}. We partition the basis set as $\Bas = \Cor \bigcup \BasFC$, where $\Cor$ and $\BasFC$ are the sets of core and active orbitals, respectively, and define the valence-only local range-separation parameter as +\begin{equation} + \label{eq:def_mur_val} + \murpsival = \frac{\sqrt{\pi}}{2} \wbasiscoalval{}, +\end{equation} +where $\wbasisval$ is the valence-only effective interaction defined as +\begin{equation} + \label{eq:wbasis_val} + \wbasisval = + \begin{cases} + \fbasisval /\twodmrdiagpsival, & \text{if $\twodmrdiagpsival \ne 0$,} +\\ + \infty, & \text{otherwise,} + \end{cases} +\end{equation} +where $\fbasisval$ and $\twodmrdiagpsival$ are defined as +\begin{equation} + \label{eq:fbasis_val} + \fbasisval + = \sum_{pq\in \Bas} \sum_{rstu \in \BasFC} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2}, +\end{equation} +and +\begin{equation} + \label{eq:twordm_val} + \twodmrdiagpsival = \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}. +\end{equation} +Notice the restrictions of the sums to the set of active orbitals in Eqs.~\eqref{eq:fbasis_val} and \eqref{eq:twordm_val}. +It is noteworthy that, with the present definition, $\wbasisval$ still tends to the usual Coulomb interaction as $\Bas \to \CBS$. + +\subsection{Generic form and properties of the approximations for $\efuncden{\den}$ } +\label{sec:functional} + +\subsubsection{Generic form of the approximate functionals} +\label{sec:functional_form} + +As originally proposed and motivated in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the complementary basis functional $\efuncden{\den}$ by using the so-called correlation energy functional with multideterminant reference (ECMD) introduced by Toulouse \textit{et al.}~\cite{TouGorSav-TCA-05}. Following the recent work in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}, we propose to use a Perdew-Burke-Ernzerhof (PBE)-like functional which uses the total density $\denr$, the spin polarisation $\zeta(\br{})=[n_\uparrow(\br{})-n_\downarrow(\br{})]/\denr$, the reduced density gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$, and the on-top pair density $\ntwo(\br{})\equiv \ntwo(\br{},\br{})$. In the present work, all these quantities are computed with the same wave function $\psibasis$ used to define $\mur \equiv\murpsi$. Therefore, a given approximation X of $\efuncden{\den}$ will have the following generic local form +\begin{equation} + \begin{aligned} + \label{eq:def_ecmdpbebasis} + &\efuncdenpbe{\argebasis} = \;\;\;\;\;\;\;\; \\ &\int \d\br{} \,\denr \ecmd(\argrebasis), + \end{aligned} +\end{equation} +where $\ecmd(\argecmd)$ is the correlation energy per particle taken as +\begin{equation} + \label{eq:def_ecmdpbe} + \ecmd(\argecmd) = \frac{\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)}{1+ \beta(\argepbe) \; \mu^3}, +\end{equation} +with +\begin{equation} + \label{eq:def_beta} + \beta(\argepbe) = \frac{3}{2\sqrt{\pi}(1 - \sqrt{2})}\frac{\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)}{\ntwo/\den}, +\end{equation} +where $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ is the usual PBE correlation energy per particle~\cite{PerBurErn-PRL-96}. Before introducing the different flavors of approximate functionals that we will use here (see Section~\ref{sec:def_func}), we would like to give some motivations for this choice of functional form. + +The functional form of $\ecmd(\argecmd)$ in Eq.~\ref{eq:def_ecmdpbe} has been originally proposed in Ref.~\onlinecite{FerGinTou-JCP-18} in the context of RSDFT. In the $\mu\to 0$ limit, it reduces to the usual PBE correlation functional +\begin{equation} + \lim_{\mu \rightarrow 0} \ecmd(\argecmd) = \varepsilon_{\text{c}}^{\text{PBE}}(\argepbe), +\end{equation} +which is relevant in the weak-correlation (or high-density) limit. In the large-$\mu$ limit, it behaves as +\begin{equation} + \label{eq:lim_mularge} + \ecmd(\argecmd) \isEquivTo{\mu\to\infty} \frac{2\sqrt{\pi}(1 - \sqrt{2})}{3 \mu^3} \frac{\ntwo}{n}, +\end{equation} +which is the exact large-$\mu$ behavior of the exact ECMD correlation energy~\cite{PazMorGorBac-PRB-06,FerGinTou-JCP-18}. Of course, for a specific system, the large-$\mu$ behavior will be exact only if one uses for $n_2$ the \textit{exact} on-top pair density of this system. This large-$\mu$ limit in Eq.~\eqref{eq:lim_mularge} is relevant in the strong-correlation (or low-density) limit. In the context of RSDFT, some of the present authors have illustrated in Ref.~\onlinecite{FerGinTou-JCP-18} that the on-top pair density involved in Eq. \eqref{eq:def_ecmdpbe} plays indeed a crucial role when reaching the strong-correlation regime. The importance of the on-top pair density in the strong-correlation regime have been also recently acknowledged by Gagliardi and coworkers~\cite{CarTruGag-JPCA-17} and Pernal and coworkers\cite{GritMeePer-PRA-18}. + +Note also that $\ecmd(\argecmd)$ vanishes when $\ntwo$ vanishes +\begin{equation} + \label{eq:lim_n2} + \lim_{\ntwo \rightarrow 0} \ecmd(\argecmd) = 0, +\end{equation} +which is expected for systems with a vanishing on-top pair density, such as the totally dissociated H$_2$ molecule which is the archetype of strongly correlated systems. Finally, the function $\ecmd(\argecmd)$ vanishes when $\mu \rightarrow \infty$ like all RSDFT short-range functionals \begin{equation} + \label{eq:lim_muinf} + \lim_{\mu \rightarrow \infty} \ecmd(\argecmd) = 0. +\end{equation} + +\subsubsection{Properties of approximate functionals} +\label{sec:functional_prop} + +Within the definitions of Eqs.~\eqref{eq:def_mur} and \eqref{eq:def_ecmdpbebasis}, any approximate complementary basis functional $\efuncdenpbe{\argebasis}$ satisfies two important properties. + +First, thanks to the properties in Eq.~\eqref{eq:cbs_mu} and~\eqref{eq:lim_muinf}, $\efuncdenpbe{\argebasis}$ vanishes in the CBS limit, whatever the wave function $\psibasis$ used to define the local range-separation parameter $\mu(\br{})$, +\begin{equation} + \label{eq:lim_ebasis} + \lim_{\basis \rightarrow \text{CBS}} \efuncdenpbe{\argebasis} = 0, \quad \forall\, \psibasis, +\end{equation} +which guarantees an unaltered CBS limit. + +Second, the fact that $\efuncdenpbe{\argebasis}$ vanishes for systems with vanishing on-top pair density guarantees the correct limit for one-electron systems and for the stretched H$_2$ molecule. This property is guaranteed independently by i) the definition of the effective interaction $\wbasis$ (see Eq.~\eqref{eq:wbasis}) together with the condition in Eq.~\eqref{eq:lim_muinf}, ii) the fact that $\ecmd(\argecmd)$ vanishes when the on-top pair density vanishes (see Eq.~\eqref{eq:lim_n2}). + +\subsection{Requirements for the approximate functionals in the strong-correlation regime} +\label{sec:requirements} + +\subsubsection{Requirements: size-consistency and spin-multiplet degeneracy} + +An important requirement for any electronic-structure method is size-consistency, i.e. the additivity of the energies of non-interacting fragments, which is mandatory to avoid any ambiguity in computing interaction energies. When two subsystems $A$ and $B$ dissociate in closed-shell systems, as in the case of weak intermolecular interactions for instance, spin-restricted Hartree-Fock (RHF) is size-consistent. When the two subsystems dissociate in open-shell systems, such as in covalent bond breaking, it is well known that the RHF approach fails and an alternative is to use a complete-active-space (CAS) wave function which, provided that the active space has been properly chosen, leads to additive energies. + +Another important requirement is spin-multiplet degeneracy, i.e. the independence of the energy with respect to the $S_z$ component of a given spin state, which is also a property of any exact wave function. Such a property is also important in the context of covalent bond breaking where the ground state of the supersystem $A+B$ is generally low spin while the ground states of the fragments $A$ and $B$ are high spin and can have multiple $S_z$ components. + +\subsubsection{Condition for the functional $\efuncdenpbe{\argebasis}$ to obtain spin-multiplet degeneracy} + +A sufficient condition to achieve spin-multiplet degeneracy is to eliminate all dependencies on $S_z$, which in the case of the functional $\ecmd(\argecmd)$ means removing the dependency on the spin polarization $\zeta(\br{})$ use the PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ (see Eq. \eqref{eq:def_ecmdpbe}). + +It has been proposed to replace in functionals the dependency on the spin polarization by the dependency on the on-top pair density. Most often, it is done by introducing an effective spin polarisation~\cite{MosSan-PRA-91,BecSavSto-TCA-95,Sav-INC-96a,Sav-INC-96,MieStoSav-MP-97,TakYamYam-CPL-02,TakYamYam-IJQC-04,GraCre-MP-05,TsuScuSav-JCP-10,LimCarLuoMaOlsTruGag-JCTC-14,GarBulHenScu-JCP-15,GarBulHenScu-PCCP-15,CarTruGag-JCTC-15,GagTruLiCarHoyBa-ACR-17} (see, also, Refs.~\onlinecite{PerSavBur-PRA-95,StaDav-CPL-01}) +\begin{equation} + \label{eq:def_effspin} + \tilde{\zeta}(n,n_{2}) = +% \begin{cases} + \sqrt{ 1 - 2 \; n_{2}/n^2 }, +% 0 & \text{otherwise.} +% \end{cases} +\end{equation} +expressed as a function of the density $n$ and the on-top pair density $n_2$, calculated from a given wave function. The advantage of this approach is that this effective spin polarisation $\tilde{\zeta}$ is independent from $S_z$, since the on-top pair density is independent from $S_z$. Nevertheless, the use of $\tilde{\zeta}$ in Eq.~\eqref{eq:def_effspin} presents the disadvantage that since this expression was derived for a single-determinant wave function, and it does not appear well justified to use it for a multideterminant wave function as well. In particular, for a multideterminant wave function, it may happen that $1 - 2 \; n_{2}/n^2 < 0 $ and thus in this cas Eq.~\eqref{eq:def_effspin} gives a complex value of $\tilde{\zeta}$~\cite{BecSavSto-TCA-95}. + +%The advantage of this approach are at least two folds: i) the effective spin polarisation $\tilde{\zeta}$ is independent from $S_z$ since the on-top pair density is independent from $S_z$, ii) it introduces an indirect dependency on the on-top pair density of the wave function $\psibasis$ which usually improves the treatment of strong correlation. +%Nevertheless, the use of $\tilde{\zeta}$ presents several disadvantages as it can become complex when $1 - 2 \; n_{2}/n^2 < 0 $ and also +%the formula of equation \eqref{eq:def_effspin} is exact only when the density $n$ and on-top pair density $\ntwo^{\psibasis}$ are obtained from a single determinant\cite{PerSavBur-PRA-95}, but it is applied to multi configurational wave functions. + +An alternative way to eliminate the $S_z$ dependency is to simply set $\zeta=0$, i.e. to always use the spin-unpolarized functional. This lowers the accuracy for open-shell systems at $\mu=0$, i.e. for the usual PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$. Nevertheless, we argue that, for $\mu$ sufficiently large, it is a viable option. Indeed, the purpose of introducing the spin polarisation in semilocal density-functional approximations is to mimic the exact on-top pair density~\cite{PerSavBur-PRA-95}, but our functional $\ecmd(\argecmd)$ already explicitly depends on the on-top pair density (see Eq.~\eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}). The dependencies on $\zeta$ and $n_2$ can thus be expected to be largely redundant. Therefore, we propose here to also test the $\ecmd$ functional with \textit{a zero spin polarisation}. This ensures a $S_z$ independence and, as will be numerically shown, very weakly affects the accuracy of the functional. + + +\subsubsection{Conditions for size consistency} + +Since $\efuncdenpbe{\argebasis}$ is a single integral over $\mathbb{R}^3$ of local quantities ($n(\br{})$, $\zeta(\br{})$, $s(\br{})$, $n_2(\br{})$,$\mu(\br{})$), in the case of non-overlapping fragments $A\ldots B$ it can be written as the sum of two local contributions: one coming from the integration over the region of the subsystem $A$ and the other one from the region of the subsystem $B$. Therefore, a sufficient condition for the size consistency is that these local quantities coincide in the isolated systems and in the subsystems of the supersystem $A\ldots B$. Since these local quantities are calculated from the wave function $\psibasis$, a sufficient condition is that the wave function is multiplicatively separable in the limit of non-interacting fragments, i.e. $\Psi_{A\ldots B}^{\basis} = \Psi_A^{\basis} \Psi_B^{\basis}$. In the case where the two subsystems $A$ and $B$ dissociate in closed-shell systems, a simple RHF wave function ensures this property, but when one or several covalent bonds are broken, the use of a properly chosen CAS wave function is sufficient to recover this property. The condition for the active space involved in the CAS wave function is that it has to lead to size-consistent energies in the limit of dissociated fragments. + + +\subsection{Different types of approximations for the functional} +\label{sec:final_def_func} +\subsubsection{Definition of the protocol to design functionals} + +As the present work focusses on the strong correlation regime, we propose here to investigate only approximate functionals which are $S_z$ independent and size-consistent in the case of covalent bond breaking. Therefore, the wave functions $\psibasis$ used throughout this paper are CAS wave functions in order to ensure size consistency of all local quantities. The difference between the different flavors of functionals are only on i) the type of spin polarisation used, and ii) the type of on-top pair density used. + +Regarding the spin polarisation that enters into $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$, two different types of $S_z$-independent formulations are used: i) the \textit{effective} spin polarization $\tilde{\zeta}$ defined in Eq.~\eqref{eq:def_effspin} and calculated from the CAS wave function, and ii) a \textit{zero} spin polarization. + +Regarding the on-top pair density entering in Eq.~\eqref{eq:def_beta}, we use two different approximations. The first one is based on the uniform-electron gas (UEG) and reads +\begin{equation} + \label{eq:def_n2ueg} + \ntwo^{\text{UEG}}(n,\zeta) \approx n^2\big(1-\zeta^2\big)g_0(n), +\end{equation} +where the pair-distribution function $g_0(n)$ is taken from Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. As the spin polarization appears in Eq.~\eqref{eq:def_n2ueg}, we use the effective spin polarization $\tilde{\zeta}$ of Eq.~\eqref{eq:def_effspin} in order to ensure $S_z$ independence. Thus, $\ntwo^{\text{UEG}}$ will depend indirectly on the on-top pair density of the CAS wave function through $\tilde{\zeta}$. + +Another approach to approximate the exact on top pair density consists in using directly the on-top pair density of the CAS wave function. Following the work of some of the previous authors~\cite{FerGinTou-JCP-18,GinSceTouLoo-JCP-19} we introduce the extrapolated on-top pair density $\ntwoextrap$ as +\begin{equation} + \label{eq:def_n2extrap} + \ntwoextrap(\ntwo,\mu) = \bigg( 1 + \frac{2}{\sqrt{\pi}\mu} \bigg)^{-1} \; \ntwo +\end{equation} +which directly follows from the large-$\mu$ extrapolation of the exact on-top pair density derived by Gori-Giorgi and Savin\cite{GorSav-PRA-06} in the context of RSDFT. When using $\ntwoextrap(\ntwo,\mu)$ in a functional, we will simply refer it as ``ot''. + + +\subsubsection{Definition of functionals with good formal properties} +\label{sec:def_func} + +We define the following functionals:\\ + +i) PBE-UEG-$\tilde{\zeta}$ which uses the UEG on-top pair density defined in Eq.~\eqref{eq:def_n2ueg} and the effective spin polarization of Eq.~\eqref{eq:def_effspin} +\begin{equation} + \label{eq:def_pbeueg} + \begin{aligned} + \bar{E}^\Bas_{\pbeuegXi} = \int \d\br{} \,\denr \ecmd(\argrpbeuegXi), + \end{aligned} +\end{equation} +ii) PBE-ot-$\tilde{\zeta}$ which uses the on-top pair density of Eq.~\eqref{eq:def_n2extrap} +\begin{equation} + \label{eq:def_pbeueg} + \begin{aligned} + \bar{E}^\Bas_{\pbeontXi} = \int \d\br{} \,\denr \ecmd(\argrpbeontXi), + \end{aligned} +\end{equation} +iii) PBE-ot-$0{\zeta}$ where uses zero spin polarization and the on-top pair density of Eq.~\eqref{eq:def_n2extrap} +\begin{equation} + \label{eq:def_pbeueg} + \begin{aligned} + \bar{E}^\Bas_{\pbeontns} = \int \d\br{} \,\denr \ecmd(\argrpbeontns). + \end{aligned} +\end{equation} + +%%%%%%%%%%%%%%%%%%%%%%%% +\section{Results for the C$_2$, N$_2$, O$_2$, F$_2$, and H$_{10}$ potential energy curves} +\label{sec:results} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{figure*} + \includegraphics[width=0.45\linewidth]{data/H10/DFT_vdzE_relat.eps} + \includegraphics[width=0.45\linewidth]{data/H10/DFT_vdzE_relat_zoom.eps} + \includegraphics[width=0.45\linewidth]{data/H10/DFT_vtzE_relat.eps} + \includegraphics[width=0.45\linewidth]{data/H10/DFT_vtzE_relat_zoom.eps} + \includegraphics[width=0.45\linewidth]{data/H10/DFT_vqzE_relat.eps} + \includegraphics[width=0.45\linewidth]{data/H10/DFT_vqzE_relat_zoom.eps} + \caption{ + Potential energy curves of the H$_{10}$ chain with equally distant atoms calculated with MRCI+Q and basis-set corrected MRCI+Q using the 1) cc-pVDZ, 2) cc-pVTZ, and 3) cc-pVQZ basis sets. The MRCI+Q energies and the estimated exact energies are from Ref. \onlinecite{h10_prx}. + \label{fig:H10}} +\end{figure*} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\subsection{Computational details} + +The purpose of the present paper being the study of the basis-set correction in the regime of strong correlation, we study the potential energy curves up to the dissociation limit of a H$_{10}$ chain with equally distant atoms and the C$_2$, N$_2$, O$_2$, and F$_2$ molecules. In a given basis set, in order to compute the approximation of the exact ground-state energy using Eq.~\eqref{eq:e0approx}, one needs an approximation to both the FCI energy $\efci$ and the basis-set correction $\efuncbasisFCI$. + +In the case of the C$_2$, N$_2$, O$_2$, and F$_2$ molecules, approximations to the FCI energies are obtained using converged frozen-core (1s orbitals are kept frozen) CIPSI calculations and the extrapolation scheme for the perturbative correction of Umrigar \textit{et. al.} (see Refs. \onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJac-JCTC-19, QP2} for more details) using the Quantum Package software~\cite{QP2}. The estimated exact potential energy curves are obtained from Ref. \onlinecite{LieCle-JCP-74a}. For all geometries and basis sets, the error with respect to the exact FCI energies are estimated to be below 0.5 mH. In the case of the H$_{10}$ chain, the approximation to the FCI energies together with the estimated exact potential energy curves are obtained from the data of Ref. \onlinecite{h10_prx} where the authors performed MRCI+Q calculations with a minimal valence active space as reference (see below for the description of the active space). + +Regarding the complementary basis functional, we first perform full-valence complete-active-space self-consistent-field (CASSCF) calculations with the GAMESS-US software\cite{gamess} to obtain the wave function $\psibasis$. Then, all density-like quantities involved in the functional (density $n(\br{})$, spin polarization $\zeta(\br{})$, reduced density gradient $s(\br{})$, and on-top pair density $n_2(\br{})$) together with the local range-separation function $\mu(\br{})$ of Eq.~\eqref{eq:def_mur} are calculated with this full-valence CASSCF wave function. These CASSCF wave functions correspond to the following active spaces: 10 electrons in 10 orbitals for H$_{10}$, 8 electrons in 8 electrons for C$_2$, 10 electrons in 8 orbitals for N$_2$, 12 electrons in 8 orbitals for O$_2$, and 14 electrons in 8 orbitals for F$_2$. + +Also, as the frozen core approximation is used in all our CIPSI calculations, we use the corresponding valence-only complementary functionals. Therefore, all density-like quantities exclude any contribution from the core 1s orbitals, and the range-separation function is taken as the one defined in Eq.~\eqref{eq:def_mur_val}. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{table*} +\label{tab:d0} +\caption{Atomization energies $D_0$ (in mH) and associated errors (in square brackets) with respect to the estimated exact values computed at different approximation levels with different basis sets.} +\begin{ruledtabular} +\begin{tabular}{llcccc} + + System & Basis set & MRCI+Q$^a$ & (MRCI+Q)+$\pbeuegXi$ & (MRCI+Q)+$\pbeontXi$ & (MRCI+Q)+$\pbeontns$ \\ +\hline +H$_{10}$ & cc-pVDZ & 622.1 [43.3] & 642.6 [22.8] & 649.2 [16.2] & 649.5 [15.9] \\ + & cc-pVTZ & 655.2 [10.2] & 661.9 [3.5] & 666.0 [-0.6] & 666.0 [-0.6] \\ + & cc-pVQZ & 661.2 [4.2] & 664.1 [1.3] & 666.4 [-1.0] & 666.5 [-1.1] \\[0.1cm] +%\cline{2-6} + &\multicolumn{5}{c}{Estimated exact$^a$: 665.4} \\[0.2cm] +\hline +System & Basis set & FCI & FCI+$\pbeuegXi$ & FCI+$\pbeontXi$ & FCI+$\pbeontns$\\ +\hline +C$_2$ & aug-cc-pVDZ & 204.6 [29.5] & 218.0 [16.1] & 217.4 [16.7] & 217.0 [17.1] \\ + & aug-cc-pVTZ & 223.4 [10.9] & 228.1 [6.0] & 228.6 [5.5] & 226.5 [5.6] \\[0.1cm] +%\cline{2-6} + & \multicolumn{5}{c}{Estimated exact$^b$: 234.1} \\[0.2cm] +\hline +System & Basis set & FCI & FCI+$\pbeuegXi$ & FCI+$\pbeontXi$ & FCI+$\pbeontns$\\ +\hline +N$_2$ & aug-cc-pVDZ & 321.9 [40.8] & 356.2 [6.5] & 355.5 [7.2] & 354.6 [8.1] \\ + & aug-cc-pVTZ & 348.5 [14.2] & 361.5 [1.2] & 363.5 [-0.5] & 363.2 [-0.3] \\[0.1cm] + & \multicolumn{5}{c}{Estimated exact$^b$: 362.7} \\[0.2cm] +\hline +System & Basis set & FCI & FCI+$\pbeuegXi$ & FCI+$\pbeontXi$ & FCI+$\pbeontns$\\ +\hline +O$_2$ & aug-cc-pVDZ & 171.4 [20.5] & 187.6 [4.3] & 187.6 [4.3] & 187.1 [4.8] \\ + & aug-cc-pVTZ & 184.5 [7.4] & 190.3 [1.6] & 191.2 [0.7] & 191.0 [0.9] \\[0.1cm] + & \multicolumn{5}{c}{Estimated exact$^b$: 191.9} \\[0.2cm] +\hline +System & Basis set & FCI & FCI+$\pbeuegXi$ & FCI+$\pbeontXi$ & FCI+$\pbeontns$\\ +\hline +F$_2$ & aug-cc-pVDZ & 49.6 [12.6] & 54.8 [7.4] & 54.9 [7.3] & 54.8 [7.4] \\ + & aug-cc-pVTZ & 59.3 [2.9] & 61.2 [1.0] & 61.5 [0.7] & 61.5 [0.7] \\[0.1cm] + & \multicolumn{5}{c}{Estimated exact$^b$: 62.2} \\ +\end{tabular} +\end{ruledtabular} +\begin{flushleft} +\vspace{-0.2cm} +$^a$ From Ref. \onlinecite{h10_prx}. \\ +$^b$ From the extrapolated valence-only non-relativistic calculations of Ref. \onlinecite{BytLaiRuedenJCP05}. +\end{flushleft} + +\label{tab:extensiv_closed} +\end{table*} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + +\subsection{Dissociation of the H$_{10}$ chain with equally distant atoms} + + +The study of the H$_{10}$ chain with equally distant atoms is a good prototype of strongly-correlated systems as it consists in the simultaneous breaking of 10 covalent $\sigma$ bonds which all interact with each other. Also, being a relatively small system, benchmark calculations at near CBS values can be obtained (see Ref. \onlinecite{h10_prx} for a detailed study of this problem). + +We report in Figure \ref{fig:H10} the potential energy curves computed using the cc-pVXZ (X=D,T,Q) basis sets for different levels of approximations. The computation of the atomization energies $D_0$ at each level of theory used here is reported in Table \ref{tab:d0}. A general trend that can be observed from these data is that, in a given basis set, the quality of the potential energy cruves are globally improved by adding the basis-set correction, whatever the level of approximation used for the functional $\efuncbasisFCI$. Also, no bizarre behaviors are found when stretching the bonds, which shows that the functionals are robust when reaching the strong correlation regime. + +More quantitatively, the values of $D_0$ are within chemical accuracy (i.e., an error below 1.4 mH) from the cc-pVTZ basis set when using the PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ functionals, whereas such an accuracy is not reached at the cc-pVQZ basis set using standard MRCI+Q. + +Regarding in more details the performance of the different types of approximate functionals, the results show that PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ are very similar (the maximal difference on $D_0$ being 0.3 mH), and they give slightly more accurate results than PBE-UEG-$\tilde{\zeta}$. These findings bring two important clues on the role of the different physical ingredients used in the functionals: i) the explicit use of the on-top pair density coming from the CASSCF wave function (see Eq.~\eqref{eq:def_n2extrap}) is preferable to the use of the on-top pair density based on the UEG (see Eq.~\eqref{eq:def_n2ueg}); ii) removing the dependency on any kind of spin polarization does not lead to significant loss of accuracy provided that one uses a qualitatively correct on-top pair density. Point ii) is important as it shows that spin polarization in density-functional approximations essentially plays the same role as that of the on-top pair density. + +\subsection{Dissociation of the C$_2$, N$_2$, O$_2$, and F$_2$ molecules} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{figure*} + \includegraphics[width=0.45\linewidth]{data/C2/DFT_avdzE_relat.eps} + \includegraphics[width=0.45\linewidth]{data/C2/DFT_avdzE_relat_zoom.eps} + \includegraphics[width=0.45\linewidth]{data/C2/DFT_avtzE_relat.eps} + \includegraphics[width=0.45\linewidth]{data/C2/DFT_avtzE_relat_zoom.eps} + \caption{ + Potential energy curves of the C$_2$ molecule calculated with near-FCI and basis-set corrected near-FCI using the 1) aug-cc-pVDZ and 2) aug-cc-pVTZ basis sets. The estimated exact energies are from Ref. \onlinecite{LieCle-JCP-74a}. + \label{fig:C2}} +\end{figure*} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{figure*} + \includegraphics[width=0.45\linewidth]{data/N2/DFT_avdzE_relat.eps} + \includegraphics[width=0.45\linewidth]{data/N2/DFT_avdzE_relat_zoom.eps} + \includegraphics[width=0.45\linewidth]{data/N2/DFT_avtzE_relat.eps} + \includegraphics[width=0.45\linewidth]{data/N2/DFT_avtzE_relat_zoom.eps} + \caption{ + Potential energy curves of the N$_2$ molecule calculated with near-FCI and basis-set corrected near-FCI using the 1) aug-cc-pVDZ and 2) aug-cc-pVTZ basis sets. The estimated exact energies are from Ref. \onlinecite{LieCle-JCP-74a}. + \label{fig:N2}} +\end{figure*} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{figure*} + \includegraphics[width=0.45\linewidth]{data/O2/DFT_avdzE_relat.eps} + \includegraphics[width=0.45\linewidth]{data/O2/DFT_avdzE_relat_zoom.eps} +% \includegraphics[width=0.45\linewidth]{data/O2/DFT_avtzE_relat.eps} +% \includegraphics[width=0.45\linewidth]{data/O2/DFT_avtzE_relat_zoom.eps} + \caption{ + Potential energy curves of the O$_2$ molecule calculated with near-FCI and basis-set corrected near-FCI using the 1) aug-cc-pVDZ and 2) aug-cc-pVTZ basis sets. The estimated exact energies are from Ref. \onlinecite{LieCle-JCP-74a}. + \label{fig:O2}} +\end{figure*} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{figure*} + \includegraphics[width=0.45\linewidth]{data/F2/DFT_avdzE_relat.eps} + \includegraphics[width=0.45\linewidth]{data/F2/DFT_avdzE_relat_zoom.eps} + \includegraphics[width=0.45\linewidth]{data/F2/DFT_avtzE_relat.eps} + \includegraphics[width=0.45\linewidth]{data/F2/DFT_avtzE_relat_zoom.eps} + \caption{ + Potential energy curves of the F$_2$ molecule calculated with near-FCI and basis-set corrected near-FCI using the 1) aug-cc-pVDZ and 2) aug-cc-pVTZ basis sets. The estimated exact energies are from Ref. \onlinecite{LieCle-JCP-74a}. + \label{fig:F2}} +\end{figure*} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +The C$_2$, N$_2$, O$_2$ and F$_2$ molecules are complementary to the H$_{10}$ system for the present study as the level of strong correlation increases while stretching the bond similarly to the case of H$_{10}$, but in addition these molecules exhibit more important and versatile types of weak correlations due to the larger number of electrons. Indeed, the short-range correlation effects are known to play a strong differential effect on the computation of $D_0$, while the shape of the curve far from the equilibrium geometry is governed by dispersion interactions which are medium to long-range weak correlation effects. Also, O$_2$ has a triplet ground state and is therefore a good check for the performance of the dependence on the spin polarization of the different types of functionals proposed here. + +We report in Figure \ref{fig:C2}, \ref{fig:N2}, \ref{fig:O2}, and \ref{fig:F2} the potential energy curves computed using the aug-cc-pVDZ and aug-cc-pVTZ basis sets of C$_2$, N$_2$, O$_2$, and N$_2$, respectively, for different approximation levels. The computation of the atomization energies $D_0$ at each level of theory used here is reported in Table \ref{tab:d0}. + +Just as in the case of H$_{10}$, the quality of $D_0$ are globally improved by adding the basis-set correction and it is remarkable that PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ give very similar results. The latter observation confirms that the dependency on the on-top pair density allows one to remove the dependency of any kind of spin polarization for a quite wide range of electron density and also for a high-spin system like O$_2$. More quantitatively, an error below 1.0 mH on the estimated exact valence-only $D_0$ is found for N$_2$, O$_2$, and F$_2$ with the aug-cc-pVTZ basis set using the PBE-ot-$0{\zeta}$ functional, whereas such a result is far from reach within the same basis set at the near-FCI level. In the case of C$_2$ with the aug-cc-pVTZ basis set, an error of about 5.5 mH is found with respect to the estimated exact $D_0$. Such an error is remarkably large with respect to the other diatomic molecules studied here and might be associated to the level of strong correlation in the C$_2$ molecule. + +Regarding now the performance of the basis-set correction along the whole potential energy curve, it is interesting to notice that it fails to provide a noticeable improvement far from the equilibrium geometry. Acknowledging that the weak correlation effects in these regions are dominated by dispersion interactions which are long-range effects, the failure of the present approximations for the complementary basis functionals can be understood easily. Indeed, the whole scheme designed here is based on the physics of correlation near the electron-electron cusp: the local range-separation function $\mu(\br{})$ is designed by looking at the electron-electron coalescence point and the ECMD functionals are suited for short-range correlation effects. Therefore, the failure of the present basis-set correction to describe dispersion interactions is expected. + +\section{Conclusion} +\label{sec:conclusion} + +In the present paper we have extended the recently proposed DFT-based basis-set correction to strongly correlated systems. We studied the H$_{10}$, C$_2$, N$_2$, O$_2$, and F$_2$ molecules up to the dissociation limit at near-FCI level in increasing basis sets, and investigated how the basis-set correction affects the convergence toward the CBS limit of the potential energy curves of these molecular systems. + +The DFT-based basis-set correction relies on three aspects: i) the definition of an effective non-divergent electron-electron interaction obtained from the expectation value over a wave function $\psibasis$ of the Coulomb interaction projected into an incomplete basis set $\basis$; ii) the fitting of this effective interaction with the long-range interaction used in RS-DFT, iii) the use of a complementary correlation functional of RS-DFT. In the present paper, we investigated points i) and iii) in the context of strong correlation and focused on potential energy curves and atomization energies. More precisely, we proposed a new scheme to design functionals fulfilling a) spin-multiplet degeneracy, and b) size consistency. To fulfil such requirements we proposed to use CASSCF wave functions leading to size-consistent energies, and to develop functionals using only $S_z$-independent density-like quantities. + +The development of new $S_z$-independent and size-consistent functionals has lead us to investigate the role of two related quantities: the spin-polarization and the on-top pair density. One important result of the present study is that by using functionals \textit{explicitly} depending on the on-top pair density, one can avoid dependence to any form of spin polarization without loss of accuracy. This avoids the commonly used effective spin polarization calculated from a multideterminant wave function originally proposed in Ref. \onlinecite{BecSavSto-TCA-95} which has the disadvantage of possibly become complex-valued for some multideterminant wave functions. From a more fundamental aspect, this shows that the spin polarization in DFT-related frameworks only mimics the role of the on-top pair density. + +Regarding the results of the present approach, the basis-set correction systematically improves the near-FCI calculations in a given basis set. More quantitatively, it is shown that with only triple-zeta quality basis sets chemically accurate atomization energies $D_0$ are obtained for all systems but C$_2$, whereas the uncorrected near-FCI results are far from that accuracy within the same basis set. In the case of C$_2$, an error of 5.5 mH is obtained with respect to the estimated exact $D_0$, and we leave for further study the detailed investigation of the reasons of this relatively unusual poor performance of the basis-set correction. + +Also, it is shown that the basis-set correction gives substantial differential contribution to potential energy curves close to the equilibrium geometries, but at long interatomic distances it cannot recover the dispersion interactions missing because of the incompleteness of the basis set. This behaviour is actually expected as the dispersion interactions are long-range correlation effects and the present approach was designed to only recover electron correlation effects near the electron-electron coalescence. + +Finally, regarding the computational cost of the present approach, it should be stressed (see supplementary information) that it is minor with respect to WFT methods for all systems and basis sets studied here. We thus believe that this approach is a significant step towards calculations near the CBS limit for strongly correlated systems. + +\bibliography{srDFT_SC,biblio} + +\end{document} diff --git a/Revised_Manuscript/srDFT_SCNotes.bib b/Revised_Manuscript/srDFT_SCNotes.bib new file mode 100644 index 0000000..881f198 --- /dev/null +++ b/Revised_Manuscript/srDFT_SCNotes.bib @@ -0,0 +1,2 @@ +@CONTROL{REVTEX41Control} +@CONTROL{aip41Control,pages="0",title=""} diff --git a/Revised_Manuscript/srDFT_SC_jt.tex b/Revised_Manuscript/srDFT_SC_jt.tex new file mode 100644 index 0000000..08e9b3c --- /dev/null +++ b/Revised_Manuscript/srDFT_SC_jt.tex @@ -0,0 +1,968 @@ +\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1} +\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amsmath,amssymb,amsfonts,physics,mhchem,xspace} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{txfonts} + +\usepackage[ + colorlinks=true, + citecolor=blue, + breaklinks=true + ]{hyperref} +\urlstyle{same} + +\newcommand{\alert}[1]{\textcolor{red}{#1}} +\definecolor{darkgreen}{HTML}{009900} +\usepackage[normalem]{ulem} +\newcommand{\titou}[1]{\textcolor{red}{#1}} +\newcommand{\jt}[1]{\textcolor{purple}{#1}} +\newcommand{\manu}[1]{\textcolor{darkgreen}{#1}} +\newcommand{\toto}[1]{\textcolor{brown}{#1}} +\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}} +\newcommand{\trashJT}[1]{\textcolor{purple}{\sout{#1}}} +\newcommand{\trashMG}[1]{\textcolor{darkgreen}{\sout{#1}}} +\newcommand{\trashAS}[1]{\textcolor{brown}{\sout{#1}}} +\newcommand{\MG}[1]{\manu{(\underline{\bf MG}: #1)}} +\newcommand{\JT}[1]{\juju{(\underline{\bf JT}: #1)}} +\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}} +\newcommand{\AS}[1]{\toto{(\underline{\bf AS}: #1)}} + +\newcommand{\ie}{\textit{i.e.}} +\newcommand{\eg}{\textit{e.g.}} +\newcommand{\cdash}{\multicolumn{1}{c}{---}} +\newcommand{\mc}{\multicolumn} +\newcommand{\fnm}{\footnotemark} +\newcommand{\fnt}{\footnotetext} +\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}} +\newcommand{\mr}{\multirow} +\newcommand{\SI}{\textcolor{blue}{supporting information}} + +% second quantized operators +\newcommand{\psix}[1]{\hat{\Psi}\left({\bf X}_{#1}\right)} +\newcommand{\psixc}[1]{\hat{\Psi}^{\dagger}\left({\bf X}_{#1}\right)} +\newcommand{\ai}[1]{\hat{a}_{#1}} +\newcommand{\aic}[1]{\hat{a}^{\dagger}_{#1}} +\newcommand{\vijkl}[0]{V_{ij}^{kl}} +\newcommand{\phix}[2]{\phi_{#1}(\bfr{#2})} +\newcommand{\phixprim}[2]{\phi_{#1}(\bfr{#2}')} + +\newcommand{\CBS}{\text{CBS}} + + +% energies +\newcommand{\Ec}{E_\text{c}} +\newcommand{\EPT}{E_\text{PT2}} +\newcommand{\EsCI}{E_\text{sCI}} +\newcommand{\EDMC}{E_\text{DMC}} +\newcommand{\EexFCI}{E_\text{exFCI}} +\newcommand{\EexFCIbasis}{E_\text{exFCI}^{\Bas}} +\newcommand{\EexFCIinfty}{E_\text{exFCI}^{\infty}} +\newcommand{\EexDMC}{E_\text{exDMC}} +\newcommand{\Ead}{\Delta E_\text{ad}} +\newcommand{\efci}[0]{E_{\text{FCI}}^{\Bas}} +\newcommand{\emodel}[0]{E_{\model}^{\Bas}} +\newcommand{\emodelcomplete}[0]{E_{\model}^{\infty}} +\newcommand{\efcicomplete}[0]{E_{\text{FCI}}^{\infty}} +\newcommand{\ecccomplete}[0]{E_{\text{CCSD(T)}}^{\infty}} +\newcommand{\ecc}[0]{E_{\text{CCSD(T)}}^{\Bas}} +\newcommand{\efuncbasisFCI}[0]{\bar{E}^\Bas[\denFCI]} +\newcommand{\efuncbasisfci}[0]{\bar{E}^\Bas[\denfci]} +\newcommand{\efuncbasis}[0]{\bar{E}^\Bas[\den]} +\newcommand{\efuncden}[1]{\bar{E}^\Bas[#1]} +\newcommand{\efuncdenpbe}[1]{\bar{E}_{\text{}}^\Bas[#1]} +\newcommand{\efuncdenpbeAB}[1]{\bar{E}_{\text{A}+\text{B}}^\Bas[#1]} +\newcommand{\ecompmodel}[0]{\bar{E}^\Bas[\denmodel]} +\newcommand{\ecmubis}[0]{\bar{E}_{\text{c,md}}^{\text{sr}}[\denr;\,\mu]} +\newcommand{\ecmubisldapbe}[0]{\bar{E}_{\text{c,md}\,\text{PBE}}^{\text{sr}}[\denr;\,\mu]} +\newcommand{\ecmuapprox}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mu]} +\newcommand{\ecmuapproxmur}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mur]} +\newcommand{\ecmuapproxmurfci}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denfci;\,\mur]} +\newcommand{\ecmuapproxmurmodel}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denmodel;\,\mur]} +\newcommand{\ecompmodellda}[0]{\bar{E}_{\text{LDA}}^{\Bas,\wf{}{\Bas}}[\denmodel]} +\newcommand{\ecompmodelldaval}[0]{\bar{E}_{\text{LDA, val}}^{\Bas,\wf{}{\Bas}}[\den]} +\newcommand{\ecompmodelpbe}[0]{\bar{E}_{\text{PBE}}^{\Bas,\wf{}{\Bas}}[\den]} +\newcommand{\ecompmodelpbeval}[0]{\bar{E}_{\text{PBE, val}}^{\Bas,\wf{}{\Bas}}[\den]} +\newcommand{\emulda}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denr;\mu({\bf r};\wf{}{\Bas})\right)} +\newcommand{\emuldamodel}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denmodelr;\mu({\bf r};\wf{}{\Bas})\right)} +\newcommand{\emuldaval}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denval ({\bf r});\murval;\wf{}{\Bas})\right)} +\newcommand{\ecmd}[0]{\bar{\varepsilon}_{\text{c,md}}^{\text{sr},\text{PBE}}} +\newcommand{\psibasis}[0]{\Psi^{\basis}} +\newcommand{\BasFC}{\mathcal{V}} + +%pbeuegxiHF +%\newcommand{\pbeuegxihf}{\text{PBE-UEG-}\zeta\text{-HF}^\Bas} +%\newcommand{\argpbeuegxihf}[0]{\den,\zeta,s,\ntwo^{\text{UEG}},\mu^{\text{HF},\basis}} +%\newcommand{\argrpbeuegxihf}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo^{\text{UEG}}(\br{}),\mu^{\text{HF},\basis}(\br{})} +%pbeuegxiCAS +%\newcommand{\pbeuegxi}{\text{PBE-UEG-}\zeta\text{-CAS}^\Bas} +%\newcommand{\argpbeuegxicas}[0]{\den,\zeta,s,\ntwo^{\text{UEG}},\mu_{\text{CAS}}^{\basis}} +%\newcommand{\argrpbeuegxicas}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo^{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})} +%pbeuegXiCAS +\newcommand{\pbeuegXi}{\text{PBE-UEG}} +\newcommand{\argpbeuegXi}[0]{\den,\tilde{\zeta},s,\ntwo^{\text{UEG}},\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeuegXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),\ntwo^{\text{UEG}}(\br{}),\mu_{\text{}}(\br{})} +%pbeontxiCAS +%\newcommand{\pbeontxi}{\text{SPBE-OT}} +%\newcommand{\argpbeontxi}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} +%\newcommand{\argrpbeontxi}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})} +%pbeontXiCAS +\newcommand{\pbeontXi}{\text{PBE-OT}} +\newcommand{\argpbeontXi}[0]{\den,\tilde{\zeta},s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeontXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{}}^{}(\br{})} +%pbeont0xiCAS +\newcommand{\pbeuegns}{\text{SU-PBE-UEG}} +\newcommand{\argpbeuegns}[0]{\den,0,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeuegns}[0]{\den(\br{}),0,s(\br{}),\ntwo^{\text{UEG}}(\br{}),\mu_{\text{}}^{}(\br{})} +%pbeont0xiCAS +\newcommand{\pbeontns}{\text{SU-PBE-OT}} +\newcommand{\argpbeontns}[0]{\den,0,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeontns}[0]{\den(\br{}),0,s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{}}^{}(\br{})} + +%%%%%% arguments + +\newcommand{\argepbe}[0]{\den,\zeta,s} +\newcommand{\argebasis}[0]{\den,\zeta,\ntwo,\mu} +\newcommand{\argecmd}[0]{\den,\zeta,s,\ntwo,\mu} +\newcommand{\argepbeueg}[0]{\den,\zeta,s,\ntwo^{\text{UEG}},\mu_{\Psi^{\basis}}} +\newcommand{\argepbeontxicas}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} +\newcommand{\argepbeuegXihf}[0]{\den,\tilde{\zeta},s,\ntwo^{\text{UEG}},\mu_{\Psi^{\basis}}} +\newcommand{\argrebasis}[0]{\denr,\zeta(\br{}),s(\br{}),\ntwo(\br{}),\mu(\br{})} +\newcommand{\argrebasisab}[0]{\denr,\zeta(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})} + + +% numbers +\newcommand{\rnum}[0]{{\rm I\!R}} +\newcommand{\bfr}[1]{{\bf r}_{#1}} +\newcommand{\dr}[1]{\text{d}\bfr{#1}} +\newcommand{\rr}[2]{\bfr{#1}, \bfr{#2}} +\newcommand{\rrrr}[4]{\bfr{#1}, \bfr{#2},\bfr{#3},\bfr{#4} } + + + +% effective interaction +\newcommand{\twodm}[4]{\mel{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}} +\newcommand{\murpsi}[0]{\mu_{\wf{}{\Bas}}({\bf r})} +\newcommand{\murpsibas}[0]{\mu_{\wf{}{\Bas}}({\bf r})} +\newcommand{\ntwo}[0]{n_{2}} +\newcommand{\ntwohf}[0]{n_2^{\text{HF}}} +\newcommand{\ntwophi}[0]{n_2^{{\phi}}} +\newcommand{\ntwoextrap}[0]{\mathring{n}_{2}^{\text{}}} +\newcommand{\ntwoextrapcas}[0]{\mathring{n}_2^{\text{}}} +\newcommand{\mur}[0]{\mu({\bf r})} +\newcommand{\murr}[1]{\mu({\bf r}_{#1})} +\newcommand{\murval}[0]{\mu_{\text{val}}({\bf r})} +\newcommand{\murpsival}[0]{\mu_{\wf{}{\Bas}}^{\text{val}}({\bf r})} +\newcommand{\murrval}[1]{\mu^{\text{val}}({\bf r}_{#1})} +\newcommand{\weeopmu}[0]{\hat{W}_{\text{ee}}^{\text{lr},\mu}} + + +\newcommand{\wbasis}[0]{W_{\wf{}{\Bas}}(\bfr{1},\bfr{2})} +\newcommand{\wbasiscoal}[0]{W_{\wf{}{\Bas}}(\bfr{},\bfr{})} +\newcommand{\wbasisval}[0]{W_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})} +\newcommand{\fbasis}[0]{f_{\wf{}{\Bas}}(\bfr{1},\bfr{2})} +\newcommand{\fbasisval}[0]{f_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})} + \newcommand{\ontop}[2]{ n^{(2)}_{#1}({\bf #2}_1)} + \newcommand{\twodmrpsi}[0]{ \ntwo^{\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})} + \newcommand{\twodmrdiagpsi}[0]{ n_{2,{\wf{}{\Bas}}}(\rr{1}{2})} + \newcommand{\twodmrdiagpsival}[0]{n_{2,\wf{}{\Bas}}^{\text{val}}(\rr{1}{2})} + \newcommand{\gammamnpq}[1]{\Gamma_{mn}^{pq}[#1]} + \newcommand{\gammamnkl}[0]{\Gamma_{mn}^{kl}} + \newcommand{\gammaklmn}[1]{\Gamma_{kl}^{mn}[#1]} +%\newcommand{\wbasiscoal}[1]{W_{\wf{}{\Bas}}({\bf r}_{#1})} +\newcommand{\ontoppsi}[1]{ n^{(2)}_{\wf{}{\Bas}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})} +\newcommand{\wbasiscoalval}{W_{\wf{}{\Bas}}^{\text{val}}({\bf r},{\bf r})} +\newcommand{\ontoppsival}[1]{ n^{(2)}_{\wf{}{\Bas}}^{\text{val}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})} + + + +\newcommand{\ex}[4]{$^{#1}#2_{#3}^{#4}$} +\newcommand{\ra}{\rightarrow} +\newcommand{\De}{D_\text{e}} + +% MODEL +\newcommand{\model}[0]{\mathcal{Y}} + +% densities +\newcommand{\denmodel}[0]{\den_{\model}^\Bas} +\newcommand{\denmodelr}[0]{\den_{\model}^\Bas ({\bf r})} +\newcommand{\denfci}[0]{\den_{\psifci}} +\newcommand{\denFCI}[0]{\den^{\Bas}_{\text{FCI}}} +\newcommand{\denbas}[0]{\den^\Bas} +\newcommand{\denhf}[0]{\den_{\text{HF}}^\Bas} +\newcommand{\denrfci}[0]{\denr_{\psifci}} +\newcommand{\dencipsir}[0]{{n}_{\text{CIPSI}}^\Bas({\bf r})} +\newcommand{\dencipsi}[0]{{n}_{\text{CIPSI}}^\Bas} +\newcommand{\den}[0]{{n}} +\newcommand{\denval}[0]{{n}^{\text{val}}} +\newcommand{\denr}[0]{{n}({\bf r})} +\newcommand{\onedmval}[0]{\rho_{ij,\sigma}^{\text{val}}} + +% wave functions +\newcommand{\psifci}[0]{\Psi^{\Bas}_{\text{FCI}}} +\newcommand{\psimu}[0]{\Psi^{\mu}} +% operators +\newcommand{\weeopbasis}[0]{\hat{W}_{\text{ee}}^\Bas} +\newcommand{\kinop}[0]{\hat{T}} + +\newcommand{\weeopbasisval}[0]{\hat{W}_{\text{ee}}^{\Basval}} +\newcommand{\weeop}[0]{\hat{W}_{\text{ee}}} + + +% units +\newcommand{\IneV}[1]{#1 eV} +\newcommand{\InAU}[1]{#1 a.u.} +\newcommand{\InAA}[1]{#1 \AA} + + +% methods +\newcommand{\UEG}{\text{UEG}} +\newcommand{\LDA}{\text{LDA}} +\newcommand{\PBE}{\text{PBE}} +\newcommand{\FCI}{\text{FCI}} +\newcommand{\CCSDT}{\text{CCSD(T)}} +\newcommand{\lr}{\text{lr}} +\newcommand{\sr}{\text{sr}} + +\newcommand{\Nel}{N} +\newcommand{\V}[2]{V_{#1}^{#2}} + + +\newcommand{\n}[2]{n_{#1}^{#2}} +\newcommand{\E}[2]{E_{#1}^{#2}} +\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}} +\newcommand{\bEc}[1]{\Bar{E}_\text{c}^{#1}} +\newcommand{\e}[2]{\varepsilon_{#1}^{#2}} +\newcommand{\be}[2]{\Bar{\varepsilon}_{#1}^{#2}} +\newcommand{\bec}[1]{\Bar{e}^{#1}} +\newcommand{\wf}[2]{\Psi_{#1}^{#2}} +\newcommand{\W}[2]{W_{#1}^{#2}} +\newcommand{\w}[2]{w_{#1}^{#2}} +\newcommand{\hn}[2]{\Hat{n}_{#1}^{#2}} +\newcommand{\rsmu}[2]{\mu_{#1}^{#2}} +\newcommand{\SO}[2]{\phi_{#1}(\br{#2})} + +\newcommand{\modX}{\text{X}} +\newcommand{\modY}{\text{Y}} + +% basis sets +\newcommand{\setdenbasis}{\mathcal{N}_{\Bas}} +\newcommand{\Bas}{\mathcal{B}} +\newcommand{\basis}{\mathcal{B}} +\newcommand{\Basval}{\mathcal{B}_\text{val}} +\newcommand{\Val}{\mathcal{V}} +\newcommand{\Cor}{\mathcal{C}} + +% operators +\renewcommand{\d}{\text{d}} +\newcommand{\hT}{\Hat{T}} +\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}} +\newcommand{\f}[2]{f_{#1}^{#2}} +\newcommand{\Gam}[2]{\Gamma_{#1}^{#2}} +\newcommand{\isEquivTo}[1]{\underset{#1}{\sim}} + +% coordinates +\newcommand{\br}[1]{{\mathbf{r}_{#1}}} +\newcommand{\bx}[1]{\mathbf{x}_{#1}} +\newcommand{\dbr}[1]{d\br{#1}} +\newcommand{\PBEspin}{PBEspin} +\newcommand{\PBEueg}{PBE-UEG-{$\tilde{\zeta}$}} + +\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique (UMR 7616), Sorbonne Universit\'e, CNRS, Paris, France} +\newcommand{\ISCD}{Institut des Sciences du Calcul et des Donn\'ees, Sorbonne Universit\'e, Paris, France} +\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} +\newcommand{\IUF}{Institut Universitaire de France, Paris, France} + + +\begin{document} + +\title{A density-based basis-set correction for strongly correlated molecular systems} + +\author{Emmanuel Giner} +\email{emmanuel.giner@lct.jussieu.fr} +\affiliation{\LCT} +\author{Barth\'el\'emy Pradines} +\affiliation{\LCT} +\affiliation{\ISCD} +\author{Anthony Scemama} +\affiliation{\LCPQ} +\author{Pierre-Fran\c{c}ois Loos} +\email{loos@irsamc.ups-tlse.fr} +\affiliation{\LCPQ} +\author{Julien Toulouse} +\email{toulouse@lct.jussieu.fr} +\affiliation{\LCT} +\affiliation{\IUF} + + +\begin{abstract} +We extend to strongly correlated molecular systems the recently introduced basis-set incompleteness correction based on density-functional theory (DFT) [E. Giner \textit{et al.}, \href{https://doi.org/10.1063/1.5052714}{J. Chem. Phys. \textbf{149}, 194301 (2018)}]. This basis-set correction relies on a mapping between wave-function calculations in a finite basis set and range-separated DFT (RSDFT) through the definition of an effective non-divergent interaction corresponding to the electron-electron Coulomb interaction projected in the finite basis set. This enables the use of RSDFT-type complementary density functionals to recover the dominant part of the short-range correlation effects missing in this finite basis set. To study both weak and strong correlation regimes we consider the potential energy curves of the \ce{H10}, \ce{N2}, \ce{O2}, and \ce{F2} molecules up to the dissociation limit, and we explore various approximations of complementary functionals fulfilling two key properties: spin-multiplet degeneracy (\ie, independence of the energy with respect to the spin projection $S_z$) and size consistency. Specifically, we investigate the dependence of the functional on different types of on-top pair densities and spin polarizations. The key result of this study is that the explicit dependence on the on-top pair density allows one to completely remove the dependence on any form of spin polarization without any significant loss of accuracy. Quantitatively, we show that the basis-set correction reaches chemical accuracy on atomization energies with triple-$\zeta$ quality basis sets for most of the systems studied here. Also, the present basis-set incompleteness correction provides smooth potential energy curves along the whole range of internuclear distances. +\end{abstract} + +\maketitle + +%%%%%%%%%%%%%%%%%%%%%%%% +\section{Introduction} +%%%%%%%%%%%%%%%%%%%%%%%% +The general goal of quantum chemistry is to provide reliable theoretical tools to explore the rich area of chemistry. More specifically, developments in quantum chemistry primarily aim at accurately computing the electronic structure of molecular systems. Despite intense developments, no definitive solution to this problem has been found. The theoretical challenge to tackle belongs to the quantum many-body problem, due to the intrinsic quantum nature of the electrons and the Coulomb repulsion between them. This so-called electronic correlation problem corresponds to finding a solution to the Schr\"odinger equation for a $N$-electron system, and two main roads have emerged to approximate this solution: wave-function theory (WFT) \cite{Pop-RMP-99} and density-functional theory (DFT). \cite{Koh-RMP-99} Although both WFT and DFT spring from the same Schr\"odinger equation, they rely on very different formalisms, as the former deals with the complicated $N$-electron wave function whereas the latter focuses on the much simpler one-electron density. In its Kohn-Sham (KS) formulation, \cite{KohSha-PR-65} the computational cost of DFT is very appealing since it is a simple mean-field procedure. Therefore, although continued efforts have been made to reduce the computational cost of WFT, DFT still remains the workhorse of quantum computational chemistry. + +The difficulty of obtaining a reliable theoretical description of a given chemical system can be roughly categorized by the strength of the electronic correlation. The so-called weakly correlated systems, such as closed-shell organic molecules near their equilibrium geometry, are typically dominated by correlation effects which do not affect the qualitative mean-field picture of the system. These weak-correlation effects can be either short range (near the electron-electron coalescence points) \cite{HatKloKohTew-CR-12} or long range (London dispersion interactions). \cite{AngDobJanGou-BOOK-20} The theoretical description of weakly correlated systems is one of the most concrete achievement of quantum chemistry, and the main remaining challenge for these systems is to push the limit of the chemical system size that can be treated. The case of the so-called strongly correlated systems, which are ubiquitous in chemistry, is more problematic as they exhibit a much more complex electronic structure. For example, transition metal complexes, low-spin open-shell systems, covalent bond breaking situations have all in common that they cannot be even qualitatively described by a single electronic configuration. It is now clear that the usual semilocal density-functional approximations of KS DFT fail to accurately describe these situations \cite{GorSeiSav-PCCP-08,GagTruLiCarHoyBa-ACR-17} and WFT is king for the treatment of strongly correlated systems. + +In practice, WFT uses a finite one-electron basis set. The exact solution of the Schr\"odinger equation within this basis set is then provided by full configuration interaction (FCI) which consists in a linear-algebra eigenvalue problem with a dimension scaling exponentially with the system size. Due to this exponential growth of the FCI computational cost, introducing approximations is necessary, with at least two difficulties for strongly correlated systems: i) the qualitative description of the wave function is determined by a primary set of electronic configurations (whose size can scale exponentially in many cases) among which near degeneracies and/or strong interactions appear in the Hamiltonian matrix; ii) the quantitative description of the system requires also to account for weak correlation effects which involve many other electronic configurations with typically much smaller weights in the wave function. Addressing simultaneously these two issues is a rather complicated task for a given approximate WFT method, especially if one adds the requirement of satisfying formal properties, such as spin-multiplet degeneracy (\ie, independence of the energy with respect to the spin projection $S_z$) and size consistency. + +Beside the difficulties of accurately describing the molecular electronic structure within a given basis set, a crucial limitation of WFT methods is the slow convergence of the energy (and related properties) with respect to the size of the one-electron basis set. As initially shown by the seminal work of Hylleraas \cite{Hyl-ZP-29} and further developed by Kutzelnigg and coworkers, \cite{Kut-TCA-85,KutKlo-JCP-91, NogKut-JCP-94} the main convergence problem originates from the divergence of the electron-electron Coulomb interaction at the coalescence point, which induces a discontinuity in the first derivative of the exact wave function (the so-called electron-electron cusp). Describing such a discontinuity with an incomplete one-electron basis set is impossible and, as a consequence, the convergence of the computed energies and properties are strongly affected. To alleviate this problem, extrapolation techniques have been developed, either based on a partial-wave expansion analysis, \cite{HelKloKocNog-JCP-97,HalHelJorKloKocOlsWil-CPL-98} or more recently based on perturbative arguments. \cite{IrmHulGru-PRL-19,IrmGru-JCP-2019} A more rigorous approach to tackle the basis-set convergence problem is provided by the so-called explicitly correlated F12 (or R12) methods \cite{Ten-TCA-12,TenNog-WIREs-12,HatKloKohTew-CR-12, KonBisVal-CR-12, GruHirOhnTen-JCP-17, MaWer-WIREs-18} which introduce a geminal function depending explicitly on the interelectronic distance. This ensures a correct representation of the Coulomb correlation hole around the electron-electron coalescence point, and leads to a much faster convergence of the energy than usual WFT methods. For instance, using the explicitly correlated version of coupled cluster with singles, doubles, and perturbative triples [CCSD(T)] in a triple-$\zeta$ basis set is equivalent to using a quintuple-$\zeta$ basis set with the usual CCSD(T) method, \cite{TewKloNeiHat-PCCP-07} although a computational overhead is introduced by the auxiliary basis set needed to compute the three-electron integrals involved in F12 theory. \cite{BarLoo-JCP-17} In addition to the computational cost, a possible drawback of F12 theory is its rather complex formalism which requires non-trivial developments for adapting it to a new method. For strongly correlated systems, several multi-reference methods have been extended to explicit correlation (see, for example, Refs.~\onlinecite{Ten-CPL-07,ShiWer-JCP-10,TorKniWer-JCP-11,DemStanMatTenPitNog-PCCP-12,GuoSivValNee-JCP-17}), including approaches based on the so-called universal F12 theory which are potentially applicable to any electronic-structure computational methods. \cite{TorVal-JCP-09,KonVal-JCP-11,HauMaoMukKlo-CPL-12,BooCleAlaTew-JCP-12} + +An alternative way to improve the convergence towards the complete-basis-set (CBS) limit is to treat the short-range correlation effects within DFT and to use WFT methods to deal only with the long-range and/or strong correlation effects. A rigorous approach achieving this mixing of DFT and WFT is range-separated DFT (RSDFT) (see Ref.~\onlinecite{TouColSav-PRA-04} and references therein) which relies on a decomposition of the electron-electron Coulomb interaction in terms of the interelectronic distance thanks to a range-separation parameter $\mu$. The advantage of this approach is at least two-fold: i) the DFT part deals primarily with the short-range part of the Coulomb interaction, and consequently the usual semilocal density-functional approximations are more accurate than for standard KS DFT; ii) the WFT part deals only with a smooth non-divergent interaction, and consequently the wave function has no electron-electron cusp \cite{GorSav-PRA-06} and the basis-set convergence is much faster. \cite{FraMusLupTou-JCP-15} A number of approximate RSDFT schemes have been developed involving single-reference \cite{AngGerSavTou-PRA-05, GolWerSto-PCCP-05, TouGerJanSavAng-PRL-09,JanHenScu-JCP-09, TouZhuSavJanAng-JCP-11, MusReiAngTou-JCP-15,KalTou-JCP-18,KalMusTou-JCP-19} and multi-reference \cite{LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, HedTouJen-JCP-18, FerGinTou-JCP-18} WFT methods. Nevertheless, there are still some open issues in RSDFT, such as remaining fractional-charge and fractional-spin errors in the short-range density functionals \cite{MusTou-MP-17} or the dependence of the quality of the results on the value of the range-separation parameter $\mu$. + +Building on the development of RSDFT, a possible solution to the basis-set convergence problem has been recently proposed by some of the present authors~\cite{GinPraFerAssSavTou-JCP-18} in which RSDFT functionals are used to recover only the correlation effects outside a given basis set. The key point here is to realize that a wave function developed in an incomplete basis set is cuspless and could also originate from a Hamiltonian with a non-divergent long-range electron-electron interaction. Therefore, a mapping with RSDFT can be performed through the introduction of an effective non-divergent interaction representing the usual electron-electron Coulomb interaction projected in an incomplete basis set. First applications to weakly correlated molecular systems have been successfully carried out, \cite{LooPraSceTouGin-JCPL-19} together with extensions of this approach to the calculations of excitation energies \cite{GinSceTouLoo-JCP-19} and ionization potentials. \cite{LooPraSceGinTou-JCTC-20} The goal of the present work is to further develop this approach for the description of strongly correlated systems. + +The paper is organized as follows. In Sec.~\ref{sec:theory}, we recall the mathematical framework of the basis-set correction and we present its extension for strongly correlated systems. In particular, our focus is primarily set on imposing two key formal properties: spin-multiplet degeneracy and size consistency. +Then, in Sec.~\ref{sec:results}, we apply the method to the calculation of the potential energy curves of the \ce{H10}, \ce{N2}, \ce{O2}, and \ce{F2} molecules up to the dissociation limit. Finally, we conclude in Sec.~\ref{sec:conclusion}. + +%%%%%%%%%%%%%%%%%%%%%%%% +\section{Theory} +\label{sec:theory} +%%%%%%%%%%%%%%%%%%%%%%%% +As the theory behind the present basis-set correction has been exposed in details in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we only briefly recall the main equations and concepts needed for this study in Secs.~\ref{sec:basic}, \ref{sec:wee}, and \ref{sec:mur}. More specifically, in Sec.~\ref{sec:basic}, we recall the basic mathematical framework of the present theory by introducing the complementary functional to a basis set. Section \ref{sec:wee} introduces the effective non-divergent interaction in the basis set, which leads us to the definition of the effective \textit{local} range-separation function in Sec.~\ref{sec:mur}. Then, Sec.~\ref{sec:functional} exposes the new approximate RSDFT-based complementary correlation functionals. The generic form of such functionals is exposed in Sec.~\ref{sec:functional_form}, their properties in the context of the basis-set correction are discussed in Sec.~\ref{sec:functional_prop}, and the specific requirements for strong correlation are discussed in Sec.~\ref{sec:requirements}. Finally, the actual functionals used in this work are introduced in Sec.~\ref{sec:def_func}. + +\subsection{Basic theory} +\label{sec:basic} + +The exact ground-state energy $E_0$ of a $N$-electron system can, in principle, be obtained in DFT by a minimization over $N$-representable one-electron densities $\denr$ +\begin{equation} + \label{eq:levy} + E_0 = \min_{\den} \bigg\{ F[\den] + \int \d \br{} v_{\text{ne}} (\br{}) \denr \bigg\}, +\end{equation} +where $v_\text{ne}(\br{})$ is the nuclei-electron potential, and $F[\den]$ is the universal Levy-Lieb density functional written with the constrained search formalism as~\cite{Lev-PNAS-79,Lie-IJQC-83} +\begin{equation} + \label{eq:levy_func} + F[\den] = \min_{\Psi \to \den} \mel{\Psi}{\kinop +\weeop}{\Psi}, +\end{equation} +where $\kinop$ and $\weeop$ are the kinetic and electron-electron Coulomb operators, and the notation $\Psi \to \den$ means that the wave function $\Psi$ yields the density $\den$. +The minimizing density $n_0$ in Eq.~\eqref{eq:levy} is the exact ground-state density. Nevertheless, in practical calculations, the accessible densities are necessarily restricted to the set of densities ``representable in a basis set $\Bas$'', \ie, densities coming from wave functions expandable in the $N$-electron Hilbert space generated by the one-electron basis set $\Bas$. In the following, we always consider only such representable-in-$\Bas$ densities. With this restriction, Eq.~\eqref{eq:levy} then gives an upper bound $E_0^\Bas$ of the exact ground-state energy. Since the density has a faster convergence with the size of the basis set than the wave function, this restriction is a rather weak one and we can consider that $E_0^\Bas$ is an acceptable approximation to the exact ground-state energy, \ie, $E_0^\Bas \approx E_0$. + +In the present context, it is important to notice that the wave functions $\Psi$ defined in Eq.~\eqref{eq:levy_func} are not restricted to a finite basis set, \ie, they should be expanded in a complete basis set. In Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, it was then proposed to decompose $F[\den]$ as, for a representable-in-$\Bas$ density $\den$, +\begin{equation} + \label{eq:def_levy_bas} + F[\den] = \min_{\wf{}{\Bas} \to \den} \mel*{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\den}, +\end{equation} +where $\wf{}{\Bas}$ are wave functions expandable in the $N$-electron Hilbert space generated by $\basis$, and +\begin{equation} + \begin{aligned} + \efuncden{\den} = \min_{\Psi \to \den} \mel*{\Psi}{\kinop +\weeop }{\Psi} + - \min_{\Psi^{\Bas} \to \den} \mel*{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \end{aligned} +\end{equation} +is the complementary density functional to the basis set $\Bas$. +Introducing the decomposition in Eq.~\eqref{eq:def_levy_bas} back into Eq.~\eqref{eq:levy} yields +\begin{multline} + \label{eq:E0basminPsiB} + E_0^\Bas = \min_{\Psi^{\Bas}} \bigg\{ \mel*{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\den_{{\Psi^{\Bas}}}} + \\ + + \int \d \br{} v_{\text{ne}} (\br{}) \den_{\Psi^{\Bas}}(\br{}) \bigg\}, +\end{multline} +where the minimization is only over wave functions $\wf{}{\Bas}$ restricted to the basis set $\basis$ and $\den_{{\Psi^{\Bas}}}(\br{})$ refers to the density generated from $\wf{}{\Bas}$. Therefore, thanks to Eq.~\eqref{eq:E0basminPsiB}, one can properly combine a WFT calculation in a finite basis set with a density functional (hereafter referred to as complementary functional) accounting for the correlation effects that are not included in the basis set. + +As a simple non-self-consistent version of this approach, we can approximate the minimizing wave function $\Psi_0^{\Bas}$ in Eq.~\eqref{eq:E0basminPsiB} by the ground-state FCI wave function $\psifci$ within $\Bas$, and we then obtain the following approximation for the exact ground-state energy [see Eqs.~(12)--(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}] +\begin{equation} + \label{eq:e0approx} + E_0 \approx E_0^\Bas \approx \efci + \efuncbasisFCI, +\end{equation} +where $\efci$ and $n_\text{FCI}^\Bas$ are the ground-state FCI energy and density, respectively. As it was originally shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} and further emphasized in Refs.~\onlinecite{LooPraSceTouGin-JCPL-19,GinSceTouLoo-JCP-19}, the main role of $\efuncbasisFCI$ is to correct for the basis-set incompleteness error, a large part of which originating from the lack of electron-electron cusp in the wave function expanded in an incomplete basis set. The whole purpose of this work is to determine approximations for $\efuncbasisFCI$ which are suitable for strongly correlated molecular systems. Two key requirements for this purpose are i) spin-multiplet degeneracy, and ii) size consistency. + +\subsection{Effective interaction in a finite basis} +\label{sec:wee} +As originally shown by Kato, \cite{Kat-CPAM-57} the electron-electron cusp of the exact wave function originates from the divergence of the Coulomb interaction at the coalescence point. Therefore, a cuspless wave function $\wf{}{\Bas}$ could also be obtained from a Hamiltonian with a non-divergent electron-electron interaction. +In other words, the impact of the basis set incompleteness can be understood as the removal of the divergence of the usual electron-electron Coulomb interaction. + +As originally derived in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} (see Sec.~II D~and Appendices), one can obtain an effective non-divergent electron-electron interaction, here referred to as $\wbasis$, which reproduces the expectation value of the electron-electron Coulomb interaction operator over a given wave function $\wf{}{\Bas}$. As we are interested in the behavior at the coalescence point, we focus on the opposite-spin part of the electron-electron interaction. More specifically, the effective electron-electron interaction associated to a given wave function $\wf{}{\Bas}$ is defined as +\begin{equation} + \label{eq:wbasis} + \wbasis = + \begin{cases} + \fbasis /\twodmrdiagpsi, & \text{if $\twodmrdiagpsi \ne 0$,} +\\ + \infty, & \text{otherwise,} + \end{cases} +\end{equation} +where +\begin{equation} + \twodmrdiagpsi = \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}, +\end{equation} +is the opposite-spin pair density associated with $\wf{}{\Bas}$, and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$ its associated tensor in a basis of spatial orthonormal orbitals $\{\SO{p}{}\}$, +\begin{equation} + \label{eq:fbasis} + \fbasis + = \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2}, +\end{equation} +and $\V{pq}{rs}= \braket{pq}{rs}$ are the usual two-electron Coulomb integrals. +With such a definition, one can show that $\wbasis$ satisfies +\begin{multline} + \frac{1}{2}\iint \dr{1} \dr{2} \wbasis \twodmrdiagpsi = + \\ + \frac{1}{2} \iint \dr{1} \dr{2} \frac{\twodmrdiagpsi}{\abs{\br{1}-\br{2}}}. +\end{multline} +As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, the effective interaction $\wbasis$ is necessarily finite at coalescence for an incomplete basis set, and tends to the usual Coulomb interaction in the CBS limit for any choice of wave function $\psibasis$, \ie, +\begin{equation} + \label{eq:cbs_wbasis} + \lim_{\Bas \to \text{CBS}} \wbasis = \frac{1}{\abs{\br{1}-\br{2}}},\quad \forall\,\psibasis. +\end{equation} +The condition in Eq.~\eqref{eq:cbs_wbasis} is fundamental as it guarantees the correct behavior of the theory in the CBS limit. + +\subsection{Local range-separation function} +\label{sec:mur} +\subsubsection{General definition} +The effective interaction within a finite basis, $\wbasis$, is bounded and resembles the long-range interaction used in RSDFT +\begin{equation} + \label{eq:weelr} + w_\text{ee}^{\lr}(\mu;r_{12}) = \frac{\text{erf}\big(\mu \,r_{12} \big)}{r_{12}}, +\end{equation} +where $\mu$ is the range-separation parameter. As originally proposed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we make the correspondence between these two interactions by using the local range-separation function +\begin{equation} + \label{eq:def_mur} + \murpsi = \frac{\sqrt{\pi}}{2} \wbasiscoal, +\end{equation} +such that the two interactions coincide at the electron-electron coalescence point for each $\br{}$ +\begin{equation} + w_\text{ee}^{\lr}(\murpsi;0) = \wbasiscoal, \quad \forall \, \br{}. +\end{equation} +Because of the very definition of $\wbasis$, one has the following property in the CBS limit [see Eq.~\eqref{eq:cbs_wbasis}] +\begin{equation} + \label{eq:cbs_mu} + \lim_{\Bas \to \text{CBS}} \murpsi = \infty, \quad \forall \,\psibasis, +\end{equation} +which is again fundamental to guarantee the correct behavior of the theory in the CBS limit. + +\subsubsection{Frozen-core approximation} +\label{sec:FC} +As all WFT calculations in this work are performed within the frozen-core approximation, we use a ``valence-only'' (or no-core) version of the various quantities needed for the complementary functional introduced in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}. We partition the basis set as $\Bas = \Cor \bigcup \BasFC$, where $\Cor$ and $\BasFC$ are the sets of core and ``valence'' (\ie, non-core) orbitals, respectively, and define the valence-only local range-separation function as +\begin{equation} + \label{eq:def_mur_val} + \murpsival = \frac{\sqrt{\pi}}{2} \wbasiscoalval{}, +\end{equation} +where +\begin{equation} + \label{eq:wbasis_val} + \wbasisval = + \begin{cases} + \fbasisval /\twodmrdiagpsival, & \text{if $\twodmrdiagpsival \ne 0$,} +\\ + \infty, & \text{otherwise,} + \end{cases} +\end{equation} +is the valence-only effective interaction and +\begin{gather} + \label{eq:fbasis_val} + \fbasisval + = \sum_{pq\in \Bas} \sum_{rstu \in \BasFC} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2}, + \\ + \label{eq:twordm_val} + \twodmrdiagpsival + = \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}. +\end{gather} +One would note the restrictions of the sums to the set $\BasFC$ in Eqs.~\eqref{eq:fbasis_val} and \eqref{eq:twordm_val}. +It is also noteworthy that, with the present definition, $\wbasisval$ still tends to the usual Coulomb interaction as $\Bas \to \CBS$. + +\subsection{General form of the complementary functional} +\label{sec:functional} + +\subsubsection{Generic approximate form} +\label{sec:functional_form} + +As originally proposed and motivated in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the complementary functional $\efuncden{\den}$ by using the so-called correlation energy functional with multideterminant reference (ECMD) introduced by Toulouse \textit{et al.}.\cite{TouGorSav-TCA-05,Tou-THESIS-05} Following the recent work in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}, we propose to consider a Perdew-Burke-Ernzerhof (PBE)-like functional which uses the one-electron density $\denr$, the spin polarization $\zeta(\br{})=[n_\uparrow(\br{})-n_\downarrow(\br{})]/\denr$ (where $n_\uparrow(\br{})$ and $n_\downarrow(\br{})$ are the spin-up and spin-down densities), the reduced density gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$, and the on-top pair density $\ntwo(\br{})\equiv \ntwo(\br{},\br{})$. In the present work, all these quantities are computed with the same wave function $\psibasis$ used to define $\mur \equiv\murpsi$. +Therefore, $\efuncden{\den}$ has the following generic form +\begin{multline} + \label{eq:def_ecmdpbebasis} + \efuncdenpbe{\argebasis} = + \\ + \int \d\br{} \,\denr \ecmd(\argrebasis), +\end{multline} +where +\begin{equation} + \label{eq:def_ecmdpbe} + \ecmd(\argecmd) = \frac{\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)}{1+ \beta(\argepbe,\ntwo) \; \mu^3} +\end{equation} +is the correlation energy per particle, with +\begin{equation} + \label{eq:def_beta} + \beta(\argepbe,\ntwo) = \frac{3}{2\sqrt{\pi}(1 - \sqrt{2})}\frac{\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)}{\ntwo/\den}, +\end{equation} +where $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ is the usual PBE correlation energy per particle. \cite{PerBurErn-PRL-96} Before introducing the different flavors of approximate functionals that we will use here (see Sec.~\ref{sec:def_func}), we would like to give some motivations for this choice of functional form. + +The form of $\ecmd(\argecmd)$ in Eq.~\eqref{eq:def_ecmdpbe} has been originally proposed in Ref.~\onlinecite{FerGinTou-JCP-18} in the context of RSDFT. In the $\mu\to 0$ limit, it reduces to the usual PBE correlation functional, \ie, +\begin{equation} + \lim_{\mu \to 0} \ecmd(\argecmd) = \varepsilon_{\text{c}}^{\text{PBE}}(\argepbe), +\end{equation} +which is relevant in the weak-correlation (or high-density) limit. In the large-$\mu$ limit, it behaves as +\begin{equation} + \label{eq:lim_mularge} + \ecmd(\argecmd) \isEquivTo{\mu\to\infty} \frac{2\sqrt{\pi}(1 - \sqrt{2})}{3 \mu^3} \frac{\ntwo}{n}, +\end{equation} +which is the exact large-$\mu$ behavior of the exact ECMD correlation energy. \cite{PazMorGorBac-PRB-06,FerGinTou-JCP-18} Of course, for a specific system, the large-$\mu$ behavior will be exact only if one uses for $n_2$ the \textit{exact} on-top pair density of this system. This large-$\mu$ limit in Eq.~\eqref{eq:lim_mularge} is relevant in the strong-correlation (or low-density) limit. In the context of RSDFT, some of the present authors have illustrated in Ref.~\onlinecite{FerGinTou-JCP-18} that the on-top pair density involved in Eq.~\eqref{eq:def_ecmdpbe} plays indeed a crucial role when reaching the strong-correlation regime. The importance of the on-top pair density in the strong-correlation regime have been also recently acknowledged by Gagliardi and coworkers \cite{CarTruGag-JPCA-17} and Pernal and coworkers.\cite{GritMeePer-PRA-18} + +Note also that $\ecmd(\argecmd)$ vanishes when $\ntwo$ vanishes, \ie, +\begin{equation} + \label{eq:lim_n2} + \lim_{\ntwo \to 0} \ecmd(\argecmd) = 0, +\end{equation} +which is expected for systems with a vanishing on-top pair density. +Finally, the function $\ecmd(\argecmd)$ vanishes when $\mu \to \infty$ like all RSDFT short-range functionals, \ie, +\begin{equation} + \label{eq:lim_muinf} + \lim_{\mu \to \infty} \ecmd(\argecmd) = 0. +\end{equation} + +\subsubsection{Two limits where the complementary functional vanishes} +\label{sec:functional_prop} + +Within the definitions of Eqs.~\eqref{eq:def_mur} and \eqref{eq:def_ecmdpbebasis}, any approximate complementary functional $\efuncdenpbe{\argebasis}$ satisfies two important properties. + +First, thanks to the properties in Eqs.~\eqref{eq:cbs_mu} and~\eqref{eq:lim_muinf}, $\efuncdenpbe{\argebasis}$ vanishes in the CBS limit, independently of the type of wave function $\psibasis$ used to define the local range-separation function $\mu(\br{})$ in a given basis set $\Bas$, +\begin{equation} + \label{eq:lim_ebasis} + \lim_{\basis \to \text{CBS}} \efuncdenpbe{\argebasis} = 0, \quad \forall\, \psibasis. +\end{equation} + +Second, $\efuncdenpbe{\argebasis}$ correctly vanishes for systems with uniformly vanishing on-top pair density, such as one-electron systems and for the stretched H$_2$ molecule, +\begin{equation} + \lim_{n_2 \to 0} \efuncdenpbe{\argebasis} = 0. +\end{equation} +This property is doubly guaranteed by i) the choice of setting $\wbasis = \infty$ for a vanishing pair density [see Eq.~\eqref{eq:wbasis}], which leads to $\mu(\br{}) \to \infty$ and thus a vanishing $\ecmd(\argecmd)$ [see Eq.~\eqref{eq:lim_muinf}], and ii) the fact that $\ecmd(\argecmd)$ vanishes anyway when the on-top pair density vanishes [see Eq.~\eqref{eq:lim_n2}]. + +\subsection{Requirements on the complementary functional for strong correlation} +\label{sec:requirements} + +An important requirement for any electronic-structure method is size consistency, \ie, the additivity of the energies of non-interacting fragments, which is mandatory to avoid any ambiguity in computing interaction energies. When two subsystems \ce{A} and \ce{B} dissociate in closed-shell systems, as in the case of weak intermolecular interactions for instance, spin-restricted Hartree-Fock (RHF) is size-consistent. When the two subsystems dissociate in open-shell systems, such as in covalent bond breaking, it is well known that the RHF approach fails and an alternative is to use a complete-active-space self-consistent-field (CASSCF) wave function which, provided that the active space has been properly chosen, leads to additive energies. + +Another important requirement is spin-multiplet degeneracy, \ie, the independence of the energy with respect to the $S_z$ component of a given spin state, which is also a property of any exact wave function. Such a property is also important in the context of covalent bond breaking where the ground state of the supersystem $\ce{A + B}$ is generally of lower spin than the corresponding ground states of the fragments (\ce{A} and \ce{B}) which can have multiple $S_z$ components. + +\subsubsection{Spin-multiplet degeneracy} + +A sufficient condition to achieve spin-multiplet degeneracy is to eliminate all dependencies on $S_z$. In the case of the function $\ecmd(\argecmd)$, this means removing the dependence on the spin polarization $\zeta(\br{})$ originating from the PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ [see Eq.~\eqref{eq:def_ecmdpbe}]. + +To do so, it has been proposed to replace the dependence on the spin polarization by the dependence on the on-top pair density. Most often, it is done by introducing an effective spin polarization~\cite{MosSan-PRA-91,BecSavSto-TCA-95,Sav-INC-96a,Sav-INC-96,MieStoSav-MP-97,TakYamYam-CPL-02,TakYamYam-IJQC-04,GraCre-MP-05,TsuScuSav-JCP-10,LimCarLuoMaOlsTruGag-JCTC-14,GarBulHenScu-JCP-15,GarBulHenScu-PCCP-15,CarTruGag-JCTC-15,GagTruLiCarHoyBa-ACR-17} (see, also, Refs.~\onlinecite{PerSavBur-PRA-95,StaDav-CPL-01}) +\begin{equation} + \label{eq:def_effspin} + \tilde{\zeta}(n,n_{2}) = \sqrt{ 1 - 2 \; n_{2}/n^2 } +\end{equation} +expressed as a function of the density $n$ and the on-top pair density $n_2$ calculated from a given wave function. The advantage of this approach is that this effective spin polarization $\tilde{\zeta}$ is independent from $S_z$ since the on-top pair density is $S_z$-independent. Nevertheless, the use of $\tilde{\zeta}$ in Eq.~\eqref{eq:def_effspin} presents some disadvantages since this expression was derived for a single-determinant wave function. Hence, it does not appear justified to use it for a multideterminant wave function. More particularly, it may happen, in the multideterminant case, that $1 - 2 \; n_{2}/n^2 < 0 $ which results in a complex-valued effective spin polarization. \cite{BecSavSto-TCA-95} +Therefore, following other authors, \cite{MieStoSav-MP-97,LimCarLuoMaOlsTruGag-JCTC-14,GarBulHenScu-JCP-15} we use the following definition +\begin{equation} + \label{eq:def_effspin-0} + \tilde{\zeta}(n,n_{2}) = + \begin{cases} + \sqrt{ 1 - 2 \; n_{2}/n^2 }, & \text{if } n^2 \ge 2 n_{2}, + \\ + 0, & \text{otherwise.} + \end{cases} +\end{equation} + +An alternative way to eliminate the $S_z$ dependence is to simply set $\zeta=0$, \ie, to resort to the spin-unpolarized functional. This lowers the accuracy for open-shell systems at $\mu=0$, \ie, for the usual PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$. Nevertheless, we argue that, for sufficiently large $\mu$, it is a viable option. Indeed, the purpose of introducing the spin polarization in semilocal density-functional approximations is to mimic the exact on-top pair density, \cite{PerSavBur-PRA-95} but our functional $\ecmd(\argecmd)$ already explicitly depends on the on-top pair density [see Eqs.~\eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}]. The dependencies on $\zeta$ and $n_2$ can thus be expected to be largely redundant. Consequently, we propose here to test the use of $\ecmd$ with \textit{a zero spin polarization}. This ensures its $S_z$ independence and, as will be numerically demonstrated, very weakly affects the complementary functional accuracy. + + +\subsubsection{Size consistency} + +Since $\efuncdenpbe{\argebasis}$ is computed via a single integral over $\mathbb{R}^3$ [see Eq.~\eqref{eq:def_ecmdpbebasis}] which involves only local quantities [$n(\br{})$, $\zeta(\br{})$, $s(\br{})$, $n_2(\br{})$, and $\mu(\br{})$], in the case of non-overlapping fragments $\text{A}+\text{B}$, it can be written as the sum of two local contributions: one coming from the integration over the region of subsystem \ce{A} and the other one from the region of subsystem \ce{B}. Therefore, a sufficient condition for size consistency is that these quantities locally coincide in the isolated fragments and in the supersystem $\text{A}+\text{B}$. Since these local quantities are calculated from the wave function $\psibasis$, a sufficient condition is that the wave function is multiplicatively separable in the limit of non-interacting fragments, \ie, $\ket*{\Psi_{\text{A}+\text{B}}^{\basis}} = \ket*{\Psi_{\ce{A}}^{\basis}} \otimes \ket*{\Psi_{\ce{B}}^{\basis}}$. We refer the interested reader to Appendix~\ref{app:sizeconsistency} for a detailed proof and discussion of the latter statement. +In the case where the two subsystems \ce{A} and \ce{B} dissociate in closed-shell systems, a simple RHF wave function ensures this property, but when one or several covalent bonds are broken, a properly chosen CASSCF wave function can be used to recover this property. The underlying active space must however be chosen in such a way that it leads to size-consistent energies in the limit of dissociated fragments. + + +\subsection{Actual approximations used for the complementary functional} +\label{sec:def_func} + +As the present work focuses on the strong-correlation regime, we propose here to investigate only approximate functionals which are $S_z$ independent and size-consistent in the case of covalent bond breaking. Therefore, the wave functions $\psibasis$ used throughout this paper are CASSCF wave functions in order to ensure size consistency of all local quantities. The difference between the different flavors of functionals are only due to the types of spin polarization and on-top pair density used. + +Regarding the spin polarization that enters into the function $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$, two different types of $S_z$-independent formulations are considered: i) the \textit{effective} spin polarization $\tilde{\zeta}$ defined in Eq.~\eqref{eq:def_effspin-0} and calculated from the CASSCF wave function, and ii) a \textit{zero} spin polarization. In the latter case, the functional is referred as to ``SU'' which stands for ``spin unpolarized''. + +Regarding the on-top pair density entering in Eq.~\eqref{eq:def_beta}, we use two different approximations. The first one is based on the uniform electron gas (UEG) and reads +\begin{equation} + \label{eq:def_n2ueg} + \ntwo^{\text{UEG}}(n,\zeta) \approx n^2\big(1-\zeta^2\big)g_0(n), +\end{equation} +where the pair-distribution function $g_0(n)$ is taken from Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. As the spin polarization appears in Eq.~\eqref{eq:def_n2ueg}, we use the effective spin polarization $\tilde{\zeta}$ of Eq.~\eqref{eq:def_effspin-0} in order to ensure $S_z$ independence. Thus, $\ntwo^{\text{UEG}}$ will depend indirectly on the on-top pair density of the CASSCF wave function through $\tilde{\zeta}$. When using $\ntwo^{\text{UEG}}(\br{}) \equiv \ntwo^{\text{UEG}}(n(\br{}),\tilde{\zeta}(\br{}))$ in a functional, we will refer to it as ``UEG''. + +The second approach to approximate the exact on-top pair density consists in using directly the on-top pair density of the CASSCF wave function. Following the work of some of the present authors, \cite{FerGinTou-JCP-18,GinSceTouLoo-JCP-19} we introduce the extrapolated on-top pair density +\begin{equation} + \label{eq:def_n2extrap} + \ntwoextrap(\ntwo,\mu) = \qty( 1 + \frac{2}{\sqrt{\pi}\mu} )^{-1} \; \ntwo, +\end{equation} +which directly follows from the large-$\mu$ extrapolation of the exact on-top pair density derived by Gori-Giorgi and Savin\cite{GorSav-PRA-06} in the context of RSDFT. Thus, the extrapolated on-top pair density $\ntwoextrap$ depends on the local range-separation function $\mu$. When using $\ntwoextrap(\br{}) \equiv \ntwoextrap(\ntwo(\br{}),\mu(\br{}))$ in a functional, we will simply refer it as ``OT'', which stands for "on-top". + +We then define three complementary functionals: +\begin{itemize} + +\item[i)] $\pbeuegXi$ which combines the effective spin polarization of Eq.~\eqref{eq:def_effspin-0} and the UEG on-top pair density defined in Eq.~\eqref{eq:def_n2ueg}: +\begin{multline} + \label{eq:def_pbeueg_i} + \bar{E}^\Bas_{\pbeuegXi} + \\ + = \int \d\br{} \,\denr \ecmd(\argrpbeuegXi), +\end{multline} +\item[ii)] $\pbeontXi$ which combines the effective spin polarization of Eq.~\eqref{eq:def_effspin-0} and the on-top pair density of Eq.~\eqref{eq:def_n2extrap}: +\begin{equation} + \label{eq:def_pbeueg_ii} + \bar{E}^\Bas_{\pbeontXi} = \int \d\br{} \,\denr \ecmd(\argrpbeontXi), +\end{equation} +%\item[iii)] $\pbeuegns$ which combines a zero spin polarization and the UEG on-top pair density of Eq.~\eqref{eq:def_n2ueg}: +%\begin{equation} +% \label{eq:def_pbeueg_iii} +% \bar{E}^\Bas_{\pbeuegns} = \int \d\br{} \,\denr \ecmd(\argrpbeuegns), +%\end{equation} +\item[iii)] $\pbeontns$ which combines a zero spin polarization and the on-top pair density of Eq.~\eqref{eq:def_n2extrap}: +\begin{equation} + \label{eq:def_pbeueg_iv} + \bar{E}^\Bas_{\pbeontns} = \int \d\br{} \,\denr \ecmd(\argrpbeontns). +\end{equation} +\end{itemize} +The performance of each of these functionals is tested in the following. Note that we did not define a spin-unpolarized version of the PBE-UEG functional because it would have been significantly inferior (in terms of performance) compared to the three other functionals. Indeed, because to the lack of knowledge on the spin polarization or on the accurate on-top pair density, such a functional would be inaccurate in particular for open-shell systems. This assumption has been numerically confirmed by calculations. + +%%%%%%%%%%%%%%%%%%%%%%%% +\section{Results} +\label{sec:results} +\begin{figure*} + \includegraphics[width=0.45\linewidth]{data/H10/DFT_vdzE_relat.eps} + \includegraphics[width=0.45\linewidth]{data/H10/DFT_vdzE_relat_zoom.eps} + \includegraphics[width=0.45\linewidth]{data/H10/DFT_vtzE_relat.eps} + \includegraphics[width=0.45\linewidth]{data/H10/DFT_vtzE_relat_zoom.eps} + \includegraphics[width=0.45\linewidth]{data/H10/DFT_vqzE_relat.eps} + \includegraphics[width=0.45\linewidth]{data/H10/DFT_vqzE_relat_zoom.eps} + \caption{ + Potential energy curves of the H$_{10}$ chain with equally-spaced atoms calculated with MRCI+Q and basis-set corrected MRCI+Q using the cc-pVDZ (top, labelled vdz), cc-pVTZ (middle, labelled as ``vtz''), and cc-pVQZ (bottom, labelled as ``vqz'') basis sets. + The MRCI+Q energies and the estimated exact energies have been extracted from Ref.~\onlinecite{h10_prx}. +% \alert{The reported energy is divided by the number of atoms in the chain.} + \label{fig:H10}} +\end{figure*} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{table*} +\caption{Atomization energies (in mHa) and associated errors (in square brackets) with respect to the estimated exact values computed at different levels of theory with various basis sets.} +\begin{ruledtabular} +\begin{tabular}{lrdddd} + + System & \tabc{Basis set} & \tabc{MRCI+Q\fnm[1]} & \tabc{(MRCI+Q)+$\pbeuegXi$} & \tabc{(MRCI+Q)+$\pbeontXi$} & \tabc{(MRCI+Q)+$\pbeontns$} \\ +\hline +\ce{H10} & cc-pVDZ & 622.1 [43.3] & 642.6 [22.8] & 649.2 [16.2] & 649.5 [15.9] \\ + & cc-pVTZ & 655.2 [10.2] & 661.9 [3.5] & 666.0 [-0.6] & 666.0 [-0.6] \\ + & cc-pVQZ & 661.2 [4.2] & 664.1 [1.3] & 666.4 [-1.0] & 666.5 [-1.1] \\[0.1cm] +%\cline{2-6} + & &\multicolumn{4}{c}{Estimated exact:\fnm[1] 665.4} \\[0.2cm] +\hline +% & & \tabc{exFCI} & \tabc{exFCI+$\pbeuegXi$} & \tabc{exFCI+$\pbeontXi$} & \tabc{exFCI+$\pbeontns$}\\ +%\hline +%\ce{C2} & aug-cc-pVDZ & 204.6 [29.5] & 218.0 [16.1] & 217.4 [16.7] & 217.0 [17.1] \\ +% & aug-cc-pVTZ & 223.4 [10.9] & 228.1 [6.0] & 228.6 [5.5] & 226.5 [5.6] \\[0.1cm] +%\cline{2-6} +% & & \multicolumn{4}{c}{Estimated exact:\fnm[2] 234.1} \\[0.2cm] +%\hline + & & \tabc{exFCI} & \tabc{exFCI+$\pbeuegXi$} & \tabc{exFCI+$\pbeontXi$} & \tabc{exFCI+$\pbeontns$}\\ +\hline +\ce{N2} & aug-cc-pVDZ & 321.9 [40.8] & 356.2 [6.5] & 355.5 [7.2] & 354.6 [8.1] \\ + & aug-cc-pVTZ & 348.5 [14.2] & 361.5 [1.2] & 363.5 [-0.5] & 363.2 [-0.3] \\ + & aug-cc-pVQZ & 356.6 [6.1 ] & 362.8 [-0.1] & 364.2 [-1.5] & 364.3 [-1.6] \\[0.1cm] + & & \multicolumn{4}{c}{Estimated exact:\fnm[2] 362.7} \\[0.2cm] +\hline + & & \tabc{exFCI} & \tabc{exFCI+$\pbeuegXi$} & \tabc{exFCI+$\pbeontXi$} & \tabc{exFCI+$\pbeontns$}\\ +\hline +\ce{O2} & aug-cc-pVDZ & 171.4 [20.5] & 187.6 [4.3] & 187.6 [4.3] & 187.1 [4.8] \\ + & aug-cc-pVTZ & 184.5 [7.4] & 190.3 [1.6] & 191.2 [0.7] & 191.0 [0.9] \\ + & aug-cc-pVQZ & 188.3 [3.6] & 190.3 [1.6] & 191.0 [0.9] & 190.9 [1.0] \\[0.1cm] + & & \multicolumn{4}{c}{Estimated exact:\fnm[2] 191.9} \\[0.2cm] +\hline + & & \tabc{exFCI} & \tabc{exFCI+$\pbeuegXi$} & \tabc{exFCI+$\pbeontXi$} & \tabc{exFCI+$\pbeontns$}\\ +\hline +\ce{F2} & aug-cc-pVDZ & 49.6 [12.6] & 54.8 [7.4] & 54.9 [7.3] & 54.8 [7.4] \\ + & aug-cc-pVTZ & 59.3 [2.9] & 61.2 [1.0] & 61.5 [0.7] & 61.5 [0.7] \\ + & aug-cc-pVQZ & 60.1 [2.1] & 61.0 [1.2] & 61.3 [0.9] & 61.3 [0.9] \\[0.1cm] + \cline{2-6}\\[-0.3cm] + & & \tabc{CEEIS\fnm[3]} & \tabc{CEEIS\fnm[3]+$\pbeuegXi$} & \tabc{CEEIS\fnm[3]+$\pbeontXi$} & \tabc{CEEIS\fnm[3]+$\pbeontns$}\\ + \cline{2-6}\\[-0.3cm] + & cc-pVDZ & 43.7 [18.5] & 51.0 [11.2] & 51.0 [11.2] & 50.7 [11.5] \\ + & cc-pVTZ & 56.3 [5.9] & 59.2 [3.0] & 59.6 [2.6] & 59.5 [2.7] \\ + & cc-pVQZ & 59.9 [2.3] & 61.3 [0.9] & 61.6 [0.6] & 61.6 [0.6] \\[0.1cm] + & & \multicolumn{4}{c}{Estimated exact:\fnm[2] 62.2} \\ +\end{tabular} +\end{ruledtabular} +\fnt[1]{From Ref.~\onlinecite{h10_prx}.} +\fnt[2]{From the CEEIS valence-only non-relativistic calculations of Ref.~\onlinecite{BytLaiRuedenJCP05}.} +\fnt[3]{From the CEEIS valence-only non-relativistic calculations of Ref.~\onlinecite{BytNagGorRue-JCP-07}.} +\label{tab:d0} +\end{table*} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\subsection{Computational details} + +We present potential energy curves of small molecules up to the dissociation limit +to investigate the performance of the basis-set correction in regimes of both weak and strong correlation. +The considered systems are the \ce{H10} linear chain with equally-spaced atoms, and the \ce{N2}, \ce{O2}, and \ce{F2} diatomics. + +The computation of the ground-state energy in Eq.~\eqref{eq:e0approx} in a given basis set requires approximations to the FCI energy $\efci$ and to the basis-set correction $\efuncbasisFCI$. +For diatomics with the aug-cc-pVDZ and aug-cc-pVTZ basis sets,~\cite{KenDunHar-JCP-92} energies are obtained using frozen-core selected-CI calculations (using the CIPSI algorithm) followed by the extrapolation scheme proposed by Holmes \textit{et al.} (see Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJac-JCTC-19, QP2} for more detail). All these calculations are performed with the latest version of \textsc{Quantum Package}, \cite{QP2} and will be labeled as exFCI in the following. In the case of \ce{F2}, we also use the correlation energy extrapolated by intrinsic scaling (CEEIS) \cite{BytNagGorRue-JCP-07} as an estimate of the FCI correlation energy with the cc-pVXZ (X $=$ D, T, and Q) basis sets.~\cite{Dun-JCP-89} The estimated exact potential energy curves are obtained from experimental data \cite{LieCle-JCP-74a} for the \ce{N2} and \ce{O2} molecules, and from CEEIS calculations in the case of \ce{F2}. For all geometries and basis sets, the error with respect to the exact FCI energies are estimated to be of the order of $0.5$~mHa. +For the three diatomics, we performed an additional exFCI calculation with the aug-cc-pVQZ basis set at the equilibrium geometry to obtain reliable estimates of the FCI/CBS dissociation energy. +In the case of the \ce{H10} chain, the approximation to the FCI energies together with the estimated exact potential energy curves are obtained from the data of Ref.~\onlinecite{h10_prx} where the authors performed MRCI+Q calculations with a minimal valence active space as reference (see below for the description of the active space). + +Regarding the complementary functional, we first perform full-valence CASSCF calculations with the GAMESS-US software~\cite{gamess} to obtain the wave function $\psibasis$. Then, all density-related quantities involved in the functional [density $n(\br{})$, effective spin polarization $\tilde{\zeta}(\br{})$, reduced density gradient $s(\br{})$, and on-top pair density $n_2(\br{})$] together with the local range-separation function $\mu(\br{})$ are calculated with this full-valence CASSCF wave function. The CASSCF calculations are performed with the following active spaces: (10e,10o) for \ce{H10}, (10e,8o) for \ce{N2}, (12e,8o) for \ce{O2}, and (14e,8o) for \ce{F2}. We note that, instead of using CASSCF wave functions for $\psibasis$, one could of course use the same selected-CI wave functions used for calculating the energy but the calculations of $n_2(\br{})$ and $\mu(\br{})$ would then be more costly. Another strategy would be to use for $\psibasis$ size-consistent truncated versions of the selected-CI wave functions but we did not explore this possibility in this work. + +Also, as the frozen-core approximation is used in all our selected-CI calculations, we use the corresponding valence-only complementary functionals (see Subsec.~\ref{sec:FC}). Therefore, all density-related quantities exclude any contribution from the 1s core orbitals, and the range-separation function follows the definition given in Eq.~\eqref{eq:def_mur_val}. + +It should be stressed that the computational cost of the basis-set correction (see Appendix~\ref{app:computational}) is negligible compared to the cost the selected-CI calculations. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{figure*} + \includegraphics[width=0.45\linewidth]{data/N2/DFT_avdzE_relat.eps} + \includegraphics[width=0.45\linewidth]{data/N2/DFT_avdzE_relat_zoom.eps} + \includegraphics[width=0.45\linewidth]{data/N2/DFT_avtzE_relat.eps} + \includegraphics[width=0.45\linewidth]{data/N2/DFT_avtzE_relat_zoom.eps} + \caption{ + Potential energy curves of the \ce{N2} molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top, labelled as ``avdz'') and aug-cc-pVTZ (bottom, labelled as ``avtz'') basis sets. The estimated exact energies are based on a fit of experimental data and obtained from Ref.~\onlinecite{LieCle-JCP-74a}. + \label{fig:N2}} +\end{figure*} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\subsection{H$_{10}$ chain} + +The \ce{H10} chain with equally-spaced atoms is a prototype of strongly correlated systems as it consists in the simultaneous breaking of 10 interacting covalent $\sigma$ bonds. +As it is a relatively small system, benchmark calculations at near-CBS values are available (see Ref.~\onlinecite{h10_prx} for a detailed study of this system). + +We report in Fig.~\ref{fig:H10} the potential energy curves computed using the cc-pVXZ (X $=$ D, T, and Q) basis sets for different levels of approximation, and the corresponding atomization energies are reported in Table \ref{tab:d0}. +As a general trend, the addition of the basis-set correction globally improves +the quality of the potential energy curves, independently of the approximation level of $\efuncbasis$. Also, no erratic behavior is found when stretching the bonds, which shows that the present procedure (\ie, the determination of the range-separation function and the definition of the functionals) is robust when reaching the strong-correlation regime. +In other words, smooth potential energy curves are obtained with the present basis-set correction. +More quantitatively, the values of the atomization energies are within chemical accuracy (\ie, an error below $1.4$ mHa) with the cc-pVTZ basis set when using the $\pbeontXi$ and $\pbeontns$ functionals, whereas such an accuracy is not yet reached at the standard MRCI+Q/cc-pVQZ level of theory. + +Analyzing more carefully the performance of the different types of approximate functionals, the results show that $\pbeontXi$ and $\pbeontns$ are very similar (the maximal difference on the atomization energy being 0.3 mHa), and that they give slightly more accurate results than $\pbeuegXi$. These findings provide two important clues on the role of the different physical ingredients included in these functionals: i) the explicit use of the on-top pair density originating from the CASSCF wave function [see Eq.~\eqref{eq:def_n2extrap}] is preferable over the use of the UEG on-top pair density [see Eq.~\eqref{eq:def_n2ueg}] which is somewhat understandable, and ii) removing the dependence on any kind of spin polarization does not lead to a significant loss of accuracy providing that one employs a qualitatively correct on-top pair density. The latter point is crucial as it confirms that the spin polarization in density-functional approximations essentially plays the same role as the on-top pair density. This could have significant implications for the construction of more robust families of density-functional approximations within DFT. + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%\begin{figure*} +% \includegraphics[width=0.45\linewidth]{data/C2/DFT_avdzE_relat.pdf} +% \includegraphics[width=0.45\linewidth]{data/C2/DFT_avdzE_relat_zoom.pdf} +% \includegraphics[width=0.45\linewidth]{data/C2/DFT_avtzE_relat.pdf} +% \includegraphics[width=0.45\linewidth]{data/C2/DFT_avtzE_relat_zoom.pdf} +% \caption{ +% Potential energy curves of the \ce{C2} molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets. The estimated exact energies are based on fit of experimental data and obtained from Ref.~\onlinecite{LieCle-JCP-74a}. +% \label{fig:C2}} +%\end{figure*} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{figure*} + \includegraphics[width=0.45\linewidth]{data/O2/DFT_avdzE_relat.eps} + \includegraphics[width=0.45\linewidth]{data/O2/DFT_avdzE_relat_zoom.eps} + \includegraphics[width=0.45\linewidth]{data/O2/DFT_avtzE_relat.pdf} + \includegraphics[width=0.45\linewidth]{data/O2/DFT_avtzE_relat_zoom.pdf} + \caption{ + Potential energy curves of the \ce{O2} molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top, labelled as ``avdz'') and aug-cc-pVTZ (bottom, labelled as ``avtz'') basis sets. The estimated exact energies are based on a fit of experimental data and obtained from Ref.~\onlinecite{LieCle-JCP-74a}. + \label{fig:O2}} +\end{figure*} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%\begin{figure*} +% \includegraphics[width=0.45\linewidth]{data/F2/DFT_vdzE_relat.pdf} +% \includegraphics[width=0.45\linewidth]{data/F2/DFT_vdzE_relat_zoom.pdf} +% \includegraphics[width=0.45\linewidth]{data/F2/DFT_vtzE_relat.pdf} +% \includegraphics[width=0.45\linewidth]{data/F2/DFT_vtzE_relat_zoom.pdf} +% \includegraphics[width=0.45\linewidth]{data/F2/DFT_vqzE_relat.pdf} +% \includegraphics[width=0.45\linewidth]{data/F2/DFT_vqzE_relat_zoom.pdf} +% \caption{ +% Potential energy curves of the \ce{F2} molecule calculated with CEEIS$^1$ and basis-set corrected CEEIS$^1$ using the cc-pVDZ (top), cc-pVTZ (middle) and cc-pVQZ (bottom) basis sets. +% The estimated exact energies are based on a fit of the valence-only extrapolated CEEIS data extracted from Ref.~\onlinecite{BytNagGorRue-JCP-07}. +% \label{fig:F2}} +%\end{figure*} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{figure*} + \includegraphics[width=0.45\linewidth]{data/F2/DFT_avdzE_relat.pdf} + \includegraphics[width=0.45\linewidth]{data/F2/DFT_avdzE_relat_zoom.pdf} + \includegraphics[width=0.45\linewidth]{data/F2/DFT_avtzE_relat.pdf} + \includegraphics[width=0.45\linewidth]{data/F2/DFT_avtzE_relat_zoom.pdf} + \caption{ + Potential energy curves of the \ce{F2} molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top, labelled as ``avdz''), aug-cc-pVTZ (bottom, labelled as ``avtz'') basis sets. + The estimated exact energies are based on a fit of the non-relativistic valence-only CEEIS data extracted from Ref.~\onlinecite{BytNagGorRue-JCP-07}. + \label{fig:F2}} +\end{figure*} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\subsection{Dissociation of diatomics} + +The \ce{N2}, \ce{O2} and \ce{F2} molecules are complementary to the \ce{H10} system for the present study. The level of strong correlation in these diatomics also increases while stretching the bonds, similarly to the case of \ce{H10}, but in addition these molecules exhibit more important and versatile types of weak correlations due to the larger number of electrons. Indeed, the short-range correlation effects are known to play a strong differential effect on the computation of the atomization energy at equilibrium, while the shape of the curve far from the equilibrium geometry is governed by dispersion interactions which are medium to long-range weak-correlation effects. \cite{AngDobJanGou-BOOK-20} The dispersion interactions in \ce{H10} play a minor role on the potential energy curve due to the much smaller number of near-neighbor electron pairs compared to \ce{N2}, \ce{O2} or \ce{F2}. Also, \ce{O2} has a triplet ground state and is therefore a good candidate for checking the spin-polarization dependence of the various functionals proposed here. + +We report in Figs.~\ref{fig:N2}, \ref{fig:O2}, and \ref{fig:F2} the potential energy curves of \ce{N2}, \ce{O2}, and \ce{F2} computed at various approximation levels using the aug-cc-pVDZ and aug-cc-pVTZ basis sets. The atomization energies for each level of theory with different basis sets are reported in Table \ref{tab:d0}. + +Just as in \ce{H10}, the accuracy of the atomization energies is globally improved by adding the basis-set correction and it is remarkable that $\pbeontXi$ and $\pbeontns$ provide again very similar results. The latter observation confirms that the dependence on the on-top pair density allows one to remove the dependence of any kind of spin polarization for a quite wide range of covalent bonds and also for an open-shell system like \ce{O2}. More quantitatively, an error below 1.0 mHa compared to the estimated exact valence-only atomization energy is found for \ce{N2}, \ce{O2}, and \ce{F2} with the aug-cc-pVTZ basis set using the $\pbeontns$ functional, whereas such a feat is far from being reached within the same basis set at the near-FCI level. In the case of \ce{F2} it is clear that the addition of diffuse functions in the double- and triple-$\zeta$ basis sets strongly improves the accuracy of the results, which could have be anticipated due to the strong breathing-orbital effect induced by the ionic valence-bond forms in this molecule. \cite{HibHumByrLen-JCP-94} +It should be also noticed that when reaching the aug-cc-pVQZ basis set for \ce{N2}, the accuracy of the atomization energy slightly deteriorates for the $\pbeontXi$ and $\pbeontns$ functionals, but it remains nevertheless more accurate than the estimated FCI atomization energy and very close to chemical accuracy. + +Regarding now the performance of the basis-set correction along the whole potential energy curve, it is interesting to notice that it fails to provide a noticeable improvement far from the equilibrium geometry. Acknowledging that the weak-correlation effects in these regions are dominated by dispersion interactions which are long-range effects, the failure of the present approximations for the complementary functional can be understood easily. Indeed, the whole scheme designed here is based on the physics of correlation near the electron-electron coalescence point: the local range-separation function $\mu(\br{})$ is based on the value of the effective electron-electron interaction at coalescence and the ECMD functionals are suited for short-range correlation effects. Therefore, the failure of the present basis-set correction to describe dispersion interactions is theoretically expected. +We hope to report further on this in the near future. + +\section{Conclusion} +\label{sec:conclusion} + +In the present paper we have extended the recently proposed DFT-based basis-set correction to strongly correlated systems. We have applied the method to the \ce{H10}, \ce{N2}, \ce{O2}, and \ce{F2} molecules up to the dissociation limit at near-FCI level in increasingly large basis sets, and investigated how the basis-set correction affects the convergence toward the CBS limit of the potential energy curves of these molecular systems. + +The density-based basis-set correction relies on three aspects: i) the definition of an effective non-divergent electron-electron interaction obtained from the expectation value over a wave function $\psibasis$ of the Coulomb electron-electron interaction projected into an incomplete basis set $\basis$; ii) the fit of this effective interaction with the long-range interaction used in RSDFT; and iii) the use of a short-range, complementary functional borrowed from RSDFT. In the present paper, we investigated i) and iii) in the context of strong correlation and focused on potential energy curves and atomization energies. More precisely, we proposed a new scheme to design functionals fulfilling spin-multiplet degeneracy and size consistency. To fulfill such requirements we proposed to use CASSCF wave functions leading to size-consistent energies, and we developed functionals using only $S_z$-independent density-like quantities. + +The development of new $S_z$-independent and size-consistent functionals has lead us to investigate the role of two related quantities: the spin polarization and the on-top pair density. One important result of the present study is that by using functionals \textit{explicitly} depending on the on-top pair density, one can eschew its spin-polarization dependence without loss of accuracy. This avoids the commonly used effective spin polarization originally proposed in Ref.~\onlinecite{BecSavSto-TCA-95} which has the disadvantage of possibly becoming complex-valued in the multideterminant case. From a more fundamental aspect, this confirms that, in a DFT framework, the spin polarization mimics the role of the on-top pair density. +Consequently, we believe that one could potentially develop new families of density-functional approximations where the spin polarization is abandoned and replaced by the on-top pair density. + +Regarding the results of the present approach, the basis-set correction systematically improves the near-FCI calculations in a given basis set. More quantitatively, it is shown that with only triple-$\zeta$ quality basis sets chemically accurate atomization energies are obtained for all systems whereas the uncorrected near-FCI results are far from this accuracy within the same basis set. + +Also, it is shown that the basis-set correction gives substantial differential contribution to potential energy curves close to the equilibrium geometries, but at long internuclear distances it cannot recover the dispersion interaction energy missing because of the basis-set incompleteness. This behavior is actually expected as dispersion interactions are of long-range nature and the present approach is designed to recover only short-range correlation effects. + +\appendix + +\section{Size consistency of the basis-set correction} +\label{app:sizeconsistency} + +\subsection{Sufficient condition for size consistency} + +The basis-set correction is expressed as an integral in real space +\begin{multline} + \label{eq:def_ecmdpbebasisAnnex} + \efuncdenpbe{\argebasis} = \\ \int \text{d}\br{} \,\denr \ecmd(\argrebasis), +\end{multline} +where all the local quantities $\argrebasis$ are obtained from the same wave function $\Psi$. In the limit of two non-overlapping and non-interacting dissociated fragments $\text{A}+\text{B}$, this integral can be rewritten as the sum of the integral over the region $\Omega_\text{A}$ and the integral over the region $\Omega_\text{B}$ +\begin{multline} + \label{eq:def_ecmdpbebasisAB} + \efuncdenpbeAB{\argebasis} = + \\ + \int_{\Omega_\text{A}} \text{d}\br{} \,\denr \ecmd(\argrebasis) + \\ + + \int_{\Omega_\text{B}} \text{d}\br{} \,\denr \ecmd(\argrebasis). +\end{multline} +Therefore, a sufficient condition to obtain size consistency is that all the local quantities $\argrebasis$ are \textit{intensive}, \ie, they \textit{locally} coincide in the supersystem $\text{A}+\text{B}$ and in each isolated fragment $\text{X}=\text{A}$ or $\text{B}$. Hence, we must have, for $\br{} \in \Omega_\text{X}$, +\begin{subequations} +\begin{gather} +n_\text{A+B}(\br{}) = n_\text{X}(\br{}), +\label{nAB} +\\ +\zeta_\text{A+B}(\br{}) = \zeta_\text{X}(\br{}), +\label{zAB} +\\ +s_\text{A+B}(\br{}) = s_\text{X}(\br{}), +\label{sAB} +\\ +n_{2,\text{A+B}}(\br{}) = n_{2,\text{X}}(\br{}), +\label{n2AB} +\\ +\mu_{\text{A+B}}(\br{}) = \mu_{\text{X}}(\br{}), +\label{muAB} +\end{gather} +\end{subequations} +where the left-hand-side quantities are for the supersystem and the right-hand-side quantities for an isolated fragment. Such conditions can be difficult to fulfill in the presence of degeneracies since the system X can be in a different mixed state (\ie, ensemble) in the supersystem $\text{A}+\text{B}$ and in the isolated fragment. \cite{Sav-CP-09} Here, we will consider the simple case where the wave function of the supersystem is multiplicatively separable, \ie, +\begin{equation} + \ket*{\wf{\text{A}+\text{B}}{}} = \ket*{\wf{\text{A}}{}} \otimes \ket*{\wf{\text{B}}{}}, +\label{PsiAB} +\end{equation} +where $\otimes$ is the antisymmetric tensor product. In this case, it is easy to shown that Eqs.~(\ref{nAB})-(\ref{sAB}) are valid, as well known, and it remains to show that Eqs.~(\ref{n2AB}) and~(\ref{muAB}) are also valid. Before showing this, we note that even though we do not explicitly consider the case of degeneracies, the lack of size consistency which could arise from spin-multiplet degeneracies can be avoided by the same strategy used for imposing the energy independence on $S_z$, \ie, by using the effective spin polarization $\tilde{\zeta}(n(\br{}),n_{2}(\br{}))$ or a zero spin polarization $\zeta(\br{}) = 0$. Moreover, the lack of size consistency which could arise from spatial degeneracies (\eg, coming from atomic $p$ states) can also be avoided by selecting the same member of the ensemble in the supersystem and in the isolated fragment. This applies to the systems treated in this work. + +\subsection{Intensivity of the on-top pair density and the local range-separation function} + +The on-top pair density can be written in an orthonormal spatial orbital basis set $\{\SO{p}{}\}$ as +\begin{equation} +\label{eq:def_n2} + n_{2{}}(\br{}) = \sum_{pqrs \in \Bas} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{}, +\end{equation} +with $\Gam{pq}{rs} = 2 \mel*{\wf{}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{}}$. As the summations run over all orbitals in the basis set $\Bas$, $n_{2{}}(\br{})$ is invariant to orbital rotations and can thus be expressed in terms of localized orbitals. For two non-overlapping fragments $\text{A}+\text{B}$, the basis set can then be partitioned into orbitals localized on the fragment A and orbitals localized on B, \ie, $\Bas=\Bas_\text{A}\cup \Bas_\text{B}$. Therefore, we see that the on-top pair density of the supersystem $\text{A}+\text{B}$ is additively separable +\begin{equation} + n_{2,\text{A}+\text{B}}(\br{}) = n_{2,\text{A}}(\br{}) + n_{2,\text{B}}(\br{}), +\end{equation} +where $n_{2,\text{X}}(\br{})$ is the on-top pair density of the fragment X +\begin{equation} + n_{2,\text{X}}(\br{}) = \sum_{pqrs \in \Bas_\text{X}} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{}, +\end{equation} +in which the elements $\Gam{pq}{rs}$ with orbital indices restricted to the fragment X are $\Gam{pq}{rs} = 2 \mel*{\wf{\text{A}+\text{B}}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{\text{A}+\text{B}}{}} = 2 \mel*{\wf{\text{X}}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{\text{X}}{}}$, owing to the multiplicative structure of the wave function [see Eq.~\eqref{PsiAB}]. This shows that the on-top pair density is a local intensive quantity. + +The local range-separation function is defined as, for $n_{2}(\br{}) \not=0$, +\begin{equation} + \label{eq:def_murAnnex} + \mur = \frac{\sqrt{\pi}}{2} \frac{f(\bfr{},\bfr{})}{n_{2}(\br{})}, +\end{equation} +where +\begin{equation} + \label{eq:def_f} + f(\bfr{},\bfr{}) = \sum_{pqrstu\in \Bas} \SO{p}{ } \SO{q}{ } \V{pq}{rs} \Gam{rs}{tu} \SO{t}{ } \SO{u}{ }. +\end{equation} +Again, $f(\bfr{},\bfr{})$ is invariant to orbital rotations and can be expressed in terms of orbitals localized on the fragments A and B. In the limit of infinitely separated fragments, the Coulomb interaction vanishes between A and B and therefore any two-electron integral $\V{pq}{rs}$ involving orbitals on both A and B vanishes. We thus see that the quantity $f(\bfr{},\bfr{})$ of the supersystem $\text{A}+\text{B}$ is additively separable +\begin{equation} + \label{eq:def_fa+b} + f_{\text{A}+\text{B}}(\bfr{},\bfr{}) = f_{\text{A}}(\bfr{},\bfr{}) + f_{\text{B}}(\bfr{},\bfr{}), +\end{equation} +with +\begin{equation} + \label{eq:def_fX} + f_\text{X}(\bfr{},\bfr{}) = \sum_{pqrstu\in \Bas_\text{X}} \SO{p}{ } \SO{q}{ } \V{pq}{rs} \Gam{rs}{tu} \SO{t}{ } \SO{u}{ }. +\end{equation} +So, $f(\bfr{},\bfr{})$ is a local intensive quantity. +As a consequence, the local range-separation function of the supersystem $\text{A}+\text{B}$ is +\begin{equation} + \label{eq:def_murAB} + \mu_{\text{A}+\text{B}}(\bfr{}) = \frac{\sqrt{\pi}}{2} \frac{f_{\text{A}}(\bfr{},\bfr{}) + f_{\text{B}}(\bfr{},\bfr{})}{n_{2,\text{A}}(\br{}) + n_{2,\text{B}}(\br{})}, +\end{equation} +which implies +\begin{equation} + \label{eq:def_murABsum} + \mu_{\text{A}+\text{B}}(\bfr{}) = \mu_{\text{X}}(\bfr{}) \;\; \text{if} \;\; \bfr{} \in \Omega_\text{X}, +\end{equation} +where $\mu_{\text{X}}(\bfr{}) = (\sqrt{\pi}/2) f_{\text{X}}(\bfr{},\bfr{})/n_{2,\text{X}}(\br{})$. +The local range-separation function is thus a local intensive quantity. + +We can therefore conclude that, if the wave function of the supersystem $\text{A}+\text{B}$ is multiplicative separable, all local quantities used in the basis-set correction functional are intensive and therefore the basis-set correction is size consistent. + +\section{Computational cost of the basis-set correction for a CASSCF wave function} +\label{app:computational} + +The computational cost of the basis-set correction is determined by the calculation of the on-top pair density $n_{2}(\br{})$ and the local range-separation function $\mur$ on the real-space grid. For a general multideterminant wave function, the computational cost is of order $O(N_\text{grid}N_{\Bas}^4)$ where $N_\text{grid}$ is the number of grid points and $N_{\Bas}$ is the number of basis functions.\cite{LooPraSceTouGin-JCPL-19} For a CASSCF wave function, a significant reduction of the scaling of the computational cost can be achieved. + +\subsection{Computation of the on-top pair density} + +For a CASSCF wave function $\Psi$, the occupied orbitals can be partitioned into a set of active orbitals $\mathcal{A}$ and a set of inactive (doubly occupied) orbitals $\mathcal{I}$. The CASSCF on-top pair density can then be written as +\begin{equation} + \label{def_n2_good} + n_{2}(\br{}) = n_{2,\mathcal{A}}(\br{}) + n_{\mathcal{A}}(\br{}) n_{\mathcal{I}}(\br{}) + \frac{n_{\mathcal{I}}(\br{})^2}{2}, +\end{equation} +where +\begin{subequations} +\begin{align} + \label{def_n2_act} + n_{2,\mathcal{A}}(\br{}) & = \sum_{pqrs \in \mathcal{A}} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{}, + \\ + n_{\mathcal{A}}(\br{}) & = \sum_{pq \in\mathcal{A}} \phi_p (\br{}) \phi_q (\br{}) + \mel*{\wf{}{}}{ \aic{p_\uparrow}\ai{q_\uparrow} + \aic{p_\downarrow}\ai{q_\downarrow} }{\wf{}{}}, + \\ + n_{\mathcal{I}}(\br{}) & = 2 \sum_{p \in \mathcal{I}} \phi_p (\br{})^2 +\end{align} +\end{subequations} +are the purely active part of the on-top pair density, the active part of the density, and the inactive part of the density, respectively. +The leading computational cost is the evaluation of $n_{2,\mathcal{A}}(\br{})$ on the grid which, according to Eq.~\eqref{def_n2_act}, scales as $O(N_\text{grid} N_\mathcal{A}^4)$ where $N_{\mathcal{A}}$ is the number of active orbitals which is much smaller than the number of basis functions $N_{\Bas}$. + +\subsection{Computation of the local range-separation function} + +In addition to the on-top pair density, the computation of $\mur$ needs the computation of $f(\bfr{},\bfr{})$ [see Eq.~\eqref{eq:def_f}] at each grid point. It can be factorized as +\begin{equation} + \label{eq:f_good} + f(\bfr{},\bfr{}) = \sum_{rs \in \Bas} V^{rs}(\bfr{}) \, \Gamma_{rs}(\bfr{}), +\end{equation} +where +\begin{subequations} +\begin{align} + V^{rs}(\bfr{}) & = \sum_{pq \in \Bas} V_{pq}^{rs} \phi_p(\br{}) \phi_q(\br{}), + \\ + \Gamma_{rs}(\bfr{}) & = \sum_{pq \in \Bas} \Gam{rs}{pq} \phi_p(\br{}) \phi_q(\br{}) . +\end{align} +\end{subequations} +For a general multideterminant wave function, the computational cost of $f(\bfr{},\bfr{})$ thus scales as $O(N_\text{grid}N_{\Bas}^4)$. + +In the case of a CASSCF wave function, $\Gam{rs}{pq}$ vanishes if one index $p,q,r,s$ does not belong to the set of inactive or active occupied orbitals $\mathcal{I}\cup \mathcal{A}$. Therefore, at a given grid point, the number of non-zero elements $\Gamma_{rs}(\bfr{})$ is only at most $(N_{\mathcal{I}}+N_{\mathcal{A}})^2$, which is usually much smaller than $N_{\Bas}^2$. As a consequence, one can also restrict the sum in the calculation of +\begin{equation} + f(\bfr{},\bfr{}) = \sum_{rs \in \mathcal{I}\cup\mathcal{A}} V^{rs}(\bfr{}) \, \Gamma_{rs}(\bfr{}). +\end{equation} +The overall computational cost is dominated by that of $V^{rs}(\bfr{})$, which scales as $O(N_\text{grid}(N_{\mathcal{I}}+N_{\mathcal{A}})^2 N_{\Bas}^2)$, which is much smaller than $O(N_\text{grid}N_{\Bas}^4)$. + +\bibliography{srDFT_SC} + +\end{document} diff --git a/Revised_Manuscript/srDFT_SC_old.tex b/Revised_Manuscript/srDFT_SC_old.tex new file mode 100644 index 0000000..c0d59c6 --- /dev/null +++ b/Revised_Manuscript/srDFT_SC_old.tex @@ -0,0 +1,753 @@ +\documentclass[aip,jcp,reprint,noshowkeys]{revtex4-1} +%\documentclass[aip,jcp,noshowkeys]{revtex4-1} +\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable} + +\usepackage{mathpazo,libertine} +\usepackage[normalem]{ulem} +\newcommand{\alert}[1]{\textcolor{red}{#1}} +\definecolor{darkgreen}{RGB}{0, 180, 0} +\newcommand{\beurk}[1]{\textcolor{darkgreen}{#1}} +\newcommand{\trash}[1]{\textcolor{red}{\sout{#1}}} +\usepackage{xspace} + +\usepackage{hyperref} +\hypersetup{ + colorlinks=true, + linkcolor=blue, + filecolor=blue, + urlcolor=blue, + citecolor=blue +} +\newcommand{\cdash}{\multicolumn{1}{c}{---}} +\newcommand{\mc}{\multicolumn} +\newcommand{\fnm}{\footnotemark} +\newcommand{\fnt}{\footnotetext} +\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}} +\newcommand{\mr}{\multirow} +\newcommand{\SI}{\textcolor{blue}{supporting information}} + +% second quantized operators +\newcommand{\psix}[1]{\hat{\Psi}\left({\bf X}_{#1}\right)} +\newcommand{\psixc}[1]{\hat{\Psi}^{\dagger}\left({\bf X}_{#1}\right)} +\newcommand{\ai}[1]{\hat{a}_{#1}} +\newcommand{\aic}[1]{\hat{a}^{\dagger}_{#1}} +\newcommand{\vijkl}[0]{V_{ij}^{kl}} +\newcommand{\phix}[2]{\phi_{#1}(\bfr{#2})} +\newcommand{\phixprim}[2]{\phi_{#1}(\bfr{#2}')} + + +%operators +\newcommand{\elemm}[3]{{\ensuremath{\bra{#1}{#2}\ket{#3}}\xspace}} +\newcommand{\ovrlp}[2]{{\ensuremath{\langle #1|#2\rangle}\xspace}} + +%\newcommand{\ket}[1]{{\ensuremath{|#1\rangle}\xspace}} +%\newcommand{\bra}[1]{{\ensuremath{\langle #1|}\xspace}} + +% + + +% energies +\newcommand{\Ec}{E_\text{c}} +\newcommand{\EPT}{E_\text{PT2}} +\newcommand{\EsCI}{E_\text{sCI}} +\newcommand{\EDMC}{E_\text{DMC}} +\newcommand{\EexFCI}{E_\text{exFCI}} +\newcommand{\EexFCIbasis}{E_\text{exFCI}^{\Bas}} +\newcommand{\EexFCIinfty}{E_\text{exFCI}^{\infty}} +\newcommand{\EexDMC}{E_\text{exDMC}} +\newcommand{\Ead}{\Delta E_\text{ad}} +\newcommand{\efci}[0]{E_{\text{FCI}}^{\Bas}} +\newcommand{\emodel}[0]{E_{\model}^{\Bas}} +\newcommand{\emodelcomplete}[0]{E_{\model}^{\infty}} +\newcommand{\efcicomplete}[0]{E_{\text{FCI}}^{\infty}} +\newcommand{\ecccomplete}[0]{E_{\text{CCSD(T)}}^{\infty}} +\newcommand{\ecc}[0]{E_{\text{CCSD(T)}}^{\Bas}} +\newcommand{\efuncbasisFCI}[0]{\bar{E}^\Bas[\denFCI]} +\newcommand{\efuncbasisfci}[0]{\bar{E}^\Bas[\denfci]} +\newcommand{\efuncbasis}[0]{\bar{E}^\Bas[\den]} +\newcommand{\efuncden}[1]{\bar{E}^\Bas[#1]} +\newcommand{\efuncdenpbe}[1]{\bar{E}_{\text{X}}^\Bas[#1]} +\newcommand{\ecompmodel}[0]{\bar{E}^\Bas[\denmodel]} +\newcommand{\ecmubis}[0]{\bar{E}_{\text{c,md}}^{\text{sr}}[\denr;\,\mu]} +\newcommand{\ecmubisldapbe}[0]{\bar{E}_{\text{c,md}\,\text{PBE}}^{\text{sr}}[\denr;\,\mu]} +\newcommand{\ecmuapprox}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mu]} +\newcommand{\ecmuapproxmur}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mur]} +\newcommand{\ecmuapproxmurfci}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denfci;\,\mur]} +\newcommand{\ecmuapproxmurmodel}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denmodel;\,\mur]} +\newcommand{\ecompmodellda}[0]{\bar{E}_{\text{LDA}}^{\Bas,\wf{}{\Bas}}[\denmodel]} +\newcommand{\ecompmodelldaval}[0]{\bar{E}_{\text{LDA, val}}^{\Bas,\wf{}{\Bas}}[\den]} +\newcommand{\ecompmodelpbe}[0]{\bar{E}_{\text{PBE}}^{\Bas,\wf{}{\Bas}}[\den]} +\newcommand{\ecompmodelpbeval}[0]{\bar{E}_{\text{PBE, val}}^{\Bas,\wf{}{\Bas}}[\den]} +\newcommand{\emulda}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denr;\mu({\bf r};\wf{}{\Bas})\right)} +\newcommand{\emuldamodel}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denmodelr;\mu({\bf r};\wf{}{\Bas})\right)} +\newcommand{\emuldaval}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denval ({\bf r});\murval;\wf{}{\Bas})\right)} +\newcommand{\ecmd}[0]{\varepsilon^{\text{c,md}}_{\text{PBE}}} +\newcommand{\psibasis}[0]{\Psi^{\basis}} + +%pbeuegxiHF +\newcommand{\pbeuegxihf}{\text{PBE-UEG-}\zeta\text{-HF}^\Bas} +\newcommand{\argpbeuegxihf}[0]{\den,\zeta,s,\ntwo_{\text{UEG}},\mu_{\text{HF}}^{\basis}} +\newcommand{\argrpbeuegxihf}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo_{\text{UEG}}(\br{}),\mu_{\text{HF}}^{\basis}(\br{})} +%pbeuegxiCAS +\newcommand{\pbeuegxicas}{\text{PBE-UEG-}\zeta\text{-CAS}^\Bas} +\newcommand{\argpbeuegxicas}[0]{\den,\zeta,s,\ntwo_{\text{UEG}},\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeuegxicas}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo_{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})} +%pbeuegXiCAS +\newcommand{\pbeuegXicas}{\text{PBE-UEG-}\tilde{\zeta}\text{-CAS}^\Bas} +\newcommand{\argpbeuegXicas}[0]{\den,\tilde{\zeta},s,\ntwo_{\text{UEG}},\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeuegXicas}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),\ntwo_{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})} +%pbeontxiCAS +\newcommand{\pbeontxicas}{\text{PBE-ONT-}\zeta\text{-CAS}^\Bas} +\newcommand{\argpbeontxicas}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeontxicas}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})} +%pbeontXiCAS +\newcommand{\pbeontXicas}{\text{PBE-ONT-}\tilde{\zeta}\text{-CAS}^\Bas} +\newcommand{\argpbeontXicas}[0]{\den,\tilde{\zeta},s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeontXicas}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})} +%pbeont0xiCAS +\newcommand{\pbeontnscas}{\text{PBE-ONT-}0\tilde{\zeta}\text{-CAS}^\Bas} +\newcommand{\argpbeontnscas}[0]{\den,0,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeontnscas}[0]{\den(\br{}),0,s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})} + +%%%%%% arguments + +\newcommand{\argepbe}[0]{\den,\zeta,s} +\newcommand{\argebasis}[0]{\den,\zeta,s,\ntwo,\mu_{\Psi^{\basis}}} +\newcommand{\argecmd}[0]{\den,\zeta,s,\ntwo,\mu} +\newcommand{\argepbeueg}[0]{\den,\zeta,s,\ntwo_{\text{UEG}},\mu_{\Psi^{\basis}}} +\newcommand{\argepbeontxicas}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} +\newcommand{\argepbeuegXihf}[0]{\den,\tilde{\zeta},s,\ntwo_{\text{UEG}},\mu_{\Psi^{\basis}}} +\newcommand{\argrebasis}[0]{\denr,\zeta(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})} +\newcommand{\argrebasisab}[0]{\denr,\zeta(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})} + + +% numbers +\newcommand{\rnum}[0]{{\rm I\!R}} +\newcommand{\bfr}[1]{{\bf r}_{#1}} +\newcommand{\dr}[1]{\text{d}\bfr{#1}} +\newcommand{\rr}[2]{\bfr{#1}, \bfr{#2}} +\newcommand{\rrrr}[4]{\bfr{#1}, \bfr{#2},\bfr{#3},\bfr{#4} } + + + +% effective interaction +\newcommand{\twodm}[4]{\elemm{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}} +\newcommand{\murpsi}[0]{\mu({\bf r};\wf{}{\Bas})} +\newcommand{\ntwo}[0]{n^{(2)}} +\newcommand{\ntwohf}[0]{n^{(2),\text{HF}}} +\newcommand{\ntwophi}[0]{n^{(2)}_{\phi}} +\newcommand{\ntwoextrap}[0]{\tilde{n}^{(2)}_{\psibasis}} +\newcommand{\ntwoextrapcas}[0]{\tilde{n}^{(2)\,\basis}_{\text{CAS}}} +\newcommand{\mur}[0]{\mu({\bf r})} +\newcommand{\murr}[1]{\mu({\bf r}_{#1})} +\newcommand{\murval}[0]{\mu_{\text{val}}({\bf r})} +\newcommand{\murpsival}[0]{\mu_{\text{val}}({\bf r};\wf{}{\Bas})} +\newcommand{\murrval}[1]{\mu_{\text{val}}({\bf r}_{#1})} +\newcommand{\weeopmu}[0]{\hat{W}_{\text{ee}}^{\text{lr},\mu}} + + +\newcommand{\wbasis}[0]{W_{\wf{}{\Bas}}(\bfr{1},\bfr{2})} +\newcommand{\wbasiscoal}[0]{W_{\wf{}{\Bas}}(\bfr{},\bfr{})} +\newcommand{\wbasisval}[0]{W_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})} +\newcommand{\fbasis}[0]{f_{\wf{}{\Bas}}(\bfr{1},\bfr{2})} +\newcommand{\fbasisval}[0]{f_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})} + \newcommand{\ontop}[2]{ n^{(2)}_{#1}({\bf #2}_1)} + \newcommand{\twodmrpsi}[0]{ \ntwo_{\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})} + \newcommand{\twodmrdiagpsi}[0]{ \ntwo_{\wf{}{\Bas}}(\rr{1}{2})} + \newcommand{\twodmrdiagpsival}[0]{ \ntwo_{\wf{}{\Bas},\,\text{val}}(\rr{1}{2})} + \newcommand{\gammamnpq}[1]{\Gamma_{mn}^{pq}[#1]} + \newcommand{\gammamnkl}[0]{\Gamma_{mn}^{kl}} + \newcommand{\gammaklmn}[1]{\Gamma_{kl}^{mn}[#1]} +%\newcommand{\wbasiscoal}[1]{W_{\wf{}{\Bas}}({\bf r}_{#1})} +\newcommand{\ontoppsi}[1]{ n^{(2)}_{\wf{}{\Bas}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})} +\newcommand{\wbasiscoalval}[1]{W_{\wf{}{\Bas}}^{\text{val}}({\bf r}_{#1})} +\newcommand{\ontoppsival}[1]{ n^{(2)}_{\wf{}{\Bas}}^{\text{val}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})} + + + +\newcommand{\ex}[4]{$^{#1}#2_{#3}^{#4}$} +\newcommand{\ra}{\rightarrow} +\newcommand{\De}{D_\text{e}} + +% MODEL +\newcommand{\model}[0]{\mathcal{Y}} + +% densities +\newcommand{\denmodel}[0]{\den_{\model}^\Bas} +\newcommand{\denmodelr}[0]{\den_{\model}^\Bas ({\bf r})} +\newcommand{\denfci}[0]{\den_{\psifci}} +\newcommand{\denFCI}[0]{\den^{\Bas}_{\text{FCI}}} +\newcommand{\denhf}[0]{\den_{\text{HF}}^\Bas} +\newcommand{\denrfci}[0]{\denr_{\psifci}} +\newcommand{\dencipsir}[0]{{n}_{\text{CIPSI}}^\Bas({\bf r})} +\newcommand{\dencipsi}[0]{{n}_{\text{CIPSI}}^\Bas} +\newcommand{\den}[0]{{n}} +\newcommand{\denval}[0]{{n}^{\text{val}}} +\newcommand{\denr}[0]{{n}({\bf r})} +\newcommand{\onedmval}[0]{\rho_{ij,\sigma}^{\text{val}}} + +% wave functions +\newcommand{\psifci}[0]{\Psi^{\Bas}_{\text{FCI}}} +\newcommand{\psimu}[0]{\Psi^{\mu}} +% operators +\newcommand{\weeopbasis}[0]{\hat{W}_{\text{ee}}^\Bas} +\newcommand{\kinop}[0]{\hat{T}} + +\newcommand{\weeopbasisval}[0]{\hat{W}_{\text{ee}}^{\Basval}} +\newcommand{\weeop}[0]{\hat{W}_{\text{ee}}} + + +% units +\newcommand{\IneV}[1]{#1 eV} +\newcommand{\InAU}[1]{#1 a.u.} +\newcommand{\InAA}[1]{#1 \AA} + + +% methods +\newcommand{\UEG}{\text{UEG}} +\newcommand{\LDA}{\text{LDA}} +\newcommand{\PBE}{\text{PBE}} +\newcommand{\FCI}{\text{FCI}} +\newcommand{\CCSDT}{\text{CCSD(T)}} +\newcommand{\lr}{\text{lr}} +\newcommand{\sr}{\text{sr}} + +\newcommand{\Nel}{N} +\newcommand{\V}[2]{V_{#1}^{#2}} + + +\newcommand{\n}[2]{n_{#1}^{#2}} +\newcommand{\E}[2]{E_{#1}^{#2}} +\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}} +\newcommand{\bEc}[1]{\Bar{E}_\text{c}^{#1}} +\newcommand{\e}[2]{\varepsilon_{#1}^{#2}} +\newcommand{\be}[2]{\Bar{\varepsilon}_{#1}^{#2}} +\newcommand{\bec}[1]{\Bar{e}^{#1}} +\newcommand{\wf}[2]{\Psi_{#1}^{#2}} +\newcommand{\W}[2]{W_{#1}^{#2}} +\newcommand{\w}[2]{w_{#1}^{#2}} +\newcommand{\hn}[2]{\Hat{n}_{#1}^{#2}} +\newcommand{\rsmu}[2]{\mu_{#1}^{#2}} +\newcommand{\SO}[2]{\phi_{#1}(\br{#2})} + +\newcommand{\modX}{\text{X}} +\newcommand{\modY}{\text{Y}} + +% basis sets +\newcommand{\setdenbasis}{\mathcal{N}_{\Bas}} +\newcommand{\Bas}{\mathcal{B}} +\newcommand{\basis}{\mathcal{B}} +\newcommand{\Basval}{\mathcal{B}_\text{val}} +\newcommand{\Val}{\mathcal{V}} +\newcommand{\Cor}{\mathcal{C}} + +% operators +\newcommand{\hT}{\Hat{T}} +\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}} +\newcommand{\f}[2]{f_{#1}^{#2}} +\newcommand{\Gam}[2]{\Gamma_{#1}^{#2}} + +% coordinates +\newcommand{\br}[1]{{\mathbf{r}_{#1}}} +\newcommand{\bx}[1]{\mathbf{x}_{#1}} +\newcommand{\dbr}[1]{d\br{#1}} +\newcommand{\PBEspin}{PBEspin} +\newcommand{\PBEueg}{PBE-UEG-{$\tilde{\zeta}$}} + +\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} +\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, France} + +\begin{document} + +\title{Mixing density functional theory and wave function theory for strong correlation: the best of both worlds} + +\begin{abstract} +bla bla bla youpi tralala +\end{abstract} + +\maketitle + +%%%%%%%%%%%%%%%%%%%%%%%% +\section{Introduction} +%%%%%%%%%%%%%%%%%%%%%%%% +The main goal of quantum chemistry is to propose reliable theoretical tools to describe the rich area of chemistry. +The accurate computation of the electronic structure of molecular systems plays a central role in the development of methods in quantum chemistry, +but despite intense developments, no definitive solution to that problem have been found. +The theoretical challenge to be overcome falls back in the category of the quantum many-body problem due the intrinsic quantum nature +of the electrons and the coulomb repulsion between them, inducing the so-called electronic correlation problem. +Tackling this problem translate to solving the Schroedinger equation for a $N$~-~electron system, and two roads have emerged to approximate the solution to this formidably complex mathematical problem: the wave function theory (WFT) and density functional theory (DFT). +Although both WFT and DFT spring from the same problem, their formalisms are very different as the former deals with the complex +$N$~-~body wave function whereas the latter handles the much simpler one~-~body density. +The computational cost of DFT is very appealing as in its Kohn-Sham (KS) formulation it can be recast in a mean-field procedure. +Therefore, although constant efforts are performed to reduce the computational cost of WFT, DFT remains still the workhorse of quantum chemistry. + +From the theoretician point of view, the complexity of description of a given chemical system can be roughly +categorized by the strength of the electronic correlation appearing in its electronic structure. +Weakly correlated systems, such as closed-shell organic molecules near their equilibrium geometry, are typically dominated by the avoidance effects when electron are near the electron coalescence point, which are often called short-range correlation effects, +or far from each other, typically dispersion forces. The theoretical description of weakly correlated systems is one of the more concrete achievement +of quantum chemistry, and the main remaining issue for these systems is to push the limit in terms of the size of the chemical systems that can be treated. +The case of the so-called strongly correlated systems, which are ubiquitous in chemistry, is much more problematic as they exhibits +a much more exotic electronic structure. +Transition metals containing systems, low-spin open shell systems, covalent bond breaking or excited states +have all in common that they cannot be even qualitatively described by a single electronic configuration. +It is now clear that the usual approximations in KS-DFT fails in giving an accurate description of these situations and WFT has become +the standard for the treatment of strongly correlated systems. +From the theoretical point of view, the complexity of the strong correlation problem is, at least, two-fold: +i) the presence of near degeneracies and/or strong interactions among a primary set of electronic configurations +(the size of which can potentially scale exponentially in some cases) determines the qualitative description of the wave function, +ii) the quantitative description of the systems must take into account weak correlation effects which requires to take into account many +other electronic configurations with typically much smaller weights in the wave function. +Fulfilling these two objectives is a rather complicated task, specially if one adds the requirement of size-extensivity and additivity of the computed energy in the case of non interacting fragments, which is a very desirable property for any approximated method. + +%To tackle this complicated problem, many methods have been proposed and an exhaustive review of the zoology of methods for strong correlation goes beyond the scope and purpose of this article. + +To tackle this problem, many WFT methods have emerged which can be categorized in two branches: the single-reference (SR) +and multi-reference (MR) methods. +The SR methods rely on a single electronic configuration as a zeroth-order wave function, typically Hartree-Fock (HF). +Then the electron correlation is introduced by increasing the rank of multiple hole-particle excitations, +preferably treated in a coupled-cluster (CC) fashion for the sake of compactness of the wave function and extensivity of the computed energies. +The advantage of these approaches rely on the rather straightforward way to improve the level of accuracy, +which consists in increasing the rank of the excitation operators used to generate the CC wave function. +Despite its appealing elegant simplicity, the computational cost of the CC methods increase drastically with the rank of the excitation +operators, even if promising alternative approaches have been proposed using stochastic techniques\cite{Thom-PRL-10,ScoTho-JCP-17,SpeNeuVigFraTho-JCP-18,DeuEmiShePie-PRL-17,DeuEmiMagShePie-JCP-18,DeuEmiYumShePie-JCP-19} or symmetry-broken approaches\cite{QiuHenZhaScu-JCP-17,QiuHenZhaScu-JCP-18,GomHenScu-JCP-19}. +In the MR approaches, the zeroth order wave function consists in a linear combination of Slater determinants which are supposed to concentrate most of strong interactions and near degeneracies inherent in the structure of the Hamiltonian for a strongly correlated system. The usual approach to build such a zeroth-order wave function is to perform a complete active space self consistent field (CASSCF) whose variational property prevent any divergence, and which can provide extensive energies. Of course, the choice of the active space is rather a subtle art and the CASSCF results might strongly depend on the level of chemical/physical knowledge of the user. +On top of this zeroth-order wave function, weak correlation is introduced by the addition of other configurations through either configuration interaction\cite{WerKno-JCP-88,KnoWer-CPL-88} (MRCI) or perturbation theory (MRPT) and even coupled cluster (MRCC), which have their strengths and weaknesses, +The advantage of MRCI approaches rely essentially in their simple linear parametrisation for the wave function together with the variational property of their energies, whose inherent drawback is the lack of size extensivity of their energies unless reaching the FCI limit. On the other hand, MRPT and MRCC can provide extensive energies but to the price of rather complicated formalisms, and these approaches might be subject to divergences and/or convergence problems due to the non linearity of the parametrisation for MRCC or a too poor choice of the zeroth-order Hamiltonian. +A natural alternative is to combine MRCI and MRPT, which falls in the category of selected CI (SCI) which goes back to the late 60's and who has received a revival of interest and applications during the last decade \cite{BenErn-PhysRev-1969,WhiHac-JCP-1969,HurMalRan-1973,EvaDauMal-ChemPhys-83,Cim-JCP-1985,Cim-JCC-1987,IllRubRic-JCP-88,PovRubIll-TCA-92,BunCarRam-JCP-06,AbrSheDav-CPL-05,MusEngels-JCC-06,BytRue-CP-09,GinSceCaf-CJC-13,CafGinScemRam-JCTC-14,GinSceCaf-JCP-15,CafAplGinScem-arxiv-16,CafAplGinSce-JCP-16,SchEva-JCP-16,LiuHofJCTC-16,HolUmrSha-JCP-17,ShaHolJeaAlaUmr-JCTC-17,HolUmrSha-JCP-17,SchEva-JCTC-17,PerCle-JCP-17,OhtJun-JCP-17,Zim-JCP-17,LiOttHolShaUmr-JCP-2018,ChiHolOttUmrShaZim-JPCA-18,SceBenJacCafLoo-JCP-18,LooSceBloGarCafJac-JCTC-18,GarSceGinCaffLoo-JCP-18,SceGarCafLoo-JCTC-18,GarGinMalSce-JCP-16,LooBogSceCafJac-JCTC-19}, and among which the CI perturbatively selected iteratively (CIPSI) can be considered as a pioneer. The main idea of the CIPSI and other related SCI algorithms is to iteratively select the most important Slater determinants thanks to perturbation theory in order to build a MRCI zeroth-order wave function which automatically concentrate the strongly interacting part of the wave function. On top of this MRCI zeroth-order wave function, a rather simple MRPT approach is used to recover the missing weak correlation and the process is iterated until reaching a given stopping criterion. It is important to notice that in the SCI algorithms, neither the SCI or the MRPT are size extensive \text{per se}, but the extensivity property is almost recovered by approaching the FCI limit. +When the SCI are affordable, their clear advantage are that they provide near FCI wave functions and energies, whatever the level of knowledge of the user on the specific physical/chemical problem considered. The drawback of SCI is certainly their \textit{intrinsic} exponential scaling due to their linear parametrisation. Nevertheless, such an exponential scaling is lowered by the smart selection of the zeroth-order wave function together with the MRPT calculation. + +Besides the difficulties of accurately describing the electronic structure within a given basis set, a crucial component of the limitations of applicability of WFT concerns the slow convergence of the energies and properties with respect to the quality of the basis set. As initially shown by the seminal work of Hylleraas\cite{Hyl-ZP-29} and further developed by Kutzelnigg \textit{et. al.}\cite{Kut-TCA-85,KutKlo-JCP-91, NogKut-JCP-94}, the main convergence problem originates from the divergence of the coulomb interaction at the electron coalescence point, which induces a discontinuity in the first-derivative of the wave function (the so-called electron-electron cusp). Describing such a discontinuity with an incomplete basis set is impossible and as a consequence, the convergence of the computed energies and properties can be strongly affected. To attenuate this problem, extrapolation techniques has been developed, either based on the Hylleraas's expansion of the coulomb operator\cite{HalHelJorKloKocOlsWil-CPL-98}, or more recently based on perturbative arguments\cite{IrmHulGru-arxiv-19}. A more rigorous approach to tackle the basis set convergence problem has been proposed by the so-called R12 and F12 methods\cite{Ten-TCA-12,TenNog-WIREs-12,HatKloKohTew-CR-12, KonBisVal-CR-12, GruHirOhnTen-JCP-17, MaWer-WIREs-18} which introduce a function explicitly depending on the interelectronic coordinates ensuring the correct cusp condition in the wave function, and the resulting correlation energies converge much faster than the usual WFT. For instance, using the explicitly correlated version of coupled cluster with single, double and perturbative triple substitution (CCSD(T)) in a triple-$\zeta$ quality basis set is equivalent to a quintuple-$\zeta$ quality of the usual CCSD(T) method\cite{TewKloNeiHat-PCCP-07}, although inherent computational overhead are introduced by the auxiliary basis sets needed to resolve the rather complex three- and four-electron integrals involved in the F12 theory. + +An alternative point of view is to leave the short-range correlation effects to DFT and to use WFT to deal only with the long-range and/or strong-correlation effects. A rigorous approach to do so is the range-separated DFT (RSDFT) formalism (see Ref.~\onlinecite{TouColSav-PRA-04} and references therein) which rely on a splitting of the coulomb interaction in terms of the interelectronic distance thanks to a range-separation parameter $\mu$. The advantage of such approach is at least two-folds: i) the DFT part deals only with the short-range part of the coulomb interaction, and therefore the usual semi-local approximations to the unknown exchange-correlation functional are more suited to that correlation regime, ii) the WFT part deals with a smooth non divergent interaction, which removes the cusp condition and therefore the basis set convergence is much faster\cite{FraMusLupTou-JCP-15}. +Therefore, a number of approximate RS-DFT schemes have been developed within single-reference \cite{AngGerSavTou-PRA-05, GolWerSto-PCCP-05, TouGerJanSavAng-PRL-09,JanHenScu-JCP-09, TouZhuSavJanAng-JCP-11, MusReiAngTou-JCP-15} or multi-reference \cite{LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, HedTouJen-JCP-18, FerGinTou-JCP-18} WFT approaches. Nevertheless, there are still some open issues in RSDFT, such as the dependence of the quality of the results on the value of the range separation $\mu$ which can be seen as an empirical parameter, together with the remaining self-interaction errors. + +Following this path, a very recent solution to the basis set convergence problem has been proposed by some of the preset authors\cite{GinPraFerAssSavTou-JCP-18} where they proposed to use RSDFT to take into account only the correlation effects outside a given basis set. The key idea in such a work is to realize that as a wave function developed in an incomplete basis set is cusp-less, it could also come from a Hamiltonian with a non divergent electron-electron interaction. Therefore, the authors proposed a mapping with RSDFT through the introduction of an effective non-divergent interaction representing the usual coulomb interaction projected in an incomplete basis set. First applications to weakly correlated molecular systems have been successfully carried recently\cite{LooPraSceTouGin-JCPL-19} together with the first attempt to generalize this approach to excited states\cite{exicted}. +The goal of the present work is to push the development of this new theory toward the description of strongly correlated systems. +The paper is organized as follows: in section \ref{sec:theory} we recall the mathematical framework of the basis set correction and we propose a practical extension for strongly correlated systems. Two key aspect are discussed: the extensivity of the correlation energies together with the $S_z$ independence of the results. +Then in section \ref{sec:results} we discuss the potential energy surfaces (PES) of N$_2$, F$_2$ and H$_{10}$ up to full dissociation as a prototype of strongly correlated problems. Finally, we conclude in section \ref{sec:conclusion} + +%%%%%%%%%%%%%%%%%%%%%%%% +\section{Theory} +\label{sec:theory} +%%%%%%%%%%%%%%%%%%%%%%%% +The theoretical framework of the basis set correction has been derived in details in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, so we recall briefly the main equations involved for the present study. +First in section \ref{sec:basic} we recall the basic mathematical framework of the present theory by introducing the density functional complementary to a basis set $\Bas$. Then in section \ref{sec:wee} we introduce an effective non divergent interaction in a basis set $\Bas$, which leads us to the definition of an effective range separation parameter varying in space in section \ref{sec:mur}. Thanks to the range separation parameter, we make a mapping with a specific class of RSDFT functionals and propose practical approximations for the unknown density functional complementary to a basis set $\Bas$, for which new approximations for the strong correlation regime are given in section \ref{sec:functional}. +\subsection{Basic formal equations} +\label{sec:basic} +The exact ground state energy $E_0$ of a $N-$electron system can be obtained by an elegant mathematical framework connecting WFT and DFT, that is the Levy-Lieb constrained search formalism which reads +\begin{equation} + \label{eq:levy} + E_0 = \min_{\denr} \bigg\{ F[\denr] + (v_{\text{ne}} (\br{}) |\denr) \bigg\}, +\end{equation} +where $(v_{ne}(\br{})|\denr)$ is the nuclei-electron interaction for a given density $\denr$ and $F[\denr]$ is the so-called Levy-Liev universal density functional +\begin{equation} + \label{eq:levy_func} + F[\denr] = \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop +\weeop }{\Psi}. +\end{equation} +The minimizing density $n_0$ of equation \eqref{eq:levy} is the exact ground state density. +Nevertheless, in practical calculations the minimization is performed over the set $\setdenbasis$ which are the densities representable in a basis set $\Bas$, we assume from thereon that the densities used in the equations belong to $\setdenbasis$. + +In the present context it is important to notice that in order to recover the \textit{exact} ground state energy, the wave functions $\Psi$ involved in the definition of eq. \eqref{eq:levy_func} must be developed in a complete basis set. +An important step proposed originally by some of the present authors in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18} +was to propose to split the minimization in the definition of $F[\denr]$ using $\wf{}{\Bas}$ which are wave functions developed in $\basis$ +\begin{equation} + \label{eq:def_levy_bas} + F[\denr] = \min_{\wf{}{\Bas} \rightarrow \denr} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\denr}, +\end{equation} +which leads to the following definition of $\efuncden{\denr}$ which is the the density functional complementary to the basis set $\Bas$ +\begin{equation} + \begin{aligned} + \efuncden{\denr} =& \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop +\weeop }{\Psi} \\  + &- \min_{\Psi^{\Bas} \rightarrow \denr} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}}. + \end{aligned} +\end{equation} +Therefore thanks to eq. \eqref{eq:def_levy_bas} one can properly connect the DFT formalism with the basis set error in WFT calculations. In other terms, the existence of $\efuncden{\denr}$ means that the correlation effects not taken into account in $\basis$ can be formulated as a density functional. + +Assuming that the density $\denFCI$ associated to the ground state FCI wave function $\psifci$ is a good approximation of the exact density, one obtains the following approximation for the exact ground state density (see equations 12-15 of Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}) +\begin{equation} + \label{eq:e0approx} + E_0 = \efci + \efuncbasisFCI +\end{equation} +where $\efci$ is the ground state FCI energy within $\Bas$. As it was originally shown in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18} and further emphasized in Ref. \onlinecite{LooPraSceTouGin-JCPL-19,GinSceTouLoo-JCP-19}, the main role of $\efuncbasisFCI$ is to correct for the basis set incompleteness errors, a large part of which originates from the lack of cusp in any wave function developed in an incomplete basis set. +The whole purpose of this paper is to determine approximations for $\efuncbasisFCI$ which are suited for treating strong correlation regimes. The two requirement for such conditions are that i) it can be defined for multi-reference wave functions, ii) it must provide size extensive energies, iii) it is invariant of the $S_z$ component of a given spin multiplicity. + +\subsection{Definition of an effective interaction within $\Bas$} +\label{sec:wee} +As it was originally shown by Kato\cite{kato}, the cusp in the exact wave function originates from the divergence of the coulomb interaction at the coalescence point. Therefore, a cusp less wave function $\wf{}{\Bas}$ could also be obtained from a Hamiltonian with a non divergent electron-electron interaction. In other words, the incompleteness of a finite basis set can be understood as the removal of the divergence of the usual coulomb interaction at the electron coalescence point. + +As it was originally derived in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18} (see section D and annexes), one can obtain an effective non divergent interaction, here referred as $\wbasis$, which reproduces the expectation value of the coulomb operator over a given wave function $\wf{}{\Bas}$. As we are interested in the behaviour at the coalescence point, we focus on the opposite spin part of the electron-electron interaction. + +More specifically, we define the effective interaction associated to a given wave function $\wf{}{\Bas}$ as +\begin{equation} + \label{eq:wbasis} + \wbasis = + \begin{cases} + \fbasis /\twodmrdiagpsi, & \text{if $\twodmrdiagpsi \ne 0$,} +\\ + \infty, & \text{otherwise,} + \end{cases} +\end{equation} +where $\twodmrdiagpsi$ is the opposite spin two-body density associated to $\wf{}{\Bas}$ +\begin{equation} + \twodmrdiagpsi = \sum_{pqrs} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}, +\end{equation} +$\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$ its associated two-body tensor, $\SO{p}{}$ are the spatial orthonormal orbitals, +\begin{equation} + \label{eq:fbasis} + \fbasis + = \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2}, +\end{equation} +and $\V{pq}{rs}=\langle pq | rs \rangle$ are the usual two-electron Coulomb integrals. +With such a definition, one can show that $\wbasis$ satisfies +\begin{equation} + \int \int \dr{1} \dr{2} \wbasis \twodmrdiagpsi = \int \int \dr{1} \dr{2} \frac{\twodmrdiagpsi}{|\br{1}-\br{2}|}. +\end{equation} +As it was shown in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, the effective interaction $\wbasis$ is necessary finite at coalescence for an incomplete basis set, and tends to the regular coulomb interaction in the limit of a complete basis set for any choice of wave function $\psibasis$, that is +\begin{equation} + \label{eq:cbs_wbasis} + \lim_{\Bas \rightarrow \text{CBS}} \wbasis = \frac{1}{|\br{1}-\br{2}|}\quad \forall\,\psibasis. +\end{equation} +The condition of equation \eqref{eq:cbs_wbasis} is fundamental as it guarantees the good behaviour of all the theory in the limit of a complete basis set. + +\subsection{Definition of a range-separation parameter varying in real space} +\label{sec:mur} +As the effective interaction within a basis set $\wbasis$ is non divergent, one can fit such a function with a long-range interaction defined in the framework of RSDFT which depends on the range-separation parameter $\mu$ +\begin{equation} + \label{eq:weelr} + w_{ee}^{\lr}(\mu;r_{12}) = \frac{\text{erf}\big(\mu \,r_{12} \big)}{r_{12}}. +\end{equation} +As originally proposed in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we introduce a range-separation parameter $\murpsi$ varying in real space +\begin{equation} + \label{eq:def_mur} + \murpsi = \frac{\sqrt{\pi}}{2} \wbasiscoal +\end{equation} +such that +\begin{equation} + w_{ee}^{\lr}(\murpsi;0) = \wbasiscoal \quad \forall \, \br{}. +\end{equation} +Because of the very definition of $\wbasis$, one has the following properties at the CBS limit (see \eqref{eq:cbs_wbasis}) +\begin{equation} + \label{eq:cbs_mu} + \lim_{\Bas \rightarrow \text{CBS}} \murpsi = \infty\quad \forall \,\psibasis, +\end{equation} +which is fundamental to guarantee the good behaviour of the theory at the CBS limit. + +\subsection{Generic form and properties of the approximations for $\efuncden{\denr}$ } +\subsubsection{Generic form of the approximated functionals} +As originally proposed and motivated in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the complementary basis set functional $\efuncden{\denr}$ by using the so-called multi-determinant correlation functional (ECMD) introduced by Toulouse and co-workers\cite{TouGorSav-TCA-05}. +Following the recent work of some of the present authors\cite{LooPraSceTouGin-JCPL-19}, we propose to use a PBE-like functional which uses the total density $\denr$, the spin polarisation $\zeta(\br{}) = n_{\alpha}(\br{}) - n_{\beta}(\br{})$, reduced density gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$ and the on-top pair density $\ntwo(\br{})$. In the present work, all the density-related quantities are computed with the same wave function $\psibasis$ used to define $\murpsi$. +Therefore, a given approximation X of $\efuncden{\denr}$ have the following generic form +\begin{equation} + \begin{aligned} + \label{eq:def_ecmdpbebasis} + \efuncdenpbe{\argebasis} = &\int d\br{} \,\denr \\ & \ecmd(\argrebasis) + \end{aligned} +\end{equation} +where $\ecmd(\argecmd)$ is the ECMD correlation energy density defined as +\begin{equation} + \label{eq:def_ecmdpbe} + \ecmd(\argecmd) = \frac{\varepsilon_{\text{c,PBE}}(\argepbe)}{1+ \mu^3 \beta(\argepbe)} +\end{equation} +with +\begin{equation} + \label{eq:def_beta} + \beta(\argebasis) = \frac{3}{2\sqrt{\pi}(1 - \sqrt{2})}\frac{\varepsilon_{\text{c,PBE}}(\argepbe)}{\ntwo/\den}, +\end{equation} +and where $\varepsilon_{\text{c,PBE}}(\argepbe)$ is the usual PBE correlation energy density\cite{PerBurErn-PRL-96}. Before introducing the different flavour of approximated functionals that we will use here (see \ref{sec:def_func}), we would like to give some motivations based on physical requirements for the such a choice of functional form. + +The actual functional form of $\ecmd(\argecmd)$ have been originally proposed by some of the present authors in the context of RSDFT~\cite{FerGinTou-JCP-18} in order to fulfill the two following limits +\begin{equation} + \lim_{\mu \rightarrow 0} \ecmd(\argecmd) = \varepsilon_{\text{c,PBE}}(\argepbe), +\end{equation} +which can be qualified as the weak correlation regime, and the large $\mu$ limit +\begin{equation} + \label{eq:lim_mularge} + \ecmd(\argecmd) = \frac{1}{\mu^3} \ntwo + o(\frac{1}{\mu^5}), +\end{equation} +which, as it was previously shown\cite{TouColSav-PRA-04, GoriSav-PRA-06,PazMorGorBac-PRB-06} by various authors, is the exact expression for the ECMD in the limit of large $\mu$, provided that $\ntwo$ is the \textit{exact} on-top pair density of the system. +In the context of RSDFT, some of the present authors have illustrated in Ref.~\onlinecite{FerGinTou-JCP-18} that the on-top pair density involved in eq. \eqref{eq:def_ecmdpbe} plays a crucial role when reaching the strong correlation regime. The importance of the on-top pair density in the strong correlation regime have been also acknowledged by Pernal and co-workers\cite{GritMeePer-PRA-18} and Gagliardi and co-workers\cite{CarTruGag-JPCA-17}. +Also, $\ecmd(\argecmd) $ vanishes when $\ntwo$ vanishes +\begin{equation} + \label{eq:lim_n2} + \lim_{\ntwo \rightarrow 0} \ecmd(\argecmd) = 0 +\end{equation} +which is exact for systems with a vanishing on-top pair density, such as the totally dissociated H$_2$ which is the archetype of strongly correlated systems. +Of course, as all RSDFT functionals the function $\ecmd(\argecmd)$ vanishes when $\mu \rightarrow \infty$ +\begin{equation} + \label{eq:lim_muinf} + \lim_{\mu \rightarrow \infty} \ecmd(\argecmd) = 0. +\end{equation} + +\subsubsection{Properties of approximated functionals} +Within the definition of \eqref{eq:def_mur} and \eqref{eq:def_ecmdpbebasis}, any approximated complementary basis set functionals $\efuncdenpbe{\argecmd}$ satisfies two important properties. +Because of the properties \eqref{eq:cbs_mu} and \eqref{eq:lim_muinf}, $\efuncdenpbe{\argecmd}$ vanishes when reaching the complete basis set limit, whatever the wave function $\psibasis$ used to define the range separation parameter $\mu_{\Psi^{\basis}}$: +\begin{equation} + \label{eq:lim_ebasis} + \lim_{\basis \rightarrow \text{CBS}} \efuncdenpbe{\argecmd} = 0\quad \forall\, \psibasis, +\end{equation} +which guarantees an unaltered limit when reaching the CBS limit. +Also, the $\efuncdenpbe{\argecmd}$ vanishes for systems with vanishing on-top pair density, which guarantees the good limit in the case of stretched H$_2$ and for one-electron system. +Such a property is guaranteed independently by i) the definition of the effective interaction $\wbasis$ (see equation \eqref{eq:wbasis}) together with the condition \eqref{eq:lim_muinf}, ii) the fact that the $\ecmd(\argecmd)$ vanishes when the on-top pair density vanishes (see equation \eqref{eq:lim_n2}). + +\subsection{Requirements for the approximated functionals in the strong correlation regime} +\subsubsection{Requirements: separability of the energies and $S_z$ invariance} +An important requirement for any electronic structure method is the extensivity of the energy, \textit{i. e.} the additivity of the energies in the case of non interacting fragments, which is particularly important to avoid any ambiguity in computing interaction energies. +When two subsystems $A$ and $B$ dissociate in closed shell systems, as in the case of weak interactions for instance, a simple HF wave function leads to extensive energies. +When the two subsystems dissociate in open shell systems, such as in covalent bond breaking, it is well known that the HF approach fail and an alternative is to use a CASSCF wave function which, provided that the active space has been properly chosen, leads to additives energies. +Another important requirement is the independence of the energy with respect to the $S_z$ component of a given spin state. +Such a property is also important in the context of covalent bond breaking where the ground state of the super system $A+B$ is in general of low spin while the ground states of the fragments $A$ and $B$ are in high spin which can have multiple $S_z$ components. + +\subsubsection{Condition for the functional $\efuncdenpbe{\argebasis}$ to obtain $S_z$ invariance} +A sufficient condition to achieve $S_z$ invariance is to eliminate all dependency to $S_z$, which in the case of $\ecmd(\argecmd)$ is the spin polarisation $\zeta(\br{})$ involved in the correlation energy density $\varepsilon_{\text{c,PBE}}(\argepbe)$ (see equation \eqref{eq:def_ecmdpbe}). +As originally shown by Perdew and co-workers\cite{PerSavBur-PRA-95}, the dependence on the spin polarisation in the KS-DFT framework can be removed by the rewriting the spin polarisation of a single Slater determinant with only the on-top pair density and the total density. In other terms, the spin density dependence usually introduced in the correlation functionals of KS-DFT tries to mimic the effect of the on-top pair density. +Based on this reasoning, a similar approach has been used in the context of multi configurational DFT in order to remove the $S_z$ dependency. +In practice, these approaches introduce the effective spin polarisation +\begin{equation} + \label{eq:def_effspin} + \tilde{\zeta}(n,\ntwo_{\psibasis}) = +% \begin{cases} + \sqrt{ n^2 - 4 \ntwo_{\psibasis} } +% 0 & \text{otherwise.} +% \end{cases} +\end{equation} +which uses the on-top pair density $\ntwo_{\psibasis}$ of a given wave function $\psibasis$. + +The advantages of this approach are at least two folds: i) the effective spin polarisation $\tilde{\zeta}$ is $S_z$ invariant, ii) it introduces an indirect dependency on the on-top pair density of the wave function $\psibasis$ which usually improves the treatment of strong correlation. +Nevertheless, the use of $\tilde{\zeta}$ presents several disadvantages as it can become complex when $n^2 - 4 \ntwo_{\psibasis}<0$ and also +the formula of equation \eqref{eq:def_effspin} is exact only when the density $n$ and on-top pair density $\ntwo_{\psibasis}$ are obtained from a single determinant\cite{PerSavBur-PRA-95}, but it is applied to multi configurational wave functions. + +An alternative to eliminate the $S_z$ dependency would be to simply set $\zeta(\br{})=0$, but this would lower the accuracy of the usual correlation functional, such as the PBE correlation functional used here $\varepsilon_{\text{c,PBE}}(\argepbe)$. Nevertheless, as the spin polarisation usually tries to mimic the on-top pair density and the function $\ecmd(\argecmd)$ explicitly depends on the on-top pair density (see equations \eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}), we propose here to use the $\ecmd$ functional with \textit{a zero spin polarisation}. This ensures a $S_z$ invariance and, as will be numerically shown, very weakly affect the accuracy of the functional. + +\subsubsection{Conditions on $\psibasis$ for the extensivity} +In the case of the present basis set correction, as $\efuncdenpbe{\argebasis}$ is an integral over $\mathbb{R}^3$ of local quantities, in the case of non overlapping fragments $A\ldots B$ it can be written as the sum of two local contributions: one coming from the integration over the region of the sub-system $A$ and the other one from the region of the sub-system $B$. +Therefore, a sufficient condition for the extensivity is that these quantities coincide in the isolated systems and in the subsystem of the super system $A\ldots B$. +As $\efuncdenpbe{\argebasis}$ depends only on quantities which are properties of the wave function $\psibasis$, a sufficient condition for the extensivity of these quantities is that the function factorise in the limit of non-interacting fragments, that is $\Psi_{A\ldots B}^{\basis} = \Psi_A^{\basis} \Psi_B^{\basis}$. +In the case where the two subsystems $A$ and $B$ dissociate in closed shell systems, a simple HF wave function ensures this property, but when one or several covalent bonds are broken, the use of a properly chosen CASSCF wave function is sufficient to recover this property, as will be numerically illustrated in section \ref{sec:separability}. +The condition for the active space involved in the CASSCF wave function is that it has to lead to extensive energies in the limit of dissociated fragments. + + +\subsection{Different types of approximations for the functional} +\subsubsection{Definition of the protocol to design functionals} +\label{sec:def_func} +As the present work proposes to investigate how different physical quantities impact the description of correlation, we propose here a general protocol and a corresponding nomenclature in order to make things as clear as possible. +% +Here we propose to investigate the dependency of the functionals $\efuncdenpbe{\argebasis}$ on: i) the wave function $\psibasis$ used to determine the $\murpsi$ and the various density related quantities, ii) the flavour of on-top pair density used, iii) the type of spin polarisation used. +Therefore, we propose to use the following notations: PBE-"on-top"-"spin polarisation"-$\psibasis$. + +Regarding the spin polarisation that enters into $\varepsilon_{\text{c,PBE}}(\argepbe)$, we will use three different types of formula: i) the usual spin polarisation $\zeta = n_{\alpha} - n_{\beta}$ which \textit{is not} $S_z$ invariant, ii) $\tilde{\zeta}$ defined in equation \eqref{eq:def_effspin} which \textit{is} $S_z$ invariant, and iii) a \textit{zero} spin polarization which of course \textit{is} $S_z$ invariant. + +For the wave function $\psibasis$, we will use either i) a simple RHF/ROHF wave function, ii) a minimal CASSCF leading to additive energies in the case of dissociated covalent bonds. + +Regarding the approximation to the \textit{exact} on-top pair density, we use two different approximations. The first one is based on the uniform electron gas (UEG) and reads +\begin{equation} + \label{eq:def_n2ueg} + \ntwo_{\text{UEG}}(n,\zeta,\br{}) = n(\br{})^2\big(1-\zeta(\br{})\big)g_0\big(n(\br{})\big) +\end{equation} +where the pair-distribution function $g_0(n)$ is taken from equation (46) of Ref. \onlinecite{GorSav-PRA-06}. The approximation of equation \eqref{eq:def_n2ueg} depends on the density and some spin polarisation. Notice that, when using a CASSCF wave function and $\tilde{\zeta}$ as spin polarization, the $\ntwo_{\text{UEG}}$ will depend indirectly on the on-top pair density as $\tilde{\zeta}$ depends on the on-top pair density. + +Another approach consists in taking advantage of the on-top pair density of the wave function $\psibasis$. Following the work of some of the previous authors\cite{FerGinTou-JCP-18,GinSceTouLoo-JCP-19} we introduce the extrapolated on-top pair density $\ntwoextrap$ as +\begin{equation} + \ntwoextrap(\ntwo_{\psibasis},\mu,\br{}) = \ntwo_{\wf{}{\Bas}}(\br{}) \bigg( 1 + \frac{2}{\sqrt{\pi}\murpsi} \bigg)^{-1} +\end{equation} +which directly follows from the large-$\mu$ extrapolation of the exact on-top pair density proposed by Gori-Giorgi and Savin\cite{GorSav-PRA-06}. +When using $\ntwoextrap(\ntwo,\mu,\br{})$ in a functional, we will refer simply refer it as "ont". + +\subsubsection{Definition of a hierarchy of functionals} +Within the convention proposed in the section \ref{sec:def_func}, the PBE-UEG-$\zeta$-HF is the functional which was introduced in Ref. \onlinecite{LooPraSceTouGin-JCPL-19} and which reads +\begin{equation} + \label{eq:def_pbeueg} + \begin{aligned} + \pbeuegxihf &\equiv \int d\br{} \,\denr \\ & \ecmd\big(\argrpbeuegxihf\big) + \end{aligned} +\end{equation} +Therefore, such a functional uses a HF wave function to define; i) the $\murpsi$, ii) the total density, reduced density gradients, usual spin polarisation $\zeta$ and uses the UEG-like on-top pair density with the usual spin polarisation $\zeta$. +Of course, because of the use of an HF wave function as $\psibasis$, the density related quantities are extensive only in the case of dissociation in closed shell system. Also, one can notice that changing the spin polarisation from $\zeta$ to $\tilde{\zeta}$ does not change the results as by definition, $\tilde{\zeta} = \zeta$ for a single Slater determinant. + +By changing the definition of $\psibasis=\text{HF}$ to $\psibasis=\text{CASSCF}$ on obtains the PBE-UEG-$\zeta$-CAS which reads +\begin{equation} + \label{eq:def_pbeueg} + \begin{aligned} + \pbeuegxicas &\equiv \int d\br{} \,\denr \\ & \ecmd\big(\argrpbeuegxicas\big) + \end{aligned} +\end{equation} +where the density, reduced density gradients, usual spin polarisation and UEG on-top pair density are computed from a CASSCF wave function. Therefore, the $\murpsi$, density, reduced density gradient are extensive in the case of dissociated covalent bonding. Nevertheless, the use of the regular spin polarisation $\zeta$ leads to non $S_z$ invariance. + +One can change the spin polarisation to the effective spin polarisation $\tilde{\zeta}$ to obtain the PBE-UEG-$\tilde{\zeta}$-CAS which is $S_z$ invariant, and therefore this functional will reads to +\begin{equation} + \label{eq:def_pbeueg} + \begin{aligned} + \pbeuegXicas = &\int d\br{} \,\denr \\ & \ecmd(\argrpbeuegXicas). + \end{aligned} +\end{equation} +One can also change the flavour of the on-top pair density by taking advantage of the on-top pair density $\ntwo_{\wf{}{\Bas}}(\br{})$ computed with $\psibasis$. +Therefore, one can define the PBE-ONT-$\zeta$-CAS as +\begin{equation} + \label{eq:def_pbeueg} + \begin{aligned} + \pbeontXicas = &\int d\br{} \,\denr \\ & \ecmd(\argrpbeontXicas). + \end{aligned} +\end{equation} +Such a functional can be further improved by using the $S_z$ invariant effective spin polarisation $\tilde{\zeta}$ to give the PBE-ONT-$\tilde{\zeta}$-CAS. + + + +%%%%%%%%%%%%%%%%%%%%%%%% +\section{Results} +\subsection{Numerical tests of extensivity} +The first numerical results investigated are the numerical tests of extensivity of the various functionals. +As mentioned before, when considering a super system $A+B$ dissociating into non interacting fragments $A\ldots B$, there are two different situations regarding the extensivity of the energy: when the subsystems $A$ and $B$ dissociate in closed or open shell systems. +Therefore, we shall consider two systems $A$ and $B$ and compare the sum of the energies obtained with the super system $A\ldots B$ in the limit of non interactive fragments. The error to additivity for a given method $Y$ is therefore defined as $E_Y(A) + E_Y(B) - E_Y(A\ldots B)$. + +\subsubsection{Dissociation to closed shell ground states} +We begin our study by giving numerical evidence for the extensivity of the present basis set correction for systems dissociating in closed shell systems. +In these cases, the use a HF wave function is sufficient to guarantee the extensivity of the basis set correction, and therefore we use the simple $\pbeuegxihf$ functional. The system under study is $A=\text{F}_2$ at experimental equilibrium geometry (F-F=1.411 angstroms) and $B=\text{Ne}$. +We report in table \ref{tab:extensiv_closed} the error to additivity for the HF energy and for $\pbeuegxihf$ using the aug-cc-pvdz basis set and using a He core to define the $\mu_{\text{HF}}^{\basis}(\br{})$ and the frozen core densities. +The numbers in table \ref{tab:extensiv_closed} clearly show that when HF energies are additive, the $\pbeuegxihf$ is also additive. +Also, the error to additivity using the usual spin polarisation $\zeta$ and the extrapolated on-top pair density are much lowered compared to that using UEG on-top pair density, highlighting the important role played by the on-top pair density of the CASSCF wave function. + +\begin{table*} +\caption{Total energies (in Hartree) for HF and $E$ in aug-cc-pvdz for the Ne atom, F$_2$ (with F-F=1.411 angstroms) and the super non interacting system Ne--F$_2$. } +\begin{tabular}{lcc} +%\hline + System & HF & $\pbeuegxihf$ \\ +\hline +Ne & -128.4963497306184 & -0.1039022285466806 \\ +F$_2$ & -198.698792752661 & -0.1596345827582842 \\ +Ne $\ldots$ F$_2$ & -201.554497420371 & -0.2635368113049532 \\ + \hline +Error to additivity & 3.4 $\times 10^{-13}$ & 1.1 $\times 10^{-14}$ \\ +\end{tabular} +\label{tab:extensiv_closed} +\end{table*} + +\subsubsection{Dissociation to open shell ground states} +The system studied to investigate the extensivity in the case of dissociation to open shell systems is the completely dissociated N$_2$ molecule which imply the breaking of three covalent bonds. +As the HF wave function does not lead to extensive energy, it is clear that it cannot be used as $\psibasis$ and therefore for N$_2$ we use a minimal valence CASSCF(6,6) involving the three bonding orbitals ($\sigma$, $\pi_x$, $\pi_y$) and corresponding anti-bonding orbitals and a ROHF wave function for the N atom. +The numerical results for the extensivity of the various flavours of functionals are given in table \ref{tab:extensiv_open}. From these numbers, one can clearly notice that only the functionals using the effective spin polarisation $\tilde{\zeta}$ are size extensive, whatever the type of on-top pair density used. + +\begin{table*} +\caption{Total energies (in Hartree) for N$_2$ in the aug-cc-pvdz basis set. } +\begin{tabular}{lccccc} +%\hline + System & ROHF/CASSCF(6,6) & $\pbeuegxicas$ & $\pbeuegXicas$ & $\pbeontxicas$ & $\pbeontXicas$ \\ +\hline +N & -128.496349730618 & -0.0230740500348705 & -0.0230740500348705 & -0.0247392466968251 & -0.0247392466968251 \\ +N$\ldots$N & -198.698792752661 & -0.0691133629633014 & -0.0461481000697329 & -0.0509457188492165 & -0.0494784933936403 \\ + \hline +Error to additivity & 1.0 $\times 10^{-13}$ & 0.02296 & 8.0 $\times 10^{-15}$ & 0.0015 & 9.9 $\times 10^{-15}$ \\ +\end{tabular} +\label{tab:extensiv_open} +\end{table*} + +\label{sec:results} +%%%%%%%%%%%%%%%%%%%%%%%% +\begin{figure} + \includegraphics[width=\linewidth]{data/N2/DFT_avdzE_relat.pdf} + \includegraphics[width=\linewidth]{data/N2/DFT_avdzE_relat_zoom.pdf} +% \includegraphics[width=\linewidth]{data/N2/DFT_avdzE_error.pdf} + \caption{ + N$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. + \label{fig:N2_avdz}} +\end{figure} + +\begin{figure} + \includegraphics[width=\linewidth]{data/N2/DFT_avtzE_relat.pdf} + \includegraphics[width=\linewidth]{data/N2/DFT_avtzE_relat_zoom.pdf} +% \includegraphics[width=\linewidth]{data/N2/DFT_avtzE_error.pdf}\\ +% \includegraphics[width=\linewidth]{fig2c} + \caption{ + N$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. + \label{fig:N2_avtz}} +\end{figure} + +\begin{table*} +\caption{Dissociation energy ($D_0$) computed at different levels in various basis sets. } +\begin{ruledtabular} +\begin{tabular}{lcccc} + +%\hline + System/basis & FCI & FCI+$\pbeuegXicas$ & FCI+$\pbeontXicas$ & FCI+$\pbeontnscas$ \\ +\hline +N$_2$, aug-cc-pvdz & 321.4$/ $42.8 & 355.6$/$8.6 & 355.0$/$9.2 & 354.0$/$10.2 \\ +N$_2$, aug-cc-pvtz & 347.8$/$16.4 & 361.0$/$3.2 & 362.7$/$1.5 & 362.4$/$1.8 \\ + \hline + & \multicolumn{4}{c}{Estimated exact} \\ + & \multicolumn{4}{c}{364.2 } \\ +\hline +F$_2$, aug-cc-pvdz & 49.2$/$11.5 & 54.1$/$6.6 & 54.3$/$6.4 & 54.1$/$6.7 \\ +F$_2$, aug-cc-pvtz & 58.6$/$2.1 & 60.6$/$0.1 & 60.9$/$-0.2 & 60.9$/$-0.2 \\ + \hline + & \multicolumn{4}{c}{Estimated exact} \\ + & \multicolumn{4}{c}{60.7 } \\ +\hline +H$_{10}$, cc-pvdz & 622.1$/$43.3 & 642.6$/$22.8 & 649.2$/$16.2 & 649.5$/$15.9 \\ +H$_{10}$, cc-pvtz & 655.2$/$10.2 & 661.9$/$3.5 & 666.0$/$-0.6 & 666.0$/$-0.6 \\ +H$_{10}$, cc-pvqz & 661.2$/$4.2 & 664.1$/$1.3 & 666.4$/$-1.0 & 666.5$/$-1.1 \\ + \hline + & \multicolumn{4}{c}{Estimated exact} \\ + & \multicolumn{4}{c}{665.4 } \\ +\end{tabular} +\end{ruledtabular} + +\label{tab:extensiv_closed} +\end{table*} + +\begin{figure} + \includegraphics[width=\linewidth]{data/F2/DFT_avdzE_relat.pdf} + \includegraphics[width=\linewidth]{data/F2/DFT_avdzE_relat_zoom.pdf} +% \includegraphics[width=\linewidth]{fig2c} + \caption{ + F$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. + \label{fig:F2_avdz}} +\end{figure} + +\begin{figure} + \includegraphics[width=\linewidth]{data/F2/DFT_avtzE_relat.pdf} + \includegraphics[width=\linewidth]{data/F2/DFT_avtzE_relat_zoom.pdf} +% \includegraphics[width=\linewidth]{fig2c} + \caption{ + F$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. + \label{fig:F2_avtz}} +\end{figure} + + +\begin{figure} +% \includegraphics[width=\linewidth]{data/H10/DFT_avdzE_relat.pdf} + \includegraphics[width=\linewidth]{data/H10/DFT_vdzE_relat.pdf}\\ + \includegraphics[width=\linewidth]{data/H10/DFT_vdzE_relat_zoom.pdf} +% \includegraphics[width=\linewidth]{data/H10/DFT_vdzE_error.pdf}\\ +% \includegraphics[width=\linewidth]{fig2c} + \caption{ + H$_{10}$, cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. + \label{fig:H10_vdz}} +\end{figure} + + +\begin{figure} + \includegraphics[width=\linewidth]{data/H10/DFT_vtzE_relat.pdf}\\ + \includegraphics[width=\linewidth]{data/H10/DFT_vtzE_relat_zoom.pdf} +% \includegraphics[width=\linewidth]{fig2c} + \caption{ + H$_{10}$, cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. + \label{fig:H10_vtz}} +\end{figure} + + +\begin{figure} + \includegraphics[width=\linewidth]{data/H10/DFT_vqzE_relat.pdf}\\ + \includegraphics[width=\linewidth]{data/H10/DFT_vqzE_relat_zoom.pdf} +% \includegraphics[width=\linewidth]{fig2c} + \caption{ + H$_{10}$, cc-pvqz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. + \label{fig:H10_vqz}} +\end{figure} + + +\section{Conclusion} +\label{sec:conclusion} + + + +\bibliography{srDFT_SC} + +\end{document}

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z!%`I=XMljK=n4jDs3jE1zPQI`j)cSu|b3abD~P+fwTA5z1DLn$^(5Me67Ne!0vzTI!%?`)Pw=G3Hbwbxw?y zNDZMnbWx9xeh}kSNl0Xx21~5F{YJT zny{&Ud|Zt-?L1zYC_LnTa$2)E38(hIX`NsSVfrdQ=i~Q^pk9A}N+T+Du0fz62thwO z*yXP=A%kpw@aYF?8!7_YYFM)@YAVi1<%qJ_z&&w?! zEhG;3pIu)1`R7-m7g;^Ne^mAU7oh{?xPU5>6hZf??*LhX%JA3*#oLsOygpLpH<3Ip zh~Qf?#SdU9G(18)+tt^++7t+iC(m^^y9;ASgO$y$!{48LhkaEaR@CHJRI@kf5wI2Y zS?L%;u=9&Uh=Z+vSj8qc0pOC~`1mh1X)qxMthD(uZ(_=I<@-OdrP!1N6NEbTh8XvT z`Ty2CSm1Z%uzn6ZOyrvEHuu{!da<{g#!y@6Tj=|0_uG4i1YJo6Z7IG9k;t> z+vAe3x*8%4gyZ4cwfR=3>zGNZoh{-O9xP8wMfp6ZE642Lqn~lBJ!qcvX2Sm4o2v3m literal 0 HcmV?d00001 diff --git a/Revised_Manuscript/SI/srDFT_SC-SI.tex b/Revised_Manuscript/SI/srDFT_SC-SI.tex new file mode 100644 index 0000000..769a5c7 --- /dev/null +++ b/Revised_Manuscript/SI/srDFT_SC-SI.tex @@ -0,0 +1,442 @@ + +\documentclass[aip,jcp,reprint,noshowkeys]{revtex4-1} + +%\documentclass[aip,jcp,noshowkeys]{revtex4-1} +\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amsmath,amssymb,amsfonts,physics,mhchem,xspace} + +\usepackage{mathpazo,libertine} +\usepackage[normalem]{ulem} +\newcommand{\alert}[1]{\textcolor{red}{#1}} +\definecolor{darkgreen}{RGB}{0, 180, 0} +\newcommand{\beurk}[1]{\textcolor{darkgreen}{#1}} +\newcommand{\trash}[1]{\textcolor{red}{\sout{#1}}} +\usepackage{xspace} + +\usepackage{hyperref} +\hypersetup{ + colorlinks=true, + linkcolor=blue, + filecolor=blue, + urlcolor=blue, + citecolor=blue +} +\newcommand{\cdash}{\multicolumn{1}{c}{---}} +\newcommand{\mc}{\multicolumn} +\newcommand{\fnm}{\footnotemark} +\newcommand{\fnt}{\footnotetext} +\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}} +\newcommand{\mr}{\multirow} +\newcommand{\SI}{\textcolor{blue}{supporting information}} + +% second quantized operators +\newcommand{\psix}[1]{\hat{\Psi}\left({\bf X}_{#1}\right)} +\newcommand{\psixc}[1]{\hat{\Psi}^{\dagger}\left({\bf X}_{#1}\right)} +\newcommand{\ai}[1]{\hat{a}_{#1}} +\newcommand{\aic}[1]{\hat{a}^{\dagger}_{#1}} +\newcommand{\vijkl}[0]{V_{ij}^{kl}} +\newcommand{\phix}[2]{\phi_{#1}(\bfr{#2})} +\newcommand{\phixprim}[2]{\phi_{#1}(\bfr{#2}')} + +\newcommand{\CBS}{\text{CBS}} + + +%operators +\newcommand{\elemm}[3]{{\ensuremath{\bra{#1}{#2}\ket{#3}}\xspace}} +\newcommand{\ovrlp}[2]{{\ensuremath{\langle #1|#2\rangle}\xspace}} + +%\newcommand{\ket}[1]{{\ensuremath{|#1\rangle}\xspace}} +%\newcommand{\bra}[1]{{\ensuremath{\langle #1|}\xspace}} + +% + + +% energies +\newcommand{\Ec}{E_\text{c}} +\newcommand{\EPT}{E_\text{PT2}} +\newcommand{\EsCI}{E_\text{sCI}} +\newcommand{\EDMC}{E_\text{DMC}} +\newcommand{\EexFCI}{E_\text{exFCI}} +\newcommand{\EexFCIbasis}{E_\text{exFCI}^{\Bas}} +\newcommand{\EexFCIinfty}{E_\text{exFCI}^{\infty}} +\newcommand{\EexDMC}{E_\text{exDMC}} +\newcommand{\Ead}{\Delta E_\text{ad}} +\newcommand{\efci}[0]{E_{\text{FCI}}^{\Bas}} +\newcommand{\emodel}[0]{E_{\model}^{\Bas}} +\newcommand{\emodelcomplete}[0]{E_{\model}^{\infty}} +\newcommand{\efcicomplete}[0]{E_{\text{FCI}}^{\infty}} +\newcommand{\ecccomplete}[0]{E_{\text{CCSD(T)}}^{\infty}} +\newcommand{\ecc}[0]{E_{\text{CCSD(T)}}^{\Bas}} +\newcommand{\efuncbasisFCI}[0]{\bar{E}^\Bas[\denFCI]} +\newcommand{\efuncbasisfci}[0]{\bar{E}^\Bas[\denfci]} +\newcommand{\efuncbasis}[0]{\bar{E}^\Bas[\den]} +\newcommand{\efuncden}[1]{\bar{E}^\Bas[#1]} +\newcommand{\efuncdenpbe}[1]{\bar{E}_{\text{}}^\Bas[#1]} +\newcommand{\efuncdenpbeAB}[1]{\bar{E}_{\text{A}+\text{B}}^\Bas[#1]} +\newcommand{\ecompmodel}[0]{\bar{E}^\Bas[\denmodel]} +\newcommand{\ecmubis}[0]{\bar{E}_{\text{c,md}}^{\text{sr}}[\denr;\,\mu]} +\newcommand{\ecmubisldapbe}[0]{\bar{E}_{\text{c,md}\,\text{PBE}}^{\text{sr}}[\denr;\,\mu]} +\newcommand{\ecmuapprox}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mu]} +\newcommand{\ecmuapproxmur}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mur]} +\newcommand{\ecmuapproxmurfci}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denfci;\,\mur]} +\newcommand{\ecmuapproxmurmodel}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denmodel;\,\mur]} +\newcommand{\ecompmodellda}[0]{\bar{E}_{\text{LDA}}^{\Bas,\wf{}{\Bas}}[\denmodel]} +\newcommand{\ecompmodelldaval}[0]{\bar{E}_{\text{LDA, val}}^{\Bas,\wf{}{\Bas}}[\den]} +\newcommand{\ecompmodelpbe}[0]{\bar{E}_{\text{PBE}}^{\Bas,\wf{}{\Bas}}[\den]} +\newcommand{\ecompmodelpbeval}[0]{\bar{E}_{\text{PBE, val}}^{\Bas,\wf{}{\Bas}}[\den]} +\newcommand{\emulda}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denr;\mu({\bf r};\wf{}{\Bas})\right)} +\newcommand{\emuldamodel}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denmodelr;\mu({\bf r};\wf{}{\Bas})\right)} +\newcommand{\emuldaval}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denval ({\bf r});\murval;\wf{}{\Bas})\right)} +\newcommand{\ecmd}[0]{\bar{\varepsilon}_{\text{c,md}}^{\text{sr},\text{PBE}}} +\newcommand{\psibasis}[0]{\Psi^{\basis}} +\newcommand{\BasFC}{\mathcal{A}} + +%pbeuegxiHF +\newcommand{\pbeuegxihf}{\text{PBE-UEG-}\zeta\text{-HF}^\Bas} +\newcommand{\argpbeuegxihf}[0]{\den,\zeta,s,n_{2}^{\text{UEG}},\mu_{\text{HF}}^{\basis}} +\newcommand{\argrpbeuegxihf}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),n_{2}^{\text{UEG}}(\br{}),\mu_{\text{HF}}^{\basis}(\br{})} +%pbeuegxiCAS +\newcommand{\pbeuegxi}{\text{PBE-UEG-}\zeta\text{-CAS}^\Bas} +\newcommand{\argpbeuegxicas}[0]{\den,\zeta,s,n_{2}^{\text{UEG}},\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeuegxicas}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),n_{2}^{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})} +%pbeuegXiCAS +\newcommand{\pbeuegXi}{\text{PBE-UEG-}\tilde{\zeta}} +\newcommand{\argpbeuegXi}[0]{\den,\tilde{\zeta},s,n_{2}^{\text{UEG}},\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeuegXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),n_{2}^{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})} +%pbeontxiCAS +\newcommand{\pbeontxi}{\text{PBE-ot-}\zeta} +\newcommand{\argpbeontxi}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeontxi}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})} +%pbeontXiCAS +\newcommand{\pbeontXi}{\text{PBE-ot-}\tilde{\zeta}} +\newcommand{\argpbeontXi}[0]{\den,\tilde{\zeta},s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeontXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})} +%pbeont0xiCAS +\newcommand{\pbeontns}{\text{PBE-ot-}0\zeta} +\newcommand{\argpbeontns}[0]{\den,0,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeontns}[0]{\den(\br{}),0,s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})} + +%%%%%% arguments + +\newcommand{\argepbe}[0]{\den,\zeta,s} +\newcommand{\argebasis}[0]{\den,\zeta,\ntwo,\mu} +\newcommand{\argecmd}[0]{\den,\zeta,s,\ntwo,\mu} +\newcommand{\argepbeueg}[0]{\den,\zeta,s,\ntwo^{\text{UEG}},\mu_{\Psi^{\basis}}} +\newcommand{\argepbeontxicas}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} +\newcommand{\argepbeuegXihf}[0]{\den,\tilde{\zeta},s,\ntwo^{\text{UEG}},\mu_{\Psi^{\basis}}} +\newcommand{\argrebasis}[0]{\denr,\zeta(\br{}),s(\br{}),\ntwo(\br{}),\mu(\br{})} +\newcommand{\argrebasisab}[0]{\denr,\zeta(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})} + + +% numbers +\newcommand{\rnum}[0]{{\rm I\!R}} +\newcommand{\bfr}[1]{{\bf r}_{#1}} +\newcommand{\dr}[1]{\text{d}\bfr{#1}} +\newcommand{\rr}[2]{\bfr{#1}, \bfr{#2}} +\newcommand{\rrrr}[4]{\bfr{#1}, \bfr{#2},\bfr{#3},\bfr{#4} } + + + +% effective interaction +\newcommand{\twodm}[4]{\elemm{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}} +\newcommand{\murpsibas}[0]{\mu_{\wf{}{\Bas}}({\bf r})} +\newcommand{\murpsi}[0]{\mu({\bf r};\wf{}{A+B})} +\newcommand{\murpsia}[0]{\mu({\bf r};\wf{}{A})} +\newcommand{\murpsib}[0]{\mu({\bf r};\wf{}{B})} +\newcommand{\ntwo}[0]{n_{2}} +\newcommand{\ntwohf}[0]{n^{(2),\text{HF}}} +\newcommand{\ntwophi}[0]{n^{(2)}_{\phi}} +\newcommand{\ntwoextrap}[0]{\mathring{n}^{(2)}_{\psibasis}} +\newcommand{\ntwoextrapcas}[0]{\mathring{n}^{(2)\,\basis}_{\text{CAS}}} +\newcommand{\mur}[0]{\mu({\bf r})} +\newcommand{\murr}[1]{\mu({\bf r}_{#1})} +\newcommand{\murval}[0]{\mu_{\text{val}}({\bf r})} +\newcommand{\murpsival}[0]{\mu_{\text{val}}({\bf r};\wf{}{\Bas})} +\newcommand{\murrval}[1]{\mu_{\text{val}}({\bf r}_{#1})} +\newcommand{\weeopmu}[0]{\hat{W}_{\text{ee}}^{\text{lr},\mu}} + + +\newcommand{\wbasis}[0]{W_{\wf{}{\Bas}}(\bfr{1},\bfr{2})} +\newcommand{\wbasiscoal}[0]{W_{\wf{}{\Bas}}(\bfr{},\bfr{})} +\newcommand{\wbasisval}[0]{W_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})} +\newcommand{\fbasis}[0]{f_{\wf{}{\Bas}}(\bfr{1},\bfr{2})} +\newcommand{\fbasisval}[0]{f_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})} + \newcommand{\ontop}[2]{ n^{(2)}_{#1}({\bf #2}_1)} + \newcommand{\twodmrpsi}[0]{ n^{2,\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})} + \newcommand{\twodmrdiagpsi}[0]{ n_{2,{\wf{}{\Bas}}}(\rr{1}{2})} + \newcommand{\twodmrdiagpsitot}[0]{ n_{2,\wf{}{A+B}}(\rr{1}{2})} + \newcommand{\twodmrdiagpsiaa}[0]{ n_{2,\wf{}{AA}}(\rr{1}{2})} + \newcommand{\twodmrdiagpsiaad}[0]{ n_{2,\wf{}{AA}}(\rr{}{})} + \newcommand{\twodmrdiagpsibb}[0]{ n_{2,\wf{}{BB}}(\rr{1}{2})} + \newcommand{\twodmrdiagpsibbd}[0]{ n_{2,\wf{}{BB}}(\rr{}{})} + \newcommand{\twodmrdiagpsiab}[0]{ n_{2\wf{}{AB}}(\rr{1}{2})} + \newcommand{\twodmrdiagpsival}[0]{ n_{2\wf{}{\Bas},\,\text{val}}(\rr{1}{2})} + \newcommand{\gammamnpq}[1]{\Gamma_{mn}^{pq}[#1]} + \newcommand{\gammamnkl}[0]{\Gamma_{mn}^{kl}} + \newcommand{\gammaklmn}[1]{\Gamma_{kl}^{mn}[#1]} +%\newcommand{\wbasiscoal}[1]{W_{\wf{}{\Bas}}({\bf r}_{#1})} +\newcommand{\ontoppsi}[1]{ n_{2,\wf{}{\Bas}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})} +\newcommand{\wbasiscoalval}[1]{W_{\wf{}{\Bas}}^{\text{val}}({\bf r}_{#1})} +\newcommand{\ontoppsival}[1]{ n^{(2)}_{\wf{}{\Bas}}^{\text{val}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})} + + + +\newcommand{\ex}[4]{$^{#1}#2_{#3}^{#4}$} +\newcommand{\ra}{\rightarrow} +\newcommand{\De}{D_\text{e}} + +% MODEL +\newcommand{\model}[0]{\mathcal{Y}} + +% densities +\newcommand{\denmodel}[0]{\den_{\model}^\Bas} +\newcommand{\denmodelr}[0]{\den_{\model}^\Bas ({\bf r})} +\newcommand{\denfci}[0]{\den_{\psifci}} +\newcommand{\denFCI}[0]{\den^{\Bas}_{\text{FCI}}} +\newcommand{\denhf}[0]{\den_{\text{HF}}^\Bas} +\newcommand{\denrfci}[0]{\denr_{\psifci}} +\newcommand{\dencipsir}[0]{{n}_{\text{CIPSI}}^\Bas({\bf r})} +\newcommand{\dencipsi}[0]{{n}_{\text{CIPSI}}^\Bas} +\newcommand{\den}[0]{{n}} +\newcommand{\denval}[0]{{n}^{\text{val}}} +\newcommand{\denr}[0]{{n}({\bf r})} +\newcommand{\onedmval}[0]{\rho_{ij,\sigma}^{\text{val}}} + +% wave functions +\newcommand{\psifci}[0]{\Psi^{\Bas}_{\text{FCI}}} +\newcommand{\psimu}[0]{\Psi^{\mu}} +% operators +\newcommand{\weeopbasis}[0]{\hat{W}_{\text{ee}}^\Bas} +\newcommand{\kinop}[0]{\hat{T}} + +\newcommand{\weeopbasisval}[0]{\hat{W}_{\text{ee}}^{\Basval}} +\newcommand{\weeop}[0]{\hat{W}_{\text{ee}}} + + +% units +\newcommand{\IneV}[1]{#1 eV} +\newcommand{\InAU}[1]{#1 a.u.} +\newcommand{\InAA}[1]{#1 \AA} + + +% methods +\newcommand{\UEG}{\text{UEG}} +\newcommand{\LDA}{\text{LDA}} +\newcommand{\PBE}{\text{PBE}} +\newcommand{\FCI}{\text{FCI}} +\newcommand{\CCSDT}{\text{CCSD(T)}} +\newcommand{\lr}{\text{lr}} +\newcommand{\sr}{\text{sr}} + +\newcommand{\Nel}{N} +\newcommand{\V}[2]{V_{#1}^{#2}} + + +\newcommand{\n}[2]{n_{#1}^{#2}} +\newcommand{\E}[2]{E_{#1}^{#2}} +\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}} +\newcommand{\bEc}[1]{\Bar{E}_\text{c}^{#1}} +\newcommand{\e}[2]{\varepsilon_{#1}^{#2}} +\newcommand{\be}[2]{\Bar{\varepsilon}_{#1}^{#2}} +\newcommand{\bec}[1]{\Bar{e}^{#1}} +\newcommand{\wf}[2]{\Psi_{#1}^{#2}} +\newcommand{\W}[2]{W_{#1}^{#2}} +\newcommand{\w}[2]{w_{#1}^{#2}} +\newcommand{\hn}[2]{\Hat{n}_{#1}^{#2}} +\newcommand{\rsmu}[2]{\mu_{#1}^{#2}} +\newcommand{\SO}[2]{\phi_{#1}(\br{#2})} + +\newcommand{\modX}{\text{X}} +\newcommand{\modY}{\text{Y}} + +% basis sets +\newcommand{\setdenbasis}{\mathcal{N}_{\Bas}} +\newcommand{\Bas}{\mathcal{B}} +\newcommand{\basis}{\mathcal{B}} +\newcommand{\Basval}{\mathcal{B}_\text{val}} +\newcommand{\Val}{\mathcal{V}} +\newcommand{\Cor}{\mathcal{C}} + +% operators +\newcommand{\hT}{\Hat{T}} +\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}} +\newcommand{\f}[2]{f_{#1}^{#2}} +\newcommand{\Gam}[2]{\Gamma_{#1}^{#2}} + +% coordinates +\newcommand{\br}[1]{{\mathbf{r}_{#1}}} +\newcommand{\bx}[1]{\mathbf{x}_{#1}} +\newcommand{\dbr}[1]{d\br{#1}} +\newcommand{\PBEspin}{PBEspin} +\newcommand{\PBEueg}{PBE-UEG-{$\tilde{\zeta}$}} + +\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} +\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, France} + +\begin{document} + +\title{A density-based basis-set correction for weak and strong correlation} + + +\begin{abstract} + +\end{abstract} + +\maketitle + + +\section{Size consistency of the basis-set correction} + +\subsection{Sufficient condition for size consistency} + +The basis-set correction is expressed as an integral in real space +\begin{equation} + \label{eq:def_ecmdpbebasis} + \begin{aligned} + & \efuncdenpbe{\argebasis} = \\ & \int \text{d}\br{} \,\denr \ecmd(\argrebasis), + \end{aligned} +\end{equation} +where all the local quantities $\argrebasis$ are obtained from the same wave function $\Psi$. In the limit of two non-overlapping and non-interacting dissociated fragments $\text{A}+\text{B}$, this integral can be rewritten as the sum of the integral over the region $\Omega_\text{A}$ and the integral over the region $\Omega_\text{B}$ +\begin{equation} + \label{eq:def_ecmdpbebasis} + \begin{aligned} + & \efuncdenpbeAB{\argebasis} = \\ & \int_{\Omega_\text{A}} \text{d}\br{} \,\denr \ecmd(\argrebasis) \\ & + \int_{\Omega_\text{B}} \text{d}\br{} \,\denr \ecmd(\argrebasis). + \end{aligned} +\end{equation} +Therefore, a sufficient condition to obtain size consistency is that all the local quantities $\argrebasis$ are \textit{intensive}, i.e. that they \textit{locally} coincide in the supersystem $\text{A}+\text{B}$ and in each isolated fragment $\text{X}=\text{A}$ or $\text{B}$. Hence, for $\br{} \in \Omega_\text{X}$, we should have +\begin{subequations} +\begin{equation} +n_\text{A+B}(\br{}) = n_\text{X}(\br{}), +\label{nAB} +\end{equation} +\begin{equation} +\zeta_\text{A+B}(\br{}) = \zeta_\text{X}(\br{}), +\label{zAB} +\end{equation} +\begin{equation} +s_\text{A+B}(\br{}) = s_\text{X}(\br{}), +\label{sAB} +\end{equation} +\begin{equation} +n_{2,\text{A+B}}(\br{}) = n_{2,\text{X}}(\br{}), +\label{n2AB} +\end{equation} +\begin{equation} +\mu_{\text{A+B}}(\br{}) = \mu_{\text{X}}(\br{}), +\label{muAB} +\end{equation} +\end{subequations} +where the left-hand-side quantities are for the supersystem and the right-hand-side quantities for an isolated fragment. Such conditions can be difficult to fulfil in the presence of degeneracies since the system X can be in a different mixed state (i.e. ensemble) in the supersystem $\text{A}+\text{B}$ and in the isolated fragment~\cite{Sav-CP-09}. Here, we will consider the simple case where the wave function of the supersystem is multiplicatively separable, i.e. +\begin{equation} + \ket{\wf{\text{A}+\text{B}}{}} = \ket{\wf{\text{A}}{}} \otimes \ket{\wf{\text{B}}{}}, +\end{equation} +where $\otimes$ is the antisymmetric tensor product. In this case, it is easy to shown that Eqs.~(\ref{nAB})-(\ref{sAB}) are valid, as well known, and it remains to show that Eqs.~(\ref{n2AB}) and~(\ref{muAB}) are also valid. Before showing this, we note that even though we do not explicity consider the case of degeneracies, the lack of size consistency which could arise from spin-multiplet degeneracies can be avoided by the same strategy used for imposing the energy independence on $S_z$, i.e. by using the effective spin polarization $\tilde{\zeta}(n(\br{}),n_{2}(\br{}))$ or a zero spin polarization $\zeta(\br{}) = 0$. Moreover, the lack of size consistency which could arise from spatial degeneracies (e.g., coming from atomic p states) can also be avoided by selecting the same member of the ensemble in the supersystem and in the isolated fragement. This applies to the systems treated in this work. + +\subsection{Intensivity of the on-top pair density and of the local range-separation parameter} + +The on-top pair density can be written in an orthonormal spatial orbital basis $\{\SO{p}{}\}$ as +\begin{equation} +\label{eq:def_n2} + n_{2{}}(\br{}) = \sum_{pqrs \in \Bas} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{}, +\end{equation} +with $\Gam{pq}{rs} = 2 \mel*{\wf{}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{}}$. As the summations run over all orbitals in the basis set $\Bas$, $n_{2{}}(\br{})$ is invariant to orbital rotations and can thus be expressed in terms of localized orbitals. For two non-overlapping fragments $\text{A}+\text{B}$, the basis set can then partitioned into orbitals localized on the fragment A and orbitals localized on B, i.e. $\Bas=\Bas_\text{A}\cup \Bas_\text{B}$. Therefore, we see that the on-top pair density of the supersystem $\text{A}+\text{B}$ is additively separable +\begin{equation} + \label{eq:def_n2} + n_{2,\text{A}+\text{B}}(\br{}) = n_{2,\text{A}}(\br{}) + n_{2,\text{B}}(\br{}), +\end{equation} +where $n_{2,\text{X}}(\br{})$ is the on-top pair density of the fragment X +\begin{equation} + \label{eq:def_n2} + n_{2,\text{X}}(\br{}) = \sum_{pqrs \in \Bas_\text{X}} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{}. +\end{equation} +This shows that the on-top pair density is a local intensive quantity. + +The local range-separation parameter is defined by +\begin{equation} + \label{eq:def_murAnnex} + \mur = \frac{\sqrt{\pi}}{2} \frac{f(\bfr{},\bfr{})}{n_{2}(\br{})}, +\end{equation} +where +\begin{equation} + \label{eq:def_f} + f(\bfr{},\bfr{}) = \sum_{pqrstu\in \Bas} \SO{p}{ } \SO{q}{ } \V{pq}{rs} \Gam{rs}{tu} \SO{t}{ } \SO{u}{ }. +\end{equation} +Again, $f(\bfr{},\bfr{})$ is invariant to orbital rotations and can be expressed in terms of orbitals localized on the fragments A and B. In the limit of infinitely separated fragments, the Coulomb interaction vanishes between A and B and therefore any two-electron integral $\V{pq}{rs}$ involving orbitals on both $A$ and $B$ vanishes. We thus see that the quantity $f(\bfr{},\bfr{})$ of the supersystem $\text{A}+\text{B}$ is additively separable +\begin{equation} + \label{eq:def_fa+b} + f_{\text{A}+\text{B}}(\bfr{},\bfr{}) = f_{\text{A}}(\bfr{},\bfr{}) + f_{\text{B}}(\bfr{},\bfr{}), +\end{equation} +with +\begin{equation} + \label{eq:def_fX} + f_\text{X}(\bfr{},\bfr{}) = \sum_{pqrstu\in \Bas_\text{X}} \SO{p}{ } \SO{q}{ } \V{pq}{rs} \Gam{rs}{tu} \SO{t}{ } \SO{u}{ }. +\end{equation} +So, $f(\bfr{},\bfr{})$ is a local intensive quantity. +As a consequence, the local range-separation parameter of the supersystem $\text{A}+\text{B}$ is +\begin{equation} + \label{eq:def_murAB} + \mu_{\text{A}+\text{B}}(\bfr{}) = \frac{\sqrt{\pi}}{2} \frac{f_{\text{A}}(\bfr{},\bfr{}) + f_{\text{B}}(\bfr{},\bfr{})}{n_{2,\text{A}}(\br{}) + n_{2,\text{B}}(\br{})}, +\end{equation} +which gives +\begin{equation} + \label{eq:def_murABsum} + \mu_{\text{A}+\text{B}}(\bfr{}) = \mu_{\text{A}}(\bfr{}) + \mu_{\text{B}}(\bfr{}), +\end{equation} +with $\mu_{\text{X}}(\bfr{}) = (\sqrt{\pi}/2) f_{\text{X}}(\bfr{},\bfr{})/n_{2,\text{X}}(\br{})$. The local range-separation parameter is thus a local intensive quantity. + +In conclusion, if the wave function of the supersystem $\text{A}+\text{B}$ is multiplicative separable, all local quantities used in the basis-set correction functional are intensive and therefore the basis-set correction is size consistent. + +\section{Computational considerations} +The computational cost of the present approach is driven by two quantities: the computation of the on-top pair density and the $\murpsibas$ on the real-space grid. Within a blind approach, for each grid point the computational cost is of order $n_{\Bas}^4$ and $n_{\Bas}^6$ for the on-top pair density $n_{2,\wf{\Bas}{}}(\br{})$ and the local range separation parameter $\murpsibas$, respectively. +Nevertheless, using CASSCF wave functions to compute these quantities leads to significant simplifications which can substantially reduce the CPU time. +\subsection{Computation of the on-top pair density for a CASSCF wave function} +Given a generic wave function developed on a basis set with $n_{\Bas}$ basis functions, the evaluation of the on-top pair density is of order $\left(n_{\Bas}\right)^4$. +Nevertheless, assuming that the wave function $\Psi^{\Bas}$ is of CASSCF type, a lot of simplifications happen. +If the active space is referred as the set of spatial orbitals $\mathcal{A}$ which are labelled by the indices $t,u,v,w$, and the doubly occupied orbitals are the set of spatial orbitals $\mathcal{C}$ labeled by the indices $i,j$, one can write the on-top pair density of a CASSCF wave function as +\begin{equation} + \label{def_n2_good} + n_{2,\wf{\Bas}{}}(\br{}) = n_{2,\mathcal{A}}(\br{}) + n_{\mathcal{C}}(\br{}) n_{\mathcal{A}}(\br{}) + \left( n_{\mathcal{C}}(\br{})\right)^2 +\end{equation} +where +\begin{equation} + \label{def_n2_act} + n_{2,\mathcal{A}}(\br{}) = \sum_{t,u,v,w \, \in \mathcal{A}} 2 \mel*{\wf{}{\Bas}}{ \aic{t_\downarrow}\aic{u_\uparrow}\ai{v_\uparrow}\ai{w_\downarrow}}{\wf{}{\Bas}} \phi_t (\br{}) \phi_u (\br{}) \phi_v (\br{}) \phi_w (\br{}) +\end{equation} +is the purely active part of the on-top pair density, +\begin{equation} + n_{\mathcal{C}}(\br{}) = \sum_{i\, \in \mathcal{C}} \left(\phi_i (\br{}) \right)^2, +\end{equation} +and +\begin{equation} + n_{\mathcal{A}}(\br{}) = \sum_{t,u\, \in \mathcal{A}} \phi_t (\br{}) \phi_u (\br{}) + \mel*{\wf{}{\Bas}}{ \aic{t_\downarrow}\ai{u_\downarrow} + \aic{t_\uparrow}\ai{u_\uparrow}}{\wf{}{\Bas}} +\end{equation} +is the purely active one-body density. +Written as in eq. \eqref{def_n2_good}, the leading computational cost is the evaluation of $n_{2,\mathcal{A}}(\br{})$ which, according to eq. \eqref{def_n2_act}, scales as $\left( n_{\mathcal{A}}\right) ^4$ where $n_{\mathcal{A}}$ is the number of active orbitals which is much smaller than the number of basis functions $n_{\Bas}$. Therefore, the final computational scaling of the on-top pair density for a CASSCF wave function over the whole real-space grid is of $\left( n_{\mathcal{A}}\right) ^4 n_G$, where $n_G$ is the number of grid points. +\subsection{Computation of $\murpsibas$} +At a given grid point, the computation of $\murpsibas$ needs the computation of $f_{\wf{}{}}(\bfr{},\bfr{}) $ defined in eq. \eqref{eq:def_f} and the on-top pair density defined in eq. \eqref{eq:def_n2}. In the previous paragraph we gave an explicit form of the on-top pair density in the case of a CASSCF wave function with a computational scaling of $\left( n_{\mathcal{A}}\right)^4$. In the present paragraph we focus on simplifications that one can obtain for the computation of $f_{\wf{}{}}(\bfr{},\bfr{}) $ in the case of a CASSCF wave function. + +One can rewrite $f_{\wf{}{}}(\bfr{},\bfr{}) $ as +\begin{equation} + \label{eq:f_good} + f_{\wf{}{}}(\bfr{},\bfr{}) = \sum_{r,s \in \Bas} \mathcal{V}_r^s(\bfr{}) \, \mathcal{N}_{r}^s(\bfr{}) +\end{equation} +where +\begin{equation} + \mathcal{V}_r^s(\bfr{}) = \sum_{p,q \in \Bas} V_{pq}^{rs} \phi_p(\br{}) \phi_q(\br{}) +\end{equation} +and +\begin{equation} + \mathcal{N}_{r}^s(\bfr{}) = \sum_{p,q \in \Bas} \Gam{pq}{rs} \phi_p(\br{}) \phi_q(\br{}) . +\end{equation} +\textit{A priori}, for a given grid point, the computational scaling of eq. \eqref{eq:f_good} is of $\left(n_{\Bas}\right)^4$ and the total computational cost over the whole grid scales therefore as $\left(n_{\Bas}\right)^4 n_G$. + +In the case of a CASSCF wave function, it is interesting to notice that $\Gam{pq}{rs}$ vanishes if one index $p,q,r,s$ does not belong +to the set of of doubly occupied or active orbitals $\mathcal{C}\cup \mathcal{A}$. Therefore, at a given grid point, the matrix $\mathcal{N}_{r}^s(\bfr{})$ has only at most $\left(n_{\mathcal{A}}+n_{\mathcal{C}}\right)^2$ non-zero elements, which is usually much smaller than $\left(n_{\Bas}\right)^2$. +As a consequence, in the case of a CASSCF wave function one can rewrite $f_{\wf{}{}}(\bfr{},\bfr{})$ as +\begin{equation} + f_{\wf{}{}}(\bfr{},\bfr{}) = \sum_{r,s \in \mathcal{C}\cup\mathcal{A}} \mathcal{V}_r^s(\bfr{}) \, \mathcal{N}_{r}^s(\bfr{}). +\end{equation} +Therefore the final computational cost of $f_{\wf{}{}}(\bfr{},\bfr{})$ is dominated by that of $\mathcal{V}_r^s(\bfr{})$, which scales as $\left(n_{\mathcal{A}}+n_{\mathcal{C}}\right)^2 \left( n_{\Bas} \right)^2 n_G$, which is much weaker than $\left(n_{\Bas}\right)^4 n_G$. +\bibliography{../srDFT_SC} + +\end{document} diff --git a/Revised_Manuscript/SI/srDFT_SC-SINotes.bib b/Revised_Manuscript/SI/srDFT_SC-SINotes.bib new file mode 100644 index 0000000..881f198 --- /dev/null +++ b/Revised_Manuscript/SI/srDFT_SC-SINotes.bib @@ -0,0 +1,2 @@ +@CONTROL{REVTEX41Control} +@CONTROL{aip41Control,pages="0",title=""} diff --git a/Revised_Manuscript/TODO b/Revised_Manuscript/TODO new file mode 100644 index 0000000..44921a2 --- /dev/null +++ b/Revised_Manuscript/TODO @@ -0,0 +1,3 @@ +Redo all the figures with the good nomenclature: ++) show the 0 \xi in the non zoom ++) change the ot for ONT diff --git a/Revised_Manuscript/biblio.bib b/Revised_Manuscript/biblio.bib new file mode 100644 index 0000000..bf6fae9 --- /dev/null +++ b/Revised_Manuscript/biblio.bib @@ -0,0 +1,11864 @@ +%% This BibTeX bibliography file was created using BibDesk. +%% http://bibdesk.sourceforge.net/ + +%% Created for Pierre-Francois Loos at 2019-12-12 03:48:36 +0100 + + +%% Saved with string encoding Unicode (UTF-8) + + + +@book{AbrSte-BOOK-72, + Address = {New York}, + Author = {M. 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Phys.}, + Number = {22}, + Pages = {9559}, + Volume = {116}, + Year = {2002}} + +@article{BoeMar-JCP-04, + Author = {A. Daniel Boese and Jan M. L. Martin}, + Journal = {J. Chem. Phys.}, + Number = {8}, + Pages = {3405}, + Volume = {121}, + Year = {2004}} + +@article{BohPin-PR-51, + Author = {D. Bohm and D. Pines}, + Journal = {Phys. Rev.}, + Pages = {625}, + Volume = {82}, + Year = {1951}} + +@article{BohPin-PR-53, + Author = {D. Bohm and D. Pines}, + Journal = {Phys. Rev.}, + Pages = {609}, + Title = {A Collective Description of Electron Interactions: III. Coulomb Interactions in a Degenerate Electron Gas}, + Volume = {{92}}, + Year = {1953}} + +@article{Bon-CPL-84, + Author = {V. E. Bondybey}, + Journal = {Chem. Phys. Lett.}, + Pages = {436}, + Volume = {109}, + Year = {1984}} + +@article{BonEng-JCP-84, + Author = {V. E. Bondybey and J. H. English}, + Journal = {J. Chem. Phys.}, + Pages = {568}, + Volume = {80}, + Year = {1984}} + +@article{BooCleThoAla-JCP-11, + Author = {G. H. Booth and D. Cleland and A. J. W. Thom and A. Alavi}, + Journal = {J. Chem. Phys.}, + Pages = {084104}, + Volume = {135}, + Year = {2011}} + +@article{BooGruKreAla-Nat-13, + Author = {G. H. Booth and A. Gr\"uneis and G. Kresse and A. Alavi}, + Journal = {Nature}, + Pages = {365}, + Title = {Towards an exact description of electronic wavefunctions in real solids}, + Volume = {493}, + Year = {2013}} + +@article{BooThoAla-JCP-09, + Author = {G. H. Booth and A. J. W. Thom and A. Alavi}, + Journal = {J. Chem. Phys.}, + Number = {{5}}, + Title = {{Fermion Monte Carlo without fixed nodes: A game of life, death, and annihilation in Slater determinant space}}, + Volume = {{131}}, + Year = {{2009}}} + +@article{BorJorNicNac-TCA-98, + Author = {P. Borowski and K. D. Jordan and J. Nichols and P. Nachtigall}, + Journal = {Theor. Chem. Acc.}, + Pages = {135}, + Volume = {99}, + Year = {1998}} + +@article{BouBenMayCaf-JCP-09, + Author = {T. Bouab\c{c}a and N. Ben Amor and D. Maynau and M. Caffarel}, + Journal = {J. Chem. Phys.}, + Pages = {114107}, + Volume = {130}, + Year = {2009}} + +@article{BouBraCaf-JCP-10, + Author = {T. Bouab\c{c}a and B. Bra\"ida and M. Caffarel}, + Journal = {J. Chem. Phys.}, + Pages = {044111}, + Title = {Multi-Jastrow trial wavefunctions for electronic structure calculations with quantum Monte Carlo}, + Volume = {133}, + Year = {2010}} + +@article{BouChiLegDucBlaJac-JCTC-14, + Author = {{P. Boulanger, S. Chibani, B. Le Guennic, I. Duchemin, X. Blase, and D. Jacquemin}}, + Journal = {J. Chem. Theory Comput.}, + Pages = {4548}, + Volume = {10}, + Year = {2014}} + +@article{BouJacDucBla-JCTC-14, + Author = {P. Boulanger and D. Jacquemin and I. Duchemin and X. Blase}, + Journal = {J. Chem. Theory Comput.}, + Pages = {1212}, + Volume = {10}, + Year = {2014}} + +@article{BowSugAld-PRB-94, + Author = {C. Bowen and G. Sugiyama and B. J. Alder}, + Journal = {Phys. Rev. B}, + Pages = {14838}, + Volume = {{50}}, + Year = {1996}} + +@article{BoyCou-JPB-73, + Author = {R. J. Boyd and C. A. Coulson}, + Journal = {J. Phys. B}, + Pages = {782}, + Volume = {6}, + Year = {1973}} + +@article{BoySarUga-JPB-88, + Author = {R. J. Boyd and C. Sarasola and J. M. Ugalde}, + Journal = {J. Phys. B}, + Pages = {2555}, + Volume = {21}, + Year = {1988}} + +@article{BraHib-02, + Author = {B. Braida and L. Thogersen and W. Wu and P. C. Hiberty}, + Journal = {J. Am. Chem. Soc.}, + Pages = {11781}, + Volume = {124}, + Year = {2002}} + +@article{BraHib-04, + Author = {B. Braida and P. C. Hiberty}, + Journal = {J. Am. Chem. Soc.}, + Pages = {14890}, + Volume = {126}, + Year = {2004}} + +@article{BraHib-08, + Author = {B. Braida and P. C. Hiberty}, + Journal = {J. Phys. Chem. A}, + Pages = {13045}, + Volume = {112}, + Year = {2008}} + +@article{BraTouCafUmr-JCP-11, + Author = {B. Bra\"ida and J. Toulouse and M. Caffarel and C. J. Umrigar}, + Journal = {J. Chem. Phys.}, + Pages = {084108}, + Title = {Quantum Monte Carlo with Jastrow-valence-bond wave functions}, + Volume = {134}, + Year = {2011}} + +@misc{BraTouCafUmr-JJJ-XX-note2, + Note = {In our BOVB wave functions, we use different sets of orbital coefficients, but a single set of basis exponents shared by all orbital sets.}} + +@misc{BraTouCafUmr-JJJ-XX-note3, + Note = {Bouab\c{c}a {\it et al.}~\cite{BouBraCaf-JCP-10} have introduced a wave function with several Jastrow factors attached to individual molecular orbitals. With such a wave function, the correlation effects can be treated differently in atomic and binding regions. In the case of the FH molecule, no atomic core Jastrow was used and two different valence Jastrow factors (one for the lone pairs paying a role in the bond and the other one for the $\sigma$-bond) were introduced. The resulting binding energy was essentially exact within error bars. Combining the various VB wave functions discussed in this work with this multi-Jastrow approach is presently under investigation.}} + +@misc{BraTouCafUmr-JJJ-XX-note, + Note = {In the condensed-matter community, ``stricly localized orbitals'' often refers to orbitals that vanish exactly at some finite distance. In the present work, ``stricly localized orbitals'' is employed in the sense usually used in the quantum chemistry community, i.e. orbitals expanded on Gaussian or Slater basis functions centered on a single atom. Thus, these orbitals vanish exactly only at infinite distance.}} + +@article{BreAda-JCP-11, + Author = {E. Br\'emond and C. Adamo}, + Journal = {J. Chem. Phys.}, + Pages = {024106}, + Volume = {135}, + Year = {2011}} + +@article{BreBou-PRB-96, + Author = {J. L. Bretonnet and M. Boulahbak}, + Journal = {Phys. Rev. B}, + Pages = {6859}, + Volume = {{53}}, + Year = {1996}} + +@incollection{BreCepRey-INC-01, + Author = {D. Bressanini and D. M. Ceperley and P. Reynolds}, + Booktitle = {Recent Advances in Quantum Monte Carlo Methods, II}, + Editor = {S. Rothstein}, + Publisher = {World Scientific}, + Title = {What do we know about wave function nodes?}, + Year = {2001}} + +@article{BreCioSanAda-ACR-16, + Author = {E. Br\'emond and I. Ciofini and J. C. Sancho-Garc\'ia and C. Adamo}, + Journal = {Acc. Chem. Res.}, + Pages = {1503}, + Volume = {49}, + Year = {2016}} + +@article{BreMelMor-PRA-98, + Author = {D. Bressanini and M. Mella and G. Morosi}, + Journal = {Phys. Rev. A}, + Pages = {1678}, + Volume = {57}, + Year = {1998}} + +@article{BreMorTar-JCP-05, + Author = {D. Bressanini and G. Morosi and S. Tarasco}, + Journal = {J. Chem. Phys.}, + Pages = {204109}, + Volume = {123}, + Year = {2005}} + +@article{BreSavPerSanAda-JCTC-18, + Author = {E. Br\'emond and M. Savarese and A. J. P\'erez-Jim\'enez and J. C. Sancho-Garc\'ia and C. Adamo}, + Journal = {J. Chem. Theory Comput.}, + Pages = {4052}, + Volume = {14}, + Year = {2018}} + +@article{BroDevLem-PRB-80, + Author = {F. Brosens and J. T. Devreese and L. F. Lemmens}, + Journal = {Phys. Rev. B}, + Pages = {1363}, + Volume = {{21}}, + Year = {1980}} + +@article{BroPanSto-PRL-09, + Author = {Christian Brouder and Gianluca Panati and Gabriel Stoltz}, + Journal = {Phys. Rev. Lett.}, + Pages = {230401}, + Volume = {103}, + Year = {2009}} + +@article{BroTraLopNee-JCP-07, + Author = {M. D. Brown and J. R. Trail and P. L\'opez R\'ios and R. J. Needs}, + Journal = {J. Chem. Phys.}, + Pages = {224110}, + Title = {Energies of the first row atoms from quantum Monte Carlo}, + Volume = {126}, + Year = {2007}} + +@article{BroYinLor-SR-13, + Author = {V. Brosco and Z.-J. Ying and J. Lorenzana}, + Journal = {Sci. Rep.}, + Pages = {2172}, + Volume = {3}, + Year = {2013}} + +@article{BruHamNea-JCP-15, + Author = {F. Bruneval and S. M. Hamed and J. B. Neaton}, + Journal = {J. Chem. Phys.}, + Pages = {244101}, + Volume = {142}, + Year = {2015}} + +@article{BruProKroBre-JCP-17, + Author = {Iulia Emilia Brumboiu and Georgia Prokopiou and Leeor Kronik and Barbara Brena}, + Journal = {J. Chem. Phys.}, + Pages = {044301}, + Volume = {147}, + Year = {2017}} + +@phdthesis{Bru-THESIS-05, + Author = {F. Bruneval}, + School = {Ecole Polytechnique}, + Year = {2005}} + +@article{BuiBae-MP-02, + Author = {M. A. Buijse and E. J. Baerends}, + Journal = {Mol. Phys.}, + Pages = {401}, + Volume = {100}, + Year = {2002}} + +@article{BurCruLam-JCP-98, + Author = {K. Burke and F. G. Cruz and K.-C. Lam}, + Journal = {J. Chem. Phys}, + Pages = {8161}, + Volume = {109}, + Year = {1998}} + +@article{BurErnPer-CPL-97, + Author = {K. Burke and M. Ernzerhof and J. P. Perdew}, + Journal = {Chem. Phys. Lett.}, + Pages = {115}, + Volume = {265}, + Year = {1997}} + +@article{BurFilDol-JCP-07, + Author = {M. Burkatzki and C. Filippi and M. Dolg}, + Journal = {J. Chem. Phys.}, + Pages = {234105}, + Volume = {126}, + Year = {2007}} + +@article{Bur-JCP-12, + Author = {K. Burke}, + Journal = {J. Chem. Phys.}, + Pages = {150901}, + Title = {{Perspective on density functional theory}}, + Volume = {136}, + Year = {2012}} + +@article{BurPerErn-JCP-98, + Author = {K. Burke and J. P. Perdew and M. Ernzerhof}, + Journal = {J. Chem. Phys}, + Pages = {3760}, + Volume = {109}, + Year = {1998}} + +@article{BurPer-IJQC-95, + Author = {K. Burke and J. P. Perdew}, + Journal = {Int. J. Quantum. Chem.}, + Pages = {199}, + Volume = {{56}}, + Year = {1995}} + +@article{BurPerLan-PRL-94, + Author = {K. Burke and J. P. Perdew and D. C. Langreth}, + Journal = {Phys. Rev. Lett.}, + Pages = {1283}, + Volume = {73}, + Year = {1994}} + +@article{BurPerLev-PRA-96, + Author = {K. Burke and J. P. Perdew and M. Levy}, + Journal = {Phys. Rev. A}, + Pages = {R2915}, + Volume = {53}, + Year = {1996}} + +@inbook{BurPerWan-INC-98, + Address = {NY}, + Author = {Kieron Burke and John P. Perdew and Y. Wang}, + Booktitle = {Electronic Density Functional Theory: Recent Progress and New Directions}, + Date-Modified = {2013-02-12 00:16:04 +0000}, + Editor = {J. F. Dobson and G. Vignale and M. P. Das}, + Organization = {Plenum}, + Pages = {81}, + Pub-Num = {34}, + Publisher = {Plenum}, + Title = {Derivation of a generalized gradient approximation: The PW91 density functional}, + Url = {http://link.springer.com/chapter/10.1007%2F978-1-4899-0316-7_7}, + Year = {1997}, + Bdsk-Url-1 = {http://link.springer.com/chapter/10.1007%2F978-1-4899-0316-7_7}} + +@article{BygAllMan-JCP-12, + Author = {P. J. Bygrave and N. L. Allan and F. R. Manby}, + Journal = {J. Chem. Phys.}, + Pages = {164102}, + Volume = {137}, + Year = {2012}} + +@article{BylKle-PRB-90, + Author = {D. M. Bylander and L. Kleinman}, + Journal = {Phys. Rev. B}, + Pages = {7868}, + Volume = {41}, + Year = {1990}} + +@article{ByrCotMon-JCP-11, + Author = {{J. N. Byrd, R. C\^ot\'e, and J. A. Montgomery, Jr}}, + Journal = {J. Chem. Phys.}, + Pages = {244307}, + Volume = {135}, + Year = {2011}} + +@article{BytRue-JCP-05, + Author = {L. Bytautas and K. Ruedenberg}, + Journal = {J. Chem. Phys.}, + Pages = {154110}, + Volume = {122}, + Year = {2005}} + +@misc{BytRue-JCP-05-note, + Note = {Note that this estimate of the exact well depth differs from the one used in Ref.~\onlinecite{UmrTouFilSorHeg-JJJ-XX} where we used instead the scalar-relativistic, valence-corrected estimate of Ref.~\onlinecite{BytRue-JCP-05} since calculations were performed with a relativistic pseudopotential.}} + +@article{CadWah-ADNDT-74, + Author = {P. E. Cade and A. C. Wahl}, + Journal = {At. Data Nucl. Data Tables}, + Pages = {340}, + Volume = {13}, + Year = {1974}} + +@article{Caf-ActNum-98, + Author = {R. E. Caflisch}, + Journal = {Acta Numerica}, + Pages = {1}, + Title = {{Monte Carlo and quasi-Monte Carlo methods}}, + Volume = {1998}, + Year = {1998}} + +@article{CafCla-JCP-88, + Author = {M. Caffarel and P. Claverie}, + Doi = {10.1063/1.454228}, + Journal = {J. Chem. Phys.}, + Keywords = {QUANTUM STATISTICAL MECHANICS; MONTE CARLO METHOD; FERMIONS; FEYNMAN PATH INTEGRAL; SCHROEDINGER EQUATION}, + Number = {2}, + Pages = {1100-1109}, + Publisher = {AIP}, + Title = {{Development of a pure diffusion quantum Monte Carlo method using a full generalized Feynman--Kac formula. II. Applications to simple systems}}, + Volume = {88}, + Year = {1988}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.454228}} + +@article{CaiRei-JCP-00, + Author = {Z.-L. Cai and J. R. Reimers}, + Journal = {J. Chem. Phys.}, + Pages = {527}, + Volume = {112}, + Year = {2000}} + +@article{CaiZanKitKocKreScr-PRA-05, + Author = {J. Caillat and J. Zanghellini and M. Kitzler and O. Koch and W. Kreuzer and A. Scrinzi}, + Journal = {Phys. Rev. A}, + Pages = {012712}, + Volume = {71}, + Year = {2005}} + +@article{CalSor-PRB-98, + Author = {{M. Calandra Buonaura and S. Sorella}}, + Journal = {Phys. Rev. B}, + Pages = {11446}, + Title = {{Numerical study of the two-dimensional Heisenberg model using a Green function Monte Carlo technique with a fixed number of walkers}}, + Volume = {57}, + Year = {1998}} + +@article{CanBoyTha-JCP-93, + Author = {N. M. Cann and R. J. Boyd and A. J. Thakkar}, + Journal = {J. Chem. Phys.}, + Pages = {7132}, + Volume = {98}, + Year = {1993}} + +@article{CanMou-NL-14, + Author = {Eric Canc\`es and Nahia Mourad}, + Journal = {Nonlinearity}, + Number = {9}, + Pages = {1999}, + Title = {A mathematical perspective on density functional perturbation theory}, + Volume = {27}, + Year = {2014}} + +@article{CanPer-JCP-08, + Author = {E. Canc\`es and K. Pernal}, + Journal = {J. Chem. Phys.}, + Pages = {134108}, + Title = {{Projected gradient algorithms for Hartree-Fock and density matrix functional theory calculations}}, + Volume = {128}, + Year = {2008}} + +@article{CapNalPar-JCP-82, + Author = {Joseph F. Capitani and Roman F. Nalewajski and Robert G. Parr}, + Journal = {The Journal of Chemical Physics}, + Keywords = {MOLECULES; MOLECULAR STRAUCTURE; FUNCTIONALS; BORN–OPPENHEIMER APPROXIMATION; GROUND STATES; HAMILTONIANS; ELECTRON DENSITY; NUCLEI; ELECTRIC FIELDS; EXCITED STATES; BOUND STATE; USES}, + Number = {1}, + Pages = {568-573}, + Publisher = {AIP}, + Title = {Non-Born--Oppenheimer density functional theory of molecular systems}, + Url = {http://link.aip.org/link/?JCP/76/568/1}, + Volume = {76}, + Year = {1982}, + Bdsk-Url-1 = {http://link.aip.org/link/?JCP/76/568/1}} + +@article{CarGod-JCP-88, + Author = {{E. A. Carter and W. A. Goddard III}}, + Journal = {J. Chem. Phys.}, + Pages = {3132}, + Volume = {88}, + Year = {1988}} + +@article{CarSilMet-JCP-79, + Author = {D. P. Carroll and H. J. Silverstone and R. M. Metzger}, + Journal = {J. Chem. Phys.}, + Pages = {4142}, + Volume = {71}, + Year = {1979}} + +@article{CarTruGag-JCTC-15, + Author = {R. K. Carlson and D. G. Truhlar and L. Gagliardi}, + Journal = {J. Chem. Theory Comput.}, + Pages = {4077}, + Volume = {11}, + Year = {2015}} + +@article{CarTruGag-JPCA-17, + Author = {R. K. Carlson and D. G. Truhlar and L. Gagliardi}, + Journal = {J. Phys. Chem. A}, + Pages = {5540}, + Volume = {121}, + Year = {2017}} + +@article{CasAttSor-JCP-04, + Author = {Michele Casula and Claudio Attaccalite and Sandro Sorella}, + Journal = {J. Chem. Phys.}, + Pages = {7110}, + Title = {Correlated geminal wave function for molecules: An efficient resonating valence bond approach}, + Volume = {121}, + Year = {2004}} + +@article{CasGutGuaGadSalDau-JCP-00, + Author = {M. E. Casida and F. Gutierrez and J. Guan and F.-X. Gadea and D. Salahub and J.-P. Daudey}, + Journal = {J. Chem. Phys.}, + Pages = {7062}, + Volume = {113}, + Year = {2000}} + +@article{CasHui-ARPC-12, + Author = {M. E. Casida and M. Huix-Rotllant}, + Journal = {Annu. Rev. Phys. Chem.}, + Pages = {287}, + Title = {Progress in Time-Dependent Density-Functional Theory}, + Volume = {63}, + Year = {2012}} + +@incollection{CasHui-TCC-15, + Author = {M. E. Casida and M. Huix-Rotllant}, + Booktitle = {Density-Functional Methods for Excited States}, + Doi = {10.1007/128_2015_632}, + Editor = {N. Ferr\'e and M. Filatov and M. Huix-Rotllant}, + Publisher = {Springer}, + Series = {Topics in Current Chemistry}, + Year = {2015}, + Bdsk-Url-1 = {https://doi.org/10.1007/128_2015_632}} + +@incollection{Cas-INC-95, + Address = {Singapore}, + Author = {M. E. Casida}, + Booktitle = {Recent Advances in Density Functional Methods, Part I}, + Editor = {D. P. Chong}, + Pages = {155}, + Publisher = {World Scientific}, + Year = {1995}} + +@article{CasJamCasSal-JCP-98, + Author = {M. E. Casida and C. Jamorski and K. C. Casida and D. R. Salahub}, + Journal = {J. Chem. Phys.}, + Pages = {4439}, + Volume = {108}, + Year = {1998}} + +@article{Cas-JCP-05, + Author = {M. E. 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Phys.}, + Pages = {244105}, + Title = {{Computing the energy of a water molecule using multideterminants: A simple, efficient algorithm}}, + Volume = {135}, + Year = {2011}} + +@article{CleRoe-ADNDT-74, + Author = {E. Clementi and C. Roetti}, + Journal = {At. Data Nucl. Data Tables}, + Pages = {177}, + Volume = {14}, + Year = {1974}} + +@article{CleVei-JCP-66, + Author = {E. Clementi and A. Veillard}, + Journal = {J. Chem. Phys.}, + Pages = {3050}, + Volume = {{44}}, + Year = {1966}} + +@article{CocAssLupTou-JCP-17, + Author = {E. Coccia and R. Assaraf and E. Luppi and J. Toulouse}, + Doi = {10.1063/1.4991563}, + Journal = {J. Chem. Phys.}, + Pages = {014106}, + Title = {Ab initio lifetime correction to scattering states for time-dependent electronic-structure calculations with incomplete basis sets}, + Volume = {147}, + Year = {2017}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.4991563}} + +@article{CocCheBarSorGui-JCTC-12, + Author = {E. Coccia and O. Chernomor and M. Barborini and S. 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Phys.}, + Pages = {5149}, + Volume = {114}, + Year = {2001}} + +@article{EmaGarRamLopFerMeiPal-JCC-03, + Author = {{I. Ema, J. M. Garc\'ia de la Vega, G. Ram\'irez, R. L\'opez, J. Fern\'andez Rico, H. Meissner and J. Paldus}}, + Journal = {J. Comput. Chem.}, + Pages = {859}, + Volume = {24}, + Year = {2003}} + +@article{EmaGarRamLopFerMeiPal-JCC-03_fr, + Author = {{I. Ema, J. M. Garc\'ia de la Vega, G. Ram\'irez, R. L\'opez, J. Fern\'andez Rico, H. Meissner et J. Paldus}}, + Journal = {J. Comput. Chem.}, + Pages = {859}, + Volume = {24}, + Year = {2003}} + +@article{EngBon-IJMPB-01, + Author = {E. Engel and A. F. Bonetti}, + Journal = {Int. J. Mod. Phys. B}, + Pages = {1703}, + Volume = {15}, + Year = {2001}} + +@article{EngDre-JCC-99, + Author = {E. Engel and R. M. Dreizler}, + Journal = {J. Comput. Chem.}, + Pages = {31}, + Volume = {20}, + Year = {1999}} + +@incollection{EngDre-TCC-96, + Address = {Berlin}, + Author = {E. Engel and R. M. 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Nalewajski}, + Publisher = {Springer-Verlag}, + Year = {1996}} + +@article{ErnPer-JCP-98, + Author = {M. Ernzerhof and J. P. Perdew}, + Journal = {J. Chem. Phys.}, + Pages = {3313}, + Volume = {{109}}, + Year = {1998}} + +@article{ErnScu-JCP-99a, + Author = {M. Ernzerhof and G. E. Scuseria}, + Journal = {J. Chem. Phys.}, + Pages = {5029}, + Volume = {110}, + Year = {1999}} + +@article{EshBatFur-TCA-12, + Affiliation = {Department of Chemistry, University of California, 1102 Natural Sciences II, Irvine, CA 92697-2025, USA}, + Author = {Eshuis, Henk and Bates, Jefferson and Furche, Filipp}, + Issue = {1}, + Journal = {Theor. Chem. 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Filippi}, + Note = {private communication}} + +@article{FilSha-CPL-00, + Author = {Michael Filatov and Sason Shaik}, + Journal = {Chem. Phys. Lett.}, + Pages = {409}, + Volume = {332}, + Year = {1999}} + +@article{FilSha-CPL-99, + Author = {M. Filatov and S. Shaik}, + Journal = {Chem. Phys. Lett.}, + Pages = {429}, + Title = {{A spin-restricted ensemble-referenced Kohn-Sham method and its application to diradicaloid situations}}, + Volume = {304}, + Year = {1999}} + +@article{FilSha-JPCA-00, + Author = {Michael Filatov and Sason Shaik}, + Journal = {J. Phys. Chem. A}, + Pages = {6628}, + Volume = {104}, + Year = {2000}} + +@article{FilSha-JPCA-99, + Author = {Michael Filatov and Sason Shaik}, + Journal = {J. Phys. Chem. A}, + Pages = {8885}, + Volume = {103}, + Year = {1999}} + +@article{FilShaWoeGriPey-CPL-00, + Author = {Michael Filatov and Sason Shaik and M. Woeller and S. Grimme and S.D. Peyerimhoff}, + Journal = {Chem. Phys. Lett.}, + Pages = {135}, + Volume = {316}, + Year = {2000}} + +@misc{FilTouUmr-JJJ-XX, + Author = {Claudia Filippi and Julien Toulouse and C. J. Umrigar}, + Title = {in preparation}} + +@article{FilUmrGon-JCP-97, + Author = {C. Filippi and C. J. Umrigar and X. Gonze}, + Journal = {J. Chem. Phys.}, + Pages = {9994}, + Title = {{Excitation energies from density functional perturbation theory}}, + Volume = {107}, + Year = {1997}} + +@article{FilUmrGon-PRA-96, + Author = {C. Filippi and C. J. Umrigar and X. Gonze}, + Journal = {Phys. Rev. A}, + Pages = {4810}, + Volume = {54}, + Year = {1996}} + +@article{FilUmr-JCP-96, + Author = {C. Filippi and C. J. Umrigar}, + Journal = {J. Chem. Phys.}, + Pages = {213}, + Title = {Multiconfiguration wave functions for quantum Monte Carlo calculations of first-row diatomic molecules}, + Volume = {{105}}, + Year = {1996}} + +@article{Fil-WIR-14, + Author = {M. Filatov}, + Journal = {WIREs Comput. Mol. Sci.}, + Pages = {146}, + Title = {{Spin-restricted ensemble-referenced Kohn-Sham method: basic principles and application to strongly correlated ground and excited states of molecules}}, + Volume = {5}, + Year = {2015}} + +@article{FilZacBud-JCTC-09, + Author = {C. Filippi and M. Zaccheddu and F. Buda}, + Journal = {J. Chem. Theory Comput.}, + Pages = {2074}, + Title = {Absorption Spectrum of the Green Fluorescent Protein Chromophore: A Difficult Case for ab Initio Methods?}, + Volume = {5}, + Year = {2009}} + +@article{FilZacBud-JJJ-XX, + Author = {Claudia Filippi and Maurizio Zaccheddu and Francesco Buda}, + Note = {unpublished}} + +@incollection{FlaCafSav-INC-97, + Address = {Singapore}, + Author = {H.-J. Flad and M. Caffarel and A. Savin}, + Booktitle = {Recent advances in quantum Monte Carlo methods}, + Editor = {W. A. Lester}, + Pages = {73}, + Publisher = {World Scientific}, + Title = {Quantum Monte Carlo calculations with multi-reference trial wave functions}, + Year = {1997}} + +@misc{Fla-PRIV-XX, + Author = {H. J. Flad}, + Title = {private communication}} + +@article{FlaSav-JCP-95, + Author = {H.-J. Flad and A. Savin}, + Journal = {J. Chem. Phys.}, + Pages = {691}, + Volume = {103}, + Year = {1995}} + +@article{FlaSavPre-JCP-92, + Author = {H.-J. Flad and A. Savin and H. Preuss}, + Journal = {J. Chem. Phys.}, + Pages = {459}, + Volume = {97}, + Year = {1992}} + +@article{FlaSavScuNicPre-CPL-94, + Author = {H.-J. Flad and A. Savin and M. Schultheiss and A. Nicklass and H. Preuss}, + Journal = {Chem. Phys. Lett.}, + Pages = {274}, + Volume = {222}, + Year = {1994}} + +@article{FloGda-JCP-05, + Author = {J. R. Flores and R. J. Gdanitz}, + Journal = {J. Chem. Phys.}, + Pages = {144316}, + Volume = {123}, + Year = {2005}} + +@article{Foc-ZP-30, + Author = {V. Fock}, + Journal = {Z. Phys.}, + Pages = {855}, + Volume = {63}, + Year = {1930}} + +@article{FouMitNeeRaj-RMP-01, + Author = {W. M. C. Foulkes and L. Mitas and R. J. Needs and G. Rajagopal}, + Journal = {Rev. Mod. Phys.}, + Pages = {33}, + Title = {{Quantum Monte Carlo simulations of solids}}, + Volume = {{73}}, + Year = {2001}} + +@article{FraDurMes-JCP-97, + Author = {X. Fradera and M. Duran and J. Mestres}, + Journal = {J. Chem. Phys.}, + Pages = {3576}, + Volume = {107}, + Year = {1997}} + +@article{FraFilAmo-JCTC-12, + Author = {F. Fracchia and C. Filippi and C. Amovilli}, + Journal = {J. Chem. Theory Comput.}, + Pages = {1943}, + Title = {Size-extensive wave functions for quantum Monte Carlo: A linear scaling generalized valence bond approach}, + Volume = {8}, + Year = {2012}} + +@misc{FraLupTou-JJJ-XX, + Author = {O. Franck and E. Luppi and J. Toulouse}, + Note = {unpublished}} + +@article{FraMusLupTou-JCP-15, + Author = {O. Franck and B. Mussard and E. Luppi and J. Toulouse}, + Journal = {J. Chem. Phys.}, + Pages = {074107}, + Title = {Basis convergence of range-separated density-functional theory}, + Volume = {142}, + Year = {2015}} + +@misc{FraMusLupTou-JJJ-XX, + Author = {O. Franck and B. Mussard and E. Luppi and J. Toulouse}, + Note = {Basis convergence of range-separated density-functional theory, arXiv:1412.2613, to appear in J. Chem. Phys.}} + +@misc{FraMusLupTou-JJJ-XX-note1, + Note = {Note that the partial-wave expansion of any {\it odd} power of $r_{12}$ contains an infinite number of terms. In contrast, the partial-wave expansion of any {\it even} power of $r_{12}$ terminates at a finite $\ell$.}} + +@misc{FraMusLupTou-JJJ-XX-note2, + Note = {For a discussion of angular and radial correlation and the shape of the correlation hole in the He atom with the long-range interaction, see Ref.~\onlinecite{PolColLeiStoWerSav-IJQC-03}.}} + +@misc{FraMusLupTou-JJJ-XX-note3, + Note = {For $\mu=1$ the difference between the long-range MP2 correlation energies for $X=5$ and $X=6$ is still as small as about 0.2 mhartree.}} + +@misc{FraMusLupTou-JJJ-XX-note4, + Note = {For $r_1=r_2$, $f\lr_\ell >0$ for $\ell=0$ and $f\lr_\ell <0$ for $\ell \geq 1$.}} + +@article{FreHuxMor-PRA-84, + Author = {{D. E. Freund, B. D. Huxtable, and J. D. Morgan III}}, + Journal = {Phys. Rev. A}, + Pages = {980}, + Volume = {29}, + Year = {1984}} + +@article{FreLev-JCP-82, + Author = {Karl F. Freed and Mel Levy}, + Journal = {The Journal of Chemical Physics}, + Keywords = {ALGORITHMS; ELECTRON DENSITY; FUNCTIONALS; SERIES EXPANSION; WAVE FUNCTIONS; GROUND STATES}, + Number = {1}, + Pages = {396-398}, + Publisher = {AIP}, + Title = {Direct first principles algorithm for the universal electron density functional}, + Url = {http://link.aip.org/link/?JCP/77/396/1}, + Volume = {77}, + Year = {1982}, + Bdsk-Url-1 = {http://link.aip.org/link/?JCP/77/396/1}} + +@article{Fre-PRB-77, + Author = {D. L. Freeman}, + Journal = {Phys. Rev. B}, + Pages = {5512}, + Volume = {{15}}, + Year = {1977}} + +@article{FroCimJen-PRA-10, + Author = {E. Fromager and R. Cimiraglia and H. J. Aa. Jensen}, + Journal = {Phys. Rev. A}, + Pages = {024502}, + Volume = {81}, + Year = {2010}} + +@article{Fro-JCP-11, + Author = {E. Fromager}, + Journal = {J. Chem. Phys.}, + Pages = {244106}, + Volume = {135}, + Year = {2011}} + +@article{FroJen-JCP-11, + Author = {E. Fromager and H. J. Aa. Jensen}, + Journal = {J. Chem. Phys.}, + Pages = {034116}, + Volume = {135}, + Year = {2011}} + +@article{FroJen-PRA-08, + Author = {E. Fromager and H. J. Aa. Jensen}, + Journal = {Phys. Rev. A}, + Pages = {022504}, + Volume = {78}, + Year = {2008}} + +@misc{FroKneJen-ARX-12, + Author = {Emmanuel Fromager and Stefan Knecht and H. J. Aa. Jensen}, + Note = {\url{http://fr.arxiv.org/abs/1211.4829v1}}} + +@article{FroKneJen-JCP-13, + Author = {E. Fromager and S. Knecht and H. J. Aa. Jensen}, + Journal = {J. Chem. Phys.}, + Pages = {084101}, + Title = {{Multi-configuration time-dependent density-functional theory based on range separation}}, + Volume = {138}, + Year = {2013}} + +@article{FroReaWahWahJen-JCP-09, + Author = {E. Fromager and F. R\'eal and P. W{\aa}hlin and U. Wahlgren and H. J. Aa. Jensen}, + Journal = {J. Chem. Phys.}, + Pages = {054107}, + Volume = {131}, + Year = {2009}} + +@article{FroTouJen-JCP-07, + Author = {E. Fromager and J. Toulouse and H. J. Aa. Jensen}, + Journal = {J. Chem. Phys.}, + Pages = {074111}, + Title = {On the universality of the long-/short-range separation in multiconfigurational density-functional theory}, + Volume = {126}, + Year = {2007}} + +@article{FucGon-PRB-02, + Author = {M. Fuchs and X. Gonze}, + Journal = {Phys. Rev. B}, + Pages = {235109}, + Volume = {65}, + Year = {2002}} + +@article{FucNiqGonBur-JCP-05, + Abstract = {We show that density functional theory within the RPA (random phase + approximation for the exchange-correlation energy) provides a correct + description of bond dissociation in H2 in a spin-restricted Kohn-<96>Sham + formalism, i.e., without artificial symmetry breaking. We present + accurate adiabatic connection curves both at equilibrium and beyond + the Coulson-Fisher point. The strong curvature at large bond length + implies important static (left-<96>right) correlation, justifying modern + hybrid functional constructions but also demonstrating their limitations. + Although exact at infinite separation and accurate near the equilibrium + bond length, the RPA dissociation curve displays unphysical repulsion + at larger but finite bond lengths. Going beyond the RPA by including + the exact exchange kernel (RPA + X), we find a similar repulsion. + We argue that this deficiency is due to the absence of double excitations + in adiabatic linear response theory. Further analyzing the H2 dissociation + limit we show that the RPA + X is not size consistent, in contrast + to the RPA.}, + Author = {Fuchs, M. and Niquet, Y. M. and Gonze, X. and Burke, K.}, + Doi = {10.1063/1.1858371}, + Journal = {J. Chem. Phys.}, + Pages = {094116}, + Title = {Describing static correlation in bond dissociation by Kohn-Sham density functional theory}, + Volume = {122}, + Year = {2005}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.1858371}} + +@article{FukIwaSaw-PR-64, + Author = {N. Fukuda and F. Iwamoto and K. Sawada}, + Journal = {Phys. Rev.}, + Pages = {A932}, + Volume = {135}, + Year = {1964}} + +@book{Ful-BOOK-93, + Address = {Berlin}, + Author = {P. Fulde}, + Publisher = {Springer}, + Title = {Electron Correlations in Molecules and Solids}, + Year = {1993}} + +@article{Fur-JCP-01, + Author = {F. Furche}, + Journal = {J. Chem. Phys.}, + Pages = {5982}, + Volume = {114}, + Year = {2001}} + +@article{Fur-JCP-08, + Author = {F. Furche}, + Journal = {J. Chem. Phys.}, + Pages = {114105}, + Volume = {129}, + Year = {2008}} + +@article{FurPer-JCP-06, + Author = {F. Furche and J. P. Perdew}, + Journal = {J. Chem. 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Phys.}, + Pages = {1985}, + Volume = {8}, + Year = {2006}} + +@article{KadVoo-JCP-10, + Author = {{B. Kaduk and T. Van Voorhis}}, + Journal = {J. Chem. Phys.}, + Pages = {061102}, + Volume = {133}, + Year = {2010}} + +@article{KalMusTou-JCP-19, + Author = {C. Kalai and B. Mussard and J. Toulouse}, + Doi = {10.1063/1.5108536}, + Journal = {J. Chem. Phys.}, + Pages = {074102}, + Title = {Range-separated double-hybrid density-functional theory with coupled-cluster and random-phase approximations}, + Volume = {151}, + Year = {2019}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.5108536}} + +@misc{KalMusTou-JJJ-19, + Author = {C. Kalai and B. Mussard and J. Toulouse}, + Note = {Range-separated double-hybrid density-functional theory with coupled-cluster and random-phase approximations \href{https://arxiv.org/abs/1905.01014}{arXiv:1905.01014}}} + +@article{KalTou-JCP-18, + Author = {C. Kalai and J. Toulouse}, + Doi = {10.1063/1.5025561}, + Journal = {J. Chem. Phys.}, + Pages = {164105}, + Title = {A general range-separated double-hybrid density-functional theory}, + Volume = {148}, + Year = {2018}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.5025561}} + +@misc{KalTou-JJJ-XX-note1, + Note = {The geometries are available in the Minnesota Database at http://comp.chem.umn.edu/db/.}} + +@misc{KalTou-JJJ-XX-note2, + Note = {The term ``range-separated double hybrid (RSDH)'' has already been used in Ref.~\onlinecite{SanCivUsvTouShaMas-JCP-15} to designate the RSH+MP2 method and in Refs.~\onlinecite{CorStoJenFro-PRA-13,CorFro-IJQC-14} to generically designate several kinds of range-separated MP2/DFT hybrids. In the present work, we use RSDH to designate the specific scheme corresponding to Eq.~(\ref{EmulRSDH}).}} + +@misc{KalTou-JJJ-XX-note3, + Note = {We use the AE49 set of Fast {\it et al.}~\cite{FasCorSanTru-JPCA-99} to facilitate comparisons with previous related works~\cite{ShaTouSav-JCP-11,SouShaTou-JCP-14,MusReiAngTou-JCP-15} where this set was used. We do not expect any special difficulties with the six molecules that have been dropped from the G2-1 set.}} + +@article{KamSekTsuHir-JCP-05, + Author = {M. Kamiya and H. Sekino and T. Tsuneda and K. Hirao}, + Journal = {J. Chem. Phys.}, + Pages = {234111}, + Volume = {122}, + Year = {2005}} + +@article{KamTsuHir-JCP-02, + Author = {M. Kamiya and T. Tsuneda and K. Hirao}, + Journal = {J. Chem. Phys.}, + Pages = {6010}, + Volume = {{117}}, + Year = {2002}} + +@article{KanBec-JCTC-10, + Author = {F. O. Kannemann and Axel D. Becke}, + Journal = {J. Chem. Theory Comput.}, + Pages = {1081}, + Title = {{van der Waals interactions in density-functional theory: Intermolecular complexes}}, + Volume = {6}, + Year = {2010}} + +@article{KarDaoMar-CPL-11, + Author = {A. Karton and S. Daon and J. M. L. Martin}, + Journal = {Chem. Phys. Lett.}, + Pages = {165}, + Volume = {510}, + Year = {2011}} + +@article{Kar-JCP-03, + Author = {V. V. Karasiev}, + Journal = {J. Chem. Phys.}, + Pages = {8576}, + Volume = {118}, + Year = {2003}} + +@article{KarKroKum-JCP-13, + Author = {A. Karolewski and L. Kronik and S. K\"ummel}, + Journal = {J. Chem. Phys.}, + Pages = {204115}, + Volume = {138}, + Year = {2013}} + +@article{KarMar-JCP-11, + Author = {Amit Karton and J. M. L. Martin}, + Journal = {J. Chem. Phys.}, + Pages = {144119}, + Volume = {135}, + Year = {2011}} + +@article{KarTarLamSchMar-JPCA-08, + Author = {Amir Karton and Alex Tarnopolsky and Jean-Francois Lam\`ere and George C. Schatz and Jan M. L. Martin}, + Journal = {J. Phys. Chem. 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Lembarki and F. Rogemond and H. Chermette}, + Journal = {Phys. Rev. A}, + Pages = {3704}, + Volume = {52}, + Year = {1995}} + +@article{LenBal-JCP-83, + Author = {J. H. van Lenthe and G. G. Balint-Kurti}, + Journal = {J. Chem. Phys.}, + Pages = {5699}, + Volume = {78}, + Year = {1983}} + +@article{Len-JCP-80, + Author = {{B. H. Lengsfield III}}, + Journal = {J. Chem. Phys.}, + Pages = {382}, + Volume = {73}, + Year = {1980}} + +@article{LenLiu-JCP-81, + Author = {{B. H. Lengsfield III and B. Liu}}, + Journal = {J. Chem. Phys.}, + Pages = {478}, + Volume = {75}, + Year = {1981}} + +@article{LenLiu-JCP-81_fr, + Author = {{B. H. Lengsfield III et B. Liu}}, + Journal = {J. Chem. Phys.}, + Pages = {478}, + Volume = {75}, + Year = {1981}} + +@article{LenVerPul-MP-91, + Author = {J. H. van Lenthe and J. Verbeek and P. Pulay}, + Journal = {Mol. Phys.}, + Pages = {1159}, + Volume = {73}, + Year = {1991}} + +@article{LepMalPel-PRA-89, + Author = {M. B. Lepetit and J. P. Malrieu and M. Pelissier}, + Journal = {Phys. Rev. A}, + Pages = {981}, + Volume = {39}, + Year = {1989}} + +@article{LesKra-JCP-66, + Author = {W. A. Lester and M. Krauss}, + Journal = {J. Chem. Phys.}, + Pages = {207}, + Volume = {44}, + Year = {1966}} + +@article{LevBer-IJQC-68, + Author = {B. Levy and G. Berthier}, + Journal = {Int. J. Quantum Chem.}, + Pages = {307}, + Volume = {2}, + Year = {1968}} + +@incollection{Lev-INC-95, + Address = {New York}, + Author = {M. Levy}, + Booktitle = {Density Functional Theory}, + Editor = {E.K.U. Gross and R.M. Dreizler}, + Publisher = {Plenum Press}, + Title = {Coordinate scaling requirements for approximating exchange and correlation}, + Year = {1995}} + +@article{LevMarHan-JCP-96, + Author = {M. Levy and N. H. March and N. C. Handy}, + Journal = {J. Chem. Phys.}, + Pages = {1989}, + Volume = {104}, + Year = {1996}} + +@article{LevNag-PRL-99, + Author = {M. Levy and A. Nagy}, + Journal = {Phys. Rev. Lett.}, + Pages = {4361}, + Volume = {83}, + Year = {1999}} + +@incollection{LevPer-INC-85, + Address = {New York}, + Author = {M. Levy and J. P. Perdew}, + Booktitle = {Density Functional Methods in Physics}, + Editor = {R. M. Dreizler and J. da Providencia}, + Publisher = {Plenum}, + Year = {1985}} + +@article{LevPer-PRA-85, + Author = {M. Levy and J. P. Perdew}, + Journal = {Phys. Rev. A}, + Pages = {2010}, + Volume = {32}, + Year = {1985}} + +@article{LevPer-PRB-93, + Author = {M. Levy and J. P. Perdew}, + Journal = {Phys. Rev. B}, + Pages = {11638}, + Volume = {48}, + Year = {1993}} + +@article{LevPerSah-PRA-84, + Author = {M. Levy and J. P. Perdew and V. Sahni}, + Journal = {Phys. Rev. A}, + Pages = {2745}, + Volume = {30}, + Year = {1984}} + +@article{Lev-PNAS-79, + Author = {M. Levy}, + Journal = {Proc. Natl. Acad. Sci. U.S.A.}, + Pages = {6062}, + Volume = {76}, + Year = {1979}} + +@article{Lev-PRA-82, + Author = {M. Levy}, + Journal = {Phys. Rev. A}, + Pages = {1200}, + Volume = {26}, + Year = {1982}} + +@article{Lev-PRA-91, + Author = {M. Levy}, + Journal = {Phys. Rev. A}, + Pages = {4637}, + Volume = {43}, + Year = {1991}} + +@article{LevSov-PRA-84, + Author = {Z. H. Levine and P. Soven}, + Journal = {Phys. Rev. A}, + Pages = {625}, + Volume = {29}, + Year = {1984}} + +@article{LevYanPar-JCP-85, + Author = {M. Levy and W. Yang and R. G. Parr}, + Journal = {J. Chem. Phys.}, + Pages = {2334}, + Volume = {83}, + Year = {1985}} + +@article{LieCle-JCP-74a, + Author = {George C. Lie and Enrico Clementi}, + Journal = {J. Chem. Phys.}, + Pages = {1275}, + Title = {Study of the electronic structure of molecules. XXI. Correlation energy corrections as a functional of the Hartree-Fock density and its application to the hydrides of the second row atoms}, + Volume = {60}, + Year = {1974}} + +@article{LieCle-JCP-74b, + Author = {George C. Lie and Enrico Clementi}, + Journal = {J. Chem. Phys.}, + Pages = {1288}, + Title = {Study of the electronic structure of molecules. XXII. Correlation energy corrections as a functional of the Hartree-Fock type density and its application to the homonuclear diatomic molecules of the second row atoms}, + Volume = {60}, + Year = {1974}} + +@article{Lie-IJQC-83, + Author = {E. H. Lieb}, + Journal = {Int. J. Quantum Chem.}, + Pages = {243}, + Volume = {{24}}, + Year = {1983}} + +@article{LieOxf-IJQC-81, + Author = {E. H. Lieb and S. Oxford}, + Journal = {Int. J. Quantum. Chem.}, + Pages = {427}, + Volume = {{19}}, + Year = {1981}} + +@misc{lifetimes-note1, + Note = {For $\ell=0$, $R_2(r)$ only diverges as $1/r$ at $r=0$ which is normalizable, i.e. $\int_0^\infty \text{d}r r^2 |R_2(r)|^2 < \infty$. However, even in this case, the expectation value of the Hamiltonian operator is infinite. For $\ell \geq 1$, $R_2(r)$ is not even normalizable.}} + +@misc{lifetimes-note2, + Note = {The general solution of the Schr\"odinger equation given in Eqs.~(\ref{solR})-(\ref{solR2}) is valid for $Z\not=0$. For $Z=0$, $R_1(r)$ in Eq.~(\ref{solR1}) becomes identically zero, and should be replaced by a function which has the same expression as $R_2(r)$ with the substitution $\sqrt{-2E} \to -\sqrt{-2E}$. The continuum scattering states are then obtained as a linear combination of these two functions. However, our analysis of complex-energy states remains unchanged. Thus, our procedure for determining the lifetimes is still valid for $Z=0$.}} + +@misc{lifetimes-note3, + Note = {Because we do not impose spherical symmetry in our calculations, we do not have exactly degenerate s, p, and d states.}} + +@article{LikMorSen-PRB-97, + Author = {C. N. Likos and S. Moroni and G. Senatore}, + Journal = {Phys. Rev. B}, + Pages = {8867}, + Volume = {{55}}, + Year = {1997}} + +@article{LilTavRotSeb-PRL-04, + Author = {O. A. von Lilienfeld and I. Tavernelli and U. Rothlisberger}, + Journal = {Phys. Rev. Lett.}, + Pages = {153004}, + Volume = {93}, + Year = {2004}} + +@article{LimCarLuoMaOlsTruGag-JCTC-14, + Author = {{G. Li Manni, R. K. Carlson, S. Luo, D. Ma, J. Olsen, D. G. Truhlar, and L. Gagliardi}}, + Journal = {J. Chem. Theory Comput.}, + Pages = {3669}, + Volume = {10}, + Year = {2014}} + +@article{LinRyuLiu-JCP-91, + Author = {R. Lindh and U. Ryu and B. Liu}, + Journal = {J. Chem. Phys.}, + Pages = {5889}, + Volume = {{95}}, + Year = {1991}} + +@article{LinSal-AQC-03, + Author = {Ingvar Lindgren and Sten Salomonson}, + Journal = {Adv. Quantum Chem.}, + Pages = {95-117}, + Title = {Differentiability in density-functional theory}, + Volume = {43}, + Year = {2003}} + +@article{LinTsaLiCha-JCP-12, + Author = {Y.-S. Lin and C.-W. Tsai and G.-D. Li and J.-D. Chai}, + Journal = {J. Chem. Phys.}, + Pages = {154109}, + Volume = {136}, + Year = {2012}} + +@article{LinZhaRap-JCP-00, + Author = {Xi Lin and Hongkai Zhang and Andrew M. Rappe}, + Journal = {J. Chem. Phys.}, + Pages = {2650}, + Volume = {112}, + Year = {2000}} + +@article{LiuEggRefKroNea-JCP-17, + Author = {Zhen-Fei Liu and David A. Egger and Sivan Refaely-Abramson and Leeor Kronik and Jeffrey B. Neaton}, + Journal = {J. Chem. Phys.}, + Pages = {092326}, + Volume = {146}, + Year = {2017}} + +@article{LiuParNag-PRA-95, + Author = {S. Liu and R. G. Parr and A. Nagy}, + Journal = {Phys. Rev. A}, + Pages = {2645}, + Volume = {52}, + Year = {1995}} + +@article{LivBae-PCCP-07, + Author = {E. Livshits and R. Baer}, + Journal = {Phys. Chem. Chem. Phys.}, + Pages = {2932}, + Volume = {9}, + Year = {2007}} + +@article{LjuKovFerFoeSan-PRB-15, + Author = {M. P. Ljungberg and P. Koval and F. Ferrari and D. Foerster and D. S\'anchez-Portal}, + Journal = {Phys. Rev. B}, + Pages = {075422}, + Volume = {92}, + Year = {2015}} + +@article{LocHea-JCP-07, + Author = {Rohini C. Lochan and Martin Head-Gordon}, + Journal = {J. Chem. Phys.}, + Pages = {164101}, + Volume = {126}, + Year = {2007}} + +@article{Lon-DFS-65, + Author = {H. C. Longuet-Higgins}, + Journal = {Discuss. Faraday Soc.}, + Pages = {7}, + Volume = {40}, + Year = {1965}} + +@article{Lon-ZP-30, + Author = {F. London}, + Journal = {Z. Physik}, + Pages = {245}, + Volume = {63}, + Year = {1930}} + +@article{LooGil-WIRES-16, + Author = {Pierre-Francois Loos and Peter M. W. Gill}, + Journal = {WIREs Comput. Mol. Sci.}, + Note = {doi: 10.1002/wcms.1257}, + Pages = {410}, + Title = {The uniform electron gas}, + Volume = {6}, + Year = {2016}} + +@article{LooPraSceGinTou-ARX-19, + Author = {P.-F. Loos and B. Pradines and A. Scemama and E. Giner and J. Toulouse}, + Journal = {arXiv:1910.12238}, + Title = {A density-based basis-set incompleteness correction for GW methods}} + +@article{LooPraSceTouGin-JPCL-19, + Author = {P.-F. Loos and B. Pradines and A. Scemama and J. Toulouse and E. Giner}, + Doi = {10.1021/acs.jpclett.9b01176}, + Journal = {J. Phys. Chem. Lett.}, + Pages = {2931}, + Title = {A density-based basis-set correction for wave function theory}, + Volume = {10}, + Year = {2019}, + Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpclett.9b01176}} + +@article{LopGov-JCTC-11, + Author = {K. Lopata and N. Govind}, + Journal = {J. Chem. Theory Comput.}, + Pages = {1344}, + Title = {Modeling Fast Electron Dynamics with Real-Time Time-Dependent Density Functional Theory: Application to Small Molecules and Chromophores}, + Volume = {7}, + Year = {2011}} + +@article{LopGov-JCTC-13, + Author = {K. Lopata and N. Govind}, + Journal = {J. Chem. Theory Comput.}, + Pages = {4939}, + Volume = {9}, + Year = {2013}} + +@article{LopMaDruTowNee-PRE-06, + Author = {{P. L\'opez Rios, A. Ma, N. D. Drummond, M. D. Towler, and R. J. Needs}}, + Journal = {Phys. Rev. E}, + Pages = {066701}, + Title = {{Inhomogeneous backflow transformations in quantum Monte Carlo calculations}}, + Volume = {74}, + Year = {2006}} + +@article{LopRamGarFer-JCP-87, + Author = {{R. L\'opez, G. Ram\'irez, J. M. Garc\'ia de la Vega and J. Fern\'andez Rico}}, + Journal = {J. chim. phys.}, + Pages = {695}, + Volume = {{84}}, + Year = {1987}} + +@article{LosFdeAguMar-JPC-07, + Author = {{A. M. Losa, I. Fdez.-Galv\'an, M. A. Aguilar, and M. E. Mart\'in}}, + Journal = {J. Phys. Chem. B}, + Pages = {9864}, + Volume = {111}, + Year = {2007}} + +@article{LotBar-JCP-11, + Author = {V. Lotrich and R. J. Bartlett}, + Journal = {J. Chem. Phys.}, + Pages = {184108}, + Volume = {134}, + Year = {2011}} + +@article{Low-AP-56, + Author = {P.-O. L\"owdin}, + Journal = {Adv. 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Lett.}, + Pages = {206411}, + Volume = {102}, + Year = {2009}} + +@article{LuNguGal-JCP-10, + Author = {Deyu Lu and Huy-Viet Nguyen and Giulia Galli}, + Journal = {J. Chem. Phys.}, + Pages = {154110}, + Volume = {133}, + Year = {2010}} + +@article{LupHea-JCP-13, + Author = {E. Luppi and M. Head-Gordon}, + Journal = {J. Chem. Phys.}, + Pages = {164121}, + Title = {{The role of Rydberg and continuum levels in computing high harmonic generation spectra of the hydrogen atom using time-dependent configuration interaction}}, + Volume = {139}, + Year = {2013}} + +@article{Lut-ZPC-23, + Author = {A. Luthy}, + Journal = {Z. Phys. Chem., Stoechiom. Verwandtschaftsl.}, + Pages = {284}, + Volume = {107}, + Year = {1923}} + +@article{LynTru-JPCA-03, + Author = {B. J. Lynch and D. G. Truhlar}, + Journal = {J. Phys. Chem. A}, + Pages = {8996}, + Title = {Small Representative Benchmarks for Thermochemical Calculations}, + Volume = {107}, + Year = {2003}} + +@misc{LynZhaTru-JJJ-XX, + Note = {The geometries are available in the Minnesota Databases for Chemistry and Solid-State Physics at http://comp.chem.umn.edu/db/.}} + +@article{MaBru-PR-68, + Author = {S.-K. Ma and K. A. Brueckner}, + Journal = {Phys. Rev.}, + Pages = {165}, + Volume = {{165}}, + Year = {1968}} + +@article{MacVos-JPC-79, + Author = {A. H. MacDonald and S. H. Vosko}, + Journal = {J. Phys. C}, + Pages = {2977}, + Volume = {12}, + Year = {1979}} + +@article{MaDruTowNee-PRE-05, + Author = {A. Ma and N. D. Drummond and M. D. Towler and R. J. Needs}, + Journal = {Phys. Rev. E}, + Pages = {066704}, + Volume = {71}, + Year = {2005}} + +@book{Mah-BOOK-09, + Address = {Princeton}, + Author = {Gerald D. Mahan}, + Publisher = {Princeton University Press}, + Title = {Quantum Mechanics in a Nutshell}, + Year = {2009}} + +@article{MaiZhaCavBur-JCP-04, + Author = {N. T. Maitra and F. Zhang and R. J. Cave and K. Burke}, + Journal = {J. Chem. Phys.}, + Pages = {5932}, + Volume = {120}, + Year = {2004}} + +@article{MalMcd-CPL-98, + Author = {N. O. J. Malcolm and J. J. W. McDouall}, + Journal = {Chem. Phys. Lett.}, + Pages = {121}, + Volume = {282}, + Year = {1998}} + +@article{MalMcd-JPC-96, + Author = {N. O. J. Malcolm and J. J. W. McDouall}, + Journal = {J. Phys. Chem.}, + Pages = {10131}, + Volume = {100}, + Year = {1996}} + +@article{MalPenYanBalHol-TCA-14, + Author = {A. Malek and D. Peng and W. Yang and R. Balawender and A. Holas}, + Journal = {Theor. Chem. Acc.}, + Pages = {1559}, + Volume = {133}, + Year = {2014}} + +@article{ManLuc-JCP-01, + Author = {S. Manten and A. L\"uchow}, + Journal = {J. Chem. Phys.}, + Pages = {5362}, + Volume = {115}, + Year = {2001}} + +@article{MarAzaCasSor-JCP-09, + Author = {M. Marchi and S. Azadi and M. Casula and S. Sorella}, + Journal = {J. Chem. Phys.}, + Pages = {154116}, + Title = {Correlated geminal wave function for molecules: An efficient resonating valence bond approach}, + Volume = {131}, + Year = {2009}} + +@article{MarCasMalMulBot-JCP-07, + Author = {M. A. L. Marques and A. Castro and G. Malloci and G. Mulas and S. Botti}, + Journal = {J. Chem. Phys.}, + Pages = {014107}, + Volume = {127}, + Year = {2007}} + +@article{Mar-CPL-96, + Author = {J. M. L. Martin}, + Journal = {Chem. Phys. Lett.}, + Pages = {669}, + Volume = {259}, + Year = {1996}} + +@article{MarGarRub-PRL-06, + Abstract = {We present a first-principles description of anisotropic materials + characterized by having both weak (dispersionlike) and strong covalent + bonds, based on the adiabatic-connection fluctuation-dissipation + theorem with density functional theory. For hexagonal boron nitride + the in-plane and out-of-plane bonding as well as vibrational dynamics + are well described both at equilibrium and when the layers are pulled + apart. Bonding in covalent and ionic solids is also described. The + formalism allows us to ping down the deficiencies of common exchange-correlation + functionals and provides insight toward the inclusion of dispersion + interactions into the correlation functional.}, + Author = {Marini, A. and Garc\'ia-Gonz\'alez, P. and Rubio, A.}, + Journal = {Phys. Rev. Lett.}, + Pages = {136404}, + Title = {First-Principles Description of Correlation Effects in Layered Materials}, + Volume = {96}, + Year = {2006}} + +@article{MarHea-JCP-16, + Author = {Narbe Mardirossian and Martin Head-Gordon}, + Journal = {J. Chem. Phys.}, + Pages = {214110}, + Volume = {144}, + Year = {2016}} + +@article{MarHea-JCP-18, + Author = {Narbe Mardirossian and Martin Head-Gordon}, + Journal = {J. Chem. Phys.}, + Pages = {241736}, + Volume = {148}, + Year = {2018}} + +@article{MarHea-PCCP-14, + Author = {Narbe Mardirossian and Martin Head-Gordon}, + Journal = {Phys. Chem. Chem. Phys.}, + Pages = {9904}, + Volume = {16}, + Year = {2014}} + +@article{MarHeiKle-WIRES-18, + Author = {Christel M. Marian and Adrian Heil and Martin Kleinschmidt}, + Journal = {WIREs Comput. Mol. Sci.}, + Note = {doi: 10.1002/wcms.1394}, + Pages = {e1394}, + Title = {The DFT/MRCI method}, + Year = {2018}} + +@article{MarLosFdeAgu-JCP-04, + Author = {{M. E. Mart\'in, A. M. Losa, I. Fdez.-Galv\'an, and M. A. Aguilar}}, + Journal = {J. Chem. Phys.}, + Pages = {3710}, + Volume = {121}, + Year = {2004}} + +@article{MarMea-JCP-78, + Author = {D. J. Margoliash and W. J. Meath}, + Journal = {J. Chem. Phys.}, + Pages = {1426}, + Volume = {68}, + Year = {1978}} + +@article{MaRohMol-JCTC-10, + Author = {Y. Ma and M. Rohlfing and C. Molteni}, + Journal = {J. Chem. Theory Comput.}, + Pages = {257}, + Volume = {6}, + Year = {2010}} + +@article{MaRohMol-PRB-09, + Author = {Y. Ma and M. Rohlfing and C. Molteni}, + Journal = {Phys. Rev. B}, + Pages = {241405}, + Volume = {80}, + Year = {2009}} + +@article{Mar-PR-58, + Author = {N. H. March}, + Journal = {Phys. Rev.}, + Pages = {604}, + Volume = {{110}}, + Year = {1958}} + +@book{MarReiCep-BOOK-16, + Author = {Richard M. Martin and Lucia Reining and David M. Ceperley}, + Publisher = {Cambridge University Press}, + Title = {Interacting Electrons: Theory and Computational Approaches}, + Year = {2016}} + +@article{MarWer-JPCA-09, + Author = {O. Marchetti and H.-J. Werner}, + Journal = {J. Phys. Chem. A}, + Pages = {11580}, + Volume = {113}, + Year = {2009}} + +@article{Math10-PROG-15, + Note = {{Wolfram Research, Inc., Mathematica, Version 10.3, Champaign, IL (2015)}}} + +@article{Math9-PROG-12, + Note = {{Wolfram Research, Inc., Mathematica, Version 9.0, Champaign, IL (2012)}}} + +@article{MatKogRomDeh-PRA-98, + Author = {H. Matsuyama and T. Koga and E. Romera and J. S. Dehesa}, + Journal = {Phys. Rev. A}, + Pages = {1759}, + Volume = {57}, + Year = {1998}} + +@article{MatKoh-JCP-01, + Author = {Ann E. Mattsson and Walter Kohn}, + Journal = {The Journal of Chemical Physics}, + Keywords = {surface energy; density functional theory; errors; surface potential; jellium}, + Number = {8}, + Pages = {3441-3443}, + Publisher = {AIP}, + Title = {An energy functional for surfaces}, + Url = {http://link.aip.org/link/?JCP/115/3441/1}, + Volume = {115}, + Year = {2001}, + Bdsk-Url-1 = {http://link.aip.org/link/?JCP/115/3441/1}} + +@article{Mat-S-02, + Author = {A. E. Mattson}, + Journal = {Science}, + Pages = {759}, + Volume = {298}, + Year = {2002}} + +@article{MccStrWis-PRA-91, + Author = {C. W. McCurdy and C. K. Stroud and M. K. Wisinski}, + Journal = {Phys. Rev. A}, + Pages = {5980}, + Volume = {43}, + Year = {1991}} + +@article{Mcd-MP-03, + Author = {J. J. W. McDouall}, + Journal = {Mol. Phys.}, + Pages = {361}, + Title = {Combining two-body density functionals with multiconfigurational wavefunctions: diatomic molecules}, + Volume = {101}, + Year = {2003}} + +@article{MclBal-RMP-64, + Author = {McLachlan, A. D. and Ball, M. A.}, + Doi = {10.1103/RevModPhys.36.844}, + Journal = {Rev. Mod. Phys.}, + Pages = {844-855}, + Title = {Time-dependent Hartree-Fock Theory for Molecules}, + Volume = {36}, + Year = {1964}, + Bdsk-Url-1 = {https://doi.org/10.1103/RevModPhys.36.844}} + +@book{McW-BOOK-92, + Address = {London}, + Author = {R. McWeeny}, + Publisher = {Academic Press}, + Title = {Methods of Molecular Quantum Mechanics. Second Edition}, + Year = {1992}} + +@article{Mcw-CCA-84, + Author = {R. McWeeny}, + Journal = {Croat. Chem. Acta}, + Pages = {865}, + Volume = {57}, + Year = {1984}} + +@article{MehCasGor-PCCP-18, + Author = {N. Mehta and M. Casanova-P\'aez and Lars Goerigk}, + Journal = {Phys. Chem. Chem. Phys.}, + Pages = {23175}, + Volume = {20}, + Year = {2018}} + +@article{MelSan-PRE-05, + Author = {Melko, R. G. and Sandvik, A. W.}, + Doi = {10.1103/PhysRevE.72.026702}, + Issue = {2}, + Journal = {Phys. Rev. E}, + Numpages = {22}, + Pages = {026702}, + Publisher = {American Physical Society}, + Title = {{Stochastic series expansion algorithm for the $S=1∕2$ $XY$ model with four-site ring exchange}}, + Url = {http://link.aps.org/doi/10.1103/PhysRevE.72.026702}, + Volume = {72}, + Year = {2005}, + Bdsk-Url-1 = {http://link.aps.org/doi/10.1103/PhysRevE.72.026702}, + Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevE.72.026702}} + +@article{MenAnd-JCP-81, + Author = {F. Mentch and J. B. 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Misquitta and B. Jeziorski and K. Szalewicz}, + Journal = {Phys. Rev. Lett.}, + Pages = {033201}, + Volume = {91}, + Year = {2003}} + +@article{Mit-PRA-81, + Author = {M. H. Mittleman}, + Journal = {Phys. Rev. A}, + Pages = {1167}, + Volume = {24}, + Year = {1981}} + +@article{MiyAryKotSchUsuTer-PRB-02, + Author = {Miyake, T. and Aryasetiawan, F. and Kotani, T. and Schilfgaarde, M. v. and Usuda, M. and Terakura, K.}, + Journal = {Phys. Rev. B.}, + Pages = {245103}, + Title = {Total energy of solids: An exchange and random-phase approximation correlation study}, + Volume = {66}, + Year = {2002}} + +@book{Moi-BOOK-11, + Address = {Cambridge}, + Author = {Nimrod Moiseyev}, + Publisher = {Cambridge University Press}, + Title = {Non-Hermitian Quantum Mechanics}, + Year = {2011}} + +@article{MolPle-PR-34, + Author = {C. M{\o}ller and M. S. Plesset}, + Journal = {Phys. Rev.}, + Pages = {618}, + Volume = {46}, + Year = {1934}} + +@misc{Molpro-PROG-02, + Address = {Birmingham, UK}, + Author = {R. D. Amos and A. Bernhardsson and A. Berning and P. Celani and D. L. Cooper and M. J. O. Deegan and A. J. Dobbyn and F. Eckert and C. Hampel and G. Hetzer and P. J. Knowles and T. Korona and R. Lindh and A. W. Lloyd and S. J. {McNicholas} and F. R. Manby and W. Meyer and M. E. Mura and A. Nicklass and P. Palmieri and R. Pitzer and G. Rauhut and M. {Sch\"{u}tz} and U. Schumann and H. Stoll and A. J. Stone and R. Tarroni and T. Thorsteinsson and H.-J. Werner}, + Title = {MOLPRO, a package of ab initio programs designed by {H.-J.~Werner} and {P.~J. Knowles}, version 2002.2}, + Year = {2002}} + +@misc{Molpro-PROG-10, + Address = {Cardiff, UK}, + Author = {H.-J. Werner and P. J. Knowles and G. Knizia and F. R. Manby and M. {Sch\"{u}tz} and P. Celani and T. Korona and R. Lindh and A. Mitrushenkov and G. Rauhut and K. R. Shamasundar and T. B. Adler and R. D. 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Manby and M. {Sch\"{u}tz} and others}, + Note = {see www.molpro.net}, + Title = {MOLPRO, version 2008.2, a package of ab initio programs}, + Year = {2008}} + +@misc{Molproshort-PROG-10, + Author = {{H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby, M. {Sch\"{u}tz}, and others}}, + Note = {Cardiff, UK, 2010, see \url{http://www.molpro.net}}, + Title = {MOLPRO, version 2010.1, a package of ab initio programs}} + +@misc{Molproshort-PROG-12, + Author = {{H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby, M. {Sch\"{u}tz}, and others}}, + Note = {Cardiff, UK, 2012, see \url{http://www.molpro.net}}, + Title = {MOLPRO, version 2012.1, a package of ab initio programs}} + +@misc{Molproshort-PROG-15, + Author = {{H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby, M. {Sch\"{u}tz}, and others}}, + Note = {{Cardiff, UK, 2015, see \url{http://www.molpro.net}}}, + Title = {{MOLPRO, version 2015.1, a package of ab initio programs}}} + +@misc{Molproshort-PROG-19, + Author = {{H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby, M. {Sch\"{u}tz}, and others}}, + Note = {{Cardiff, UK, 2019, see \url{http://www.molpro.net}}}, + Title = {{MOLPRO, version 2019.1, a package of ab initio programs}}} + +@misc{Molproweb-PROG-XX, + Note = {\url{http://www.molpro.net}}, + Title = {MOLPRO, a package of ab initio programs}} + +@article{MomShi-JCP-87, + Author = {T. Momose and T. Shida}, + Journal = {J. Chem. Phys.}, + Pages = {2832}, + Volume = {87}, + Year = {1987}} + +@article{MorCepSen-PRL-95, + Author = {S. Moroni and D. M. Ceperley and G. Senatore}, + Journal = {Phys. Rev. Lett.}, + Pages = {689}, + Volume = {75}, + Year = {1995}} + +@article{MorCohYan-JCP-06a, + Author = {Paula Mori-S\'anchez and Aron J. Cohen and Weitao Yang}, + Journal = {J. Chem. Phys.}, + Pages = {091102}, + Volume = {124}, + Year = {2006}} + +@article{MorCohYan-JCP-06, + Author = {P. Mori-S{\'a}nchez and A. J. Cohen and W. Yang}, + Journal = {J. Chem. Phys.}, + Pages = {201102}, + Volume = {125}, + Year = {2006}} + +@article{MorCohYan-PRA-12, + Author = {P. Mori-S{\'a}nchez and A. J. Cohen and W. Yang}, + Journal = {Phys. Rev. A}, + Pages = {042507}, + Volume = {85}, + Year = {2012}} + +@article{MorCohYan-PRL-08, + Author = {Paula Mori-S\'anchez and Aron J. Cohen and Weitao Yang}, + Journal = {Phys. Rev. Lett.}, + Pages = {146401}, + Volume = {100}, + Year = {2008}} + +@article{MorCohYan-PRL-09, + Author = {Mori-S\'anchez, Paula and Cohen, Aron and Yang, Weitao}, + Journal = {Phys. Rev. Lett.}, + Pages = {066403}, + Title = {Discontinuous Nature of the Exchange-Correlation Functional in Strongly Correlated Systems}, + Volume = {102}, + Year = {2009}} + +@article{MorMcmClaKimScu-JCTC-12, + Author = {M. A. Morales and J. McMinis and B. K. Clark and J. Kim and G. E. Scuseria}, + Journal = {J. Chem. Theory Comput.}, + Pages = {2181}, + Volume = {8}, + Year = {2012}} + +@article{Mor-PR-29, + Author = {P. M. Morse}, + Journal = {Phys. Rev.}, + Pages = {57}, + Volume = {34}, + Year = {1929}} + +@article{MorSen-PRB-91, + Author = {S. Moroni and G. Senatore}, + Journal = {Phys. Rev. B}, + Pages = {9864}, + Volume = {{44}}, + Year = {1991}} + +@article{MorWuYan-JCP-05, + Author = {P. Mori-S{\'a}nchez and Q. Wu and W. Yang}, + Journal = {J. Chem. Phys.}, + Pages = {062204}, + Volume = {123}, + Year = {2005}} + +@article{MosDep-JCTC-19, + Author = {{M. Mostafanejad and A. E. DePrince III}}, + Journal = {J. Chem. Theory Comput.}, + Pages = {290}, + Volume = {15}, + Year = {2019}} + +@article{MosJezSza-IJQC-93, + Author = {R. Moszynski and B. Jeziorski and K. Szalewicz}, + Journal = {Int. J. Quantum. Chem.}, + Pages = {409}, + Volume = {45}, + Year = {1993}} + +@article{MosSan-IJQC-91, + Author = {F. Moscardo and E. San-Fabian}, + Journal = {Int. J. Quantum. Chem.}, + Pages = {23}, + Volume = {{40}}, + Year = {1991}} + +@article{MosSan-PRA-91, + Author = {F. Moscard\'o and E. San-Fabi\'an}, + Journal = {Phys. Rev. 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This study is completed by illustrating the well-known isomorphism between the exchange hole and the square of the dominant localized orbital lying in the space region of the reference point of the exchange hole function. The implications of these relationships for the interpretation of common chemical concepts, such as delocalization, are discussed. }, + Author = {Bastien Mussard and J\'anos G. \'Angy\'an}, + Doi = {http://dx.doi.org/10.1016/j.comptc.2014.10.039}, + Issn = {2210-271X}, + Journal = {Computational and Theoretical Chemistry}, + Keywords = {Topological analysis}, + Note = {Special Issue: Understanding structure and reactivity from topology and beyond}, + Number = {0}, + Pages = {44 - 52}, + Title = {Relationships between charge density response functions, exchange holes and localized orbitals}, + Url = {http://www.sciencedirect.com/science/article/pii/S2210271X14004952}, + Volume = {1053}, + Year = {2015}, + Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S2210271X14004952}, + Bdsk-Url-2 = {http://dx.doi.org/10.1016/j.comptc.2014.10.039}} + +@article{MusAng-TCA-15, + Author = {B. Mussard and J. G. \'Angy\'an}, + Doi = {http://dx.doi.org/10.1007/s00214-015-1751-2}, + Journal = {Theor. Chem. Acc.}, + Pages = {148}, + Volume = {134}, + Year = {2015}, + Bdsk-Url-1 = {http://dx.doi.org/10.1007/s00214-015-1751-2}} + +@incollection{MusCocAssOttUmrTou-AQC-18, + Author = {B. Mussard and E. Coccia and R. Assaraf and M. Otten and C. J. Umrigar and J. Toulouse}, + Booktitle = {Novel Electronic Structure Theory: General Innovations and Strongly Correlated Systems}, + Doi = {10.1016/bs.aiq.2017.05.005}, + Editor = {P. E. Hoggan}, + Pages = {255-270}, + Publisher = {Academic Press}, + Series = {Advances in Quantum Chemistry Vol. 76}, + Title = {Time-dependent linear-response variational Monte Carlo}, + Year = {2018}, + Bdsk-Url-1 = {https://doi.org/10.1016/bs.aiq.2017.05.005}} + +@article{MusReiAngTou-JCP-15, + Author = {B. Mussard and P. Reinhardt and J. G. \'Angy\'an and J. Toulouse}, + Journal = {J. Chem. Phys.}, + Note = {Erratum: J. Chem. Phys. {\bf 142}, 219901 (2015)}, + Pages = {154123}, + Title = {Spin-unrestricted random-phase approximation with range separation: Benchmark on atomization energies and reaction barrier heights}, + Volume = {142}, + Year = {2015}} + +@misc{MusReiAngTou-JJJ-XX-note, + Note = {See the supplementary material for the detailed results of the calculations on the AE49 and DBH24/08 datasets.}} + +@article{MusRocJanAng-JCTC-16, + Author = {B. Mussard and D. Rocca and G. Jansen and J. G. \'Angy\'an}, + Doi = {10.1021/acs.jctc.5b01129}, + Journal = {J. Chem. Theory Comput.}, + Pages = {2191-2202}, + Title = {Dielectric Matrix Formulation of Correlation Energies in the Random Phase Approximation: Inclusion of Exchange Effects}, + Volume = {12}, + Year = {2016}, + Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.5b01129}} + +@article{MusRocJanAng-JJJ-XX, + Author = {B. Mussard and D. Rocca and G. Jansen and J. G. \'Angy\'an}, + Note = {unpublished}} + +@article{MusSzaAng-JCTC-14, + Author = {B. 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Phys.}, + Pages = {184104}, + Volume = {125}, + Year = {2006}} + +@article{NagLev-PRA-01, + Author = {A. Nagy and M. Levy}, + Journal = {Phys. Rev. A}, + Pages = {052502}, + Volume = {63}, + Year = {2001}} + +@article{nagy:134107, + Author = {A. Nagy and S. Liu and L. Bartolloti}, + Eid = {134107}, + Journal = {The Journal of Chemical Physics}, + Keywords = {density functional theory; potential energy functions}, + Number = {13}, + Numpages = {5}, + Pages = {134107}, + Publisher = {AIP}, + Title = {Generalized density functional theory for degenerate states}, + Url = {http://link.aip.org/link/?JCP/122/134107/1}, + Volume = {122}, + Year = {2005}, + Bdsk-Url-1 = {http://link.aip.org/link/?JCP/122/134107/1}} + +@article{NalPar-JCP-82, + Author = {Roman F. Nalewajski and Robert G. Parr}, + Journal = {J. Chem. Phys.}, + Note = {[In Eq.~(70) of this paper the minimization over potentials should actually be a maximization.]}, + Pages = {399-407}, + Publisher = {AIP}, + Title = {Legendre transforms and Maxwell relations in density functional theory}, + Volume = {77}, + Year = {1982}} + +@article{NalPar-JCP-fr-82, + Author = {Roman F. Nalewajski and Robert G. Parr}, + Journal = {J. Chem. Phys.}, + Note = {[Dans l'\'equation~(70) de cet article la minimisation sur le potentiel est en r\'ealit\'e une maximisation.]}, + Pages = {399-407}, + Publisher = {AIP}, + Title = {Legendre transforms and Maxwell relations in density functional theory}, + Volume = {77}, + Year = {1982}} + +@article{NeeSchKosSchGri-JCTC-09, + Author = {F. Neese and T. Schwabe and S. Kossmann and B. Schirmer and S. Grimme}, + Journal = {J. Chem. Theory Comput.}, + Pages = {3060}, + Volume = {5}, + Year = {2009}} + +@article{NemTowNee-JCP-10, + Author = {N. Nemec and M. D. Towler and R. J. Needs}, + Journal = {J. Chem. 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B.}, + Pages = {3052}, + Volume = {{10}}, + Year = {1974}} + +@article{NiqFucGon-JCP-03, + Author = {Y. M. Niquet and M. Fuchs and X. Gonze}, + Journal = {J. Chem. Phys.}, + Pages = {9504}, + Volume = {118}, + Year = {2003}} + +@article{NiqFucGon-PRL-03, + Author = {Y. M. Niquet and M. Fuchs and X. Gonze}, + Journal = {Phys. Rev. Lett.}, + Pages = {219301}, + Volume = {90}, + Year = {2003}} + +@article{NiqGon-PRB-04, + Author = {Y. M. Niquet and X. Gonze}, + Journal = {Phys. Rev. B}, + Pages = {245115}, + Volume = {70}, + Year = {2004}} + +@misc{Nis-BOOK-05, + Title = {NIST Chemistry WebBook, NIST Standard Reference Database Number 69, June 2005 Release}, + Url = {http://webbook.nist.gov/chemistry York}, + Bdsk-Url-1 = {http://webbook.nist.gov/chemistry%20York}} + +@book{NIST-BOOK-05, + Address = {Gaithersburg}, + Editor = {P. J. Linstrom and W. G. Mallard}, + Publisher = {NIST}, + Title = {NIST Chemistry WebBook, NIST Standard Reference Database Number 69}, + Year = {June 2005}} + +@misc{NIST-BOOK-15, + Author = {{NIST Computational Chemistry Comparison and Benchmark Database}}, + Editor = {{Russell D. Johnson III}}, + Title = {{NIST Standard Reference Database Number 101, Release 17b}}, + Url = {http://cccbdb.nist.gov/}, + Year = {September 2015}, + Bdsk-Url-1 = {http://cccbdb.nist.gov/}} + +@article{NogHiyAkiKog-JCP-14, + Author = {Y. Noguchi and M. Hiyama and H. Akiyama and N. Koga}, + Journal = {J. Chem. Phys.}, + Pages = {044309}, + Volume = {141}, + Year = {2014}} + +@article{NooShaMuk-MP-05, + Author = {M. Nooijen and K. R. Shamasundary and D. Mukherjee}, + Journal = {Mol. Phys.}, + Pages = {2277}, + Volume = {103}, + Year = {2005}} + +@article{NorJieSer-JCP-03, + Author = {P. Norman and A. Jiemchooroj and Bo E. Sernelius}, + Journal = {J. Chem. Phys.}, + Pages = {9167}, + Volume = {118}, + Year = {2003}} + +@article{NozPin-PR-58, + Author = {P. Nozi\`eres and D. Pines}, + Journal = {Phys. Rev.}, + Pages = {442}, + Volume = {111}, + Year = {1958}} + +@article{Odd-AQC-78, + Author = {Jens Oddershede}, + Journal = {Adv. Quantum Chem.}, + Pages = {275}, + Title = {Polarization Propagator Calculations}, + Volume = {11}, + Year = {1978}} + +@article{OddJor-JCP-77, + Author = {J. Oddershede and P. J{\o}rgensen}, + Journal = {J. Chem. Phys.}, + Pages = {1541}, + Volume = {66}, + Year = {1977}} + +@article{OlePalOniDel-PRB-99, + Author = {Valerio Olevano and Maurizia Palummo and Giovanni Onida and Rodolfo Del Sole}, + Journal = {Phys. Rev. B}, + Pages = {14224}, + Volume = {{60}}, + Year = {1999}} + +@article{OleTouSch-JCP-19, + Author = {Valerio Olevano and Julien Toulouse and Peter Schuck}, + Doi = {10.1063/1.5080330}, + Journal = {J. Chem. Phys.}, + Pages = {084112}, + Title = {A formally exact one-frequency-only Bethe-Salpeter-like equation. Similarities and differences between GW+BSE and self-consistent RPA}, + Volume = {150}, + Year = {2019}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.5080330}} + +@article{OliBotMar-PCCP-11, + Author = {M. J. T. Oliveira and S. Botti and M. A. L. Marques}, + Journal = {Phys. Chem. Chem. Phys.}, + Pages = {15055}, + Volume = {13}, + Year = {2011}} + +@article{OliPer-PRA-79, + Author = {G. L. Oliver and J. P. Perdew}, + Journal = {Phys. Rev. A}, + Pages = {397}, + Volume = {20}, + Year = {1979}} + +@article{OniReiRub-RMP-02, + Author = {G. Onida and L. Reining and A. Rubio}, + Journal = {Rev. Mod. Phys.}, + Pages = {601}, + Title = {{Electronic excitations: density-functional versus many-body Green's-function approaches}}, + Volume = {74}, + Year = {2002}} + +@article{OrtBal-PRB-94, + Author = {G. Ortiz and P. Ballone}, + Journal = {Phys. Rev. B}, + Pages = {1391}, + Volume = {{50}}, + Year = {1994}} + +@article{OrtBal-PRB-err-97, + Author = {G. Ortiz and P. Ballone}, + Journal = {Phys. Rev. B}, + Pages = {9970}, + Volume = {{56}}, + Year = {1997}} + +@misc{Ort-PROG-XX, + Author = {P. Reinhardt}, + Note = {Different programs to perform ab-initio calculations within highly localized orbitals (Toulouse-Dresden-Paris, 1998--), unpublished}} + +@article{OsiGisSniBae-JCP-97, + Author = {{V. P. Osinga, S. J. A. van Gisbergen, J. G. Snijders, and E. J. Baerends}}, + Journal = {J. Chem. Phys.}, + Pages = {5091}, + Volume = {106}, + Year = {1997}} + +@article{OstKar-CPL-71, + Author = {N. Ostlund and M. Karplus}, + Journal = {Chem. Phys. Lett.}, + Pages = {450}, + Volume = {11}, + Year = {1971}} + +@misc{OthMonMar-ARX-16, + Author = {{A. A. Othman, M. de Montigny, and F. Marsiglio}}, + Note = {\url{http://arxiv.org/abs/1612.06706}}, + Title = {The Coulomb potential in quantum mechanics revisited}} + +@article{Ove-CJP-95, + Author = {A. W. Overhauser}, + Journal = {Can. J. Phys.}, + Pages = {683}, + Volume = {{73}}, + Year = {1995}} + +@article{PacBye-JCP-66, + Author = {{R. T. Pack and W. Byers-Brown}}, + Journal = {J. Chem. Phys.}, + Pages = {556}, + Title = {{Cusp conditions for molecular wavefunctions}}, + Volume = {45}, + Year = {1966}} + +@article{PaiJanHenScuGruKre-JCP-10, + Author = {J. Paier and B. G. Janesko and T. M. Henderson and G. E. Scuseria and A. Gr\"uneis and G. Kresse}, + Journal = {J. Chem. Phys.}, + Pages = {094103}, + Volume = {132}, + Year = {2010}} + +@article{PaiMarHumKreGerAng-JCP-06, + Author = {J. Paier and M. Marsman and K. Hummer and G. Kresse and I. C. Gerber and J. G. \'Angy\'an}, + Journal = {J. Chem. Phys.}, + Pages = {154709}, + Volume = {124}, + Year = {2006}} + +@article{PaiMarKre-JCP-07, + Author = {Joachim Paier and Martijn Marsman and Georg Kresse}, + Doi = {10.1063/1.2747249}, + Eid = {024103}, + Journal = {The Journal of Chemical Physics}, + Keywords = {HF calculations; density functional theory; heat of formation; lattice constants; electron gas; electron correlations; elastic moduli}, + Number = {2}, + Numpages = {10}, + Pages = {024103}, + Publisher = {AIP}, + Title = {Why does the B3LYP hybrid functional fail for metals?}, + Url = {http://link.aip.org/link/?JCP/127/024103/1}, + Volume = {127}, + Year = {2007}, + Bdsk-Url-1 = {http://link.aip.org/link/?JCP/127/024103/1}, + Bdsk-Url-2 = {https://doi.org/10.1063/1.2747249}} + +@article{PaiRenRinScuGruKreSch-NJP-12, + Author = {Joachim Paier and Xinguo Ren and Patrick Rinke and Gustavo E. Scuseria and Andreas Gr\"uneis and Georg Kresse and Matthias Scheffler}, + Journal = {New J. Phys.}, + Pages = {043002}, + Volume = {14}, + Year = {2012}} + +@article{PalCiz-JCP-70, + Author = {J. Paldus and J. {\v C}i{\v z}ek}, + Journal = {J. Chem. Phys.}, + Pages = {2919}, + Volume = {52}, + Year = {1970}} + +@article{PalCis-JCP-71, + Author = {J. Paldus and J. {\v C}i{\v z}ek}, + Journal = {J. Chem. Phys.}, + Pages = {2293}, + Volume = {54}, + Year = {1971}} + +@article{PalCiz-PRA-70, + Author = {J. Paldus and J. {\v C}i{\v z}ek}, + Journal = {Phys. Rev. A}, + Pages = {2268}, + Volume = {2}, + Year = {1970}} + +@article{Pal-JCP-77, + Author = {J. Paldus}, + Journal = {J. Chem. Phys.}, + Pages = {303}, + Volume = {67}, + Year = {1977}} + +@article{PalOniDelCorRei-PRB-99, + Author = {Maurizia Palummo and Giovanni Onida and Rodolfo Del Sole and Massimiliano Corradini and Lucia Reining}, + Journal = {Phys. Rev. B}, + Pages = {11329}, + Volume = {{60}}, + Year = {1999}} + +@article{PalPavHubSch-EPJB-11, + Author = {G. Pal and Y. Pavlyukh and W. H\"ubner and H. C. Schneider}, + Journal = {Eur. Phys. J. B.}, + Pages = {327}, + Volume = {79}, + Year = {2011}} + +@article{Pan-CPL-93, + Author = {I. Panas}, + Journal = {Chem. Phys. Lett.}, + Pages = {255}, + Volume = {201}, + Year = {1993}} + +@article{pan:5642, + Author = {Xiao-Yin Pan and Viraht Sahni}, + Issue = {12}, + Journal = {The Journal of Chemical Physics}, + Keywords = {ground states; molecular configurations; density functional theory; Schrodinger equation; ionisation potential; wave functions; eigenvalues and eigenfunctions; electron correlations; hydrogen}, + Page = {5642-5649}, + Publisher = {AIP}, + Title = {Quantal density functional theory of the hydrogen molecule}, + Url = {http://link.aip.org/link/?JCP/120/5642/1}, + Volume = {120}, + Year = {2004}, + Bdsk-Url-1 = {http://link.aip.org/link/?JCP/120/5642/1}} + +@article{PaqTou-JCP-18, + Author = {Julien Paquier and Julien Toulouse}, + Doi = {10.1063/1.5049773}, + Journal = {J. Chem. Phys.}, + Pages = {174110}, + Title = {Four-component relativistic range-separated density-functional theory: Short-range exchange local-density approximation}, + Volume = {149}, + Year = {2018}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.5049773}} + +@misc{PaqTou-JJJ-XX-note, + Note = {See Supplementary Information for details on the calculation of the sums over spins in Eqs.~(\ref{epsxC}) and (\ref{epsxB}), details on the calculation of the large-$\ct$ expansions in Eqs.~(\ref{epsCsrasymp}) and~(\ref{epsBsrasymp}), and for a Mathematica notebook containing the complete expressions of the large-$\ct$ expansions in Eqs.~(\ref{epsCsrasymp}) and~(\ref{epsBsrasymp}) and of the Pad\'e approximants in Eqs.~(\ref{PadeC}) and~(\ref{PadeB}).}} + +@article{parr:3801, + Author = {Robert G. Parr and Robert A. Donnelly and Mel Levy and William E. Palke}, + Journal = {The Journal of Chemical Physics}, + Number = {8}, + Pages = {3801-3807}, + Publisher = {AIP}, + Title = {Electronegativity: The density functional viewpoint}, + Url = {http://link.aip.org/link/?JCP/68/3801/1}, + Volume = {68}, + Year = {1978}, + Bdsk-Url-1 = {http://link.aip.org/link/?JCP/68/3801/1}} + +@book{ParYan-BOOK-89, + Address = {New York}, + Author = {R. G. Parr and W. Yang}, + Publisher = {Oxford University Press}, + Title = {Density-Functional Theory of Atoms and Molecules}, + Year = {1989}} + +@article{PasGidPer-PRA-13, + Author = {E. Pastorczak and N. I. Gidopoulos and K. Pernal}, + Journal = {Phys. Rev. A}, + Pages = {062501}, + Volume = {87}, + Year = {2013}} + +@article{PatCenJezJezSza-JPCA-07, + Author = {Konrad Patkowski and Wojciech Cencek and M. Jeziorska and B. Jeziorski and Krzysztof Szalewicz}, + Journal = {J. Phys. Chem. A}, + Pages = {7611}, + Volume = {111}, + Year = {2007}} + +@article{PatVas-PRB-72, + Author = {K. N. Pathak and P. Vashishta}, + Journal = {Phys. Rev. B.}, + Pages = {3649}, + Volume = {{7}}, + Year = {1972}} + +@book{Pauncz-BOOK-79, + Address = {New York}, + Author = {R. Pauncz}, + Publisher = {Plenum Press}, + Title = {Spin Eigenfunctions}, + Year = {1979}} + +@article{PauPilTouEll-JCP-10, + Author = {Fran\c{c}oise Pauzat and Julien Pilm\'e and Julien Toulouse and Yves Ellinger}, + Journal = {J. Chem. Phys.}, + Pages = {054301}, + Title = {About the collapse of the 3.3 µm CH stretching band with ionization in polycyclic aromatic hydrocarbons: Configuration interaction and quantum Monte Carlo studies of the CH fragment}, + Volume = {133}, + Year = {2010}} + +@article{PazMorGorBac-PRB-06, + Author = {Simone Paziani and Saverio Moroni and Paola Gori-Giorgi and Giovanni B. Bachelet}, + Journal = {Phys. Rev. B}, + Pages = {155111}, + Volume = {73}, + Year = {2006}} + +@article{PazMorGorBac-xxx-xx, + Author = {S. Paziani and S. Moroni and P. Gori-Giorgi and G. B. Bachelet}, + Journal = {in preparation}, + Volume = {{}}, + Year = {2005}} + +@article{PeaBenHelToz-JCP-08, + Author = {M. J. Peach and P. Benfield and T. Helgaker and D. J. Tozer}, + Journal = {J. Chem. Phys.}, + Pages = {044118}, + Volume = {128}, + Year = {2008}} + +@article{PeaCohToz-PCCP-06, + Author = {M. J. G. Peach and A. J. Cohen and D. J. Tozer}, + Journal = {Phys. Chem. Chem. Phys.}, + Pages = {4543}, + Volume = {8}, + Year = {2006}} + +@article{PeaHelSalKeaLutTozHan-PCCP-06, + Author = {M. J. G. Peach and T. Helgaker and P. Salek and T. W. Keal and O. B. Lutn{\ae}s and D. J. Tozer and N. C. Handy}, + Journal = {Phys. Chem. Chem. Phys.}, + Pages = {558}, + Volume = {8}, + Year = {2006}} + +@article{PeaTeaHelToz-JCTC-15, + Author = {M. J. G. Peach and A. M. Teale and T. Helgaker and D. J. Tozer}, + Journal = {J. Chem. Theory Comput.}, + Pages = {5262}, + Volume = {11}, + Year = {2015}} + +@article{PeaToz-JPCA-12, + Author = {M. J. G. Peach and D. J. Tozer}, + Journal = {J. Phys. Chem. A}, + Pages = {9783}, + Volume = {116}, + Year = {2012}} + +@misc{PedJen-JJJ-XX, + Author = {J. K. Pedersen and H. J. A. Jensen}, + Note = {(unpublished)}, + Title = {{A second order MCSCF-DFT hybrid algorithm}}} + +@misc{PedJen-JJJ-XX_fr, + Author = {J. K. Pedersen and H. J. A. Jensen}, + Note = {non publi{\'e}}} + +@phdthesis{Ped-THESIS-04, + Address = {Odense}, + Author = {J. K. Pedersen}, + School = {University of Southern Denmark}, + Title = {Description of correlation and relativistic effects in calculations of molecular properties}, + Type = {{PhD thesis}}, + Year = {2004}} + +@article{PeiNecWar-PRA-03, + Author = {K. Peirs and D. Van Neck and M. Waroquier}, + Eid = {012505}, + Issue = {1}, + Journal = {Physical Review A (Atomic, Molecular, and Optical Physics)}, + Keywords = {density functional theory; atomic structure; Green's function methods; ab initio calculations}, + Numpages = {12}, + Page = {012505}, + Publisher = {APS}, + Title = {Algorithm to derive exact exchange-correlation potentials from correlated densities in atoms}, + Url = {http://link.aps.org/abstract/PRA/v67/e012505}, + Volume = {67}, + Year = {2003}, + Bdsk-Url-1 = {http://link.aps.org/abstract/PRA/v67/e012505}} + +@article{PerBurErn-PRL-96, + Author = {J. P. Perdew and K. Burke and M. Ernzerhof}, + Journal = {Phys. Rev. Lett.}, + Pages = {3865}, + Title = {Generalized Gradient Approximation Made Simple}, + Volume = {77}, + Year = {1996}} + +@article{PerCheVosJacPedSinFio-PRB-92, + Author = {J. P. Perdew and J. A. Chevary and S. H. Vosko and K. A. Jackson and M. R. Pederson and D. J. Singh and C. Fiolhais}, + Journal = {Phys. Rev. B}, + Pages = {6671}, + Volume = {{46}}, + Year = {1992}} + +@book{PerEngDreGroGodNogCasMar-BOOK-03, + Address = {Berlin}, + Editor = {C. Fiolhais and F. Nogueira and M. A. L. Marques}, + Publisher = {Springer}, + Series = {Vol. 620 of Lecture Notes in Physics}, + Title = {A Primer in Density Functional Theory}, + Year = {2003}} + +@article{PerErnBur-JCP-96, + Author = {J. P. Perdew and M. Ernzerhof and K. Burke}, + Journal = {J. Chem. Phys}, + Pages = {9982}, + Volume = {105}, + Year = {1996}} + +@article{PerErnBurSav-IJQC-97, + Author = {J. P. Perdew and M. Ernzerhof and K. Burke and A. Savin}, + Journal = {Int. J. Quantum Chem.}, + Pages = {197}, + Volume = {{61}}, + Year = {1997}} + +@article{PerGieGriBae-JCP-07, + Author = {K. Pernal and K. Giesbertz and O. Gritsenko and E. J. Baerends}, + Journal = {J. Chem. Phys.}, + Pages = {214101}, + Volume = {127}, + Year = {2007}} + +@article{PerGriBae-PRA-07, + Author = {K. Pernal and O. Gritsenko and E. J. Baerends}, + Journal = {Phys. Rev. A}, + Pages = {012506}, + Volume = {75}, + Year = {2007}} + +@article{PerGriMee-JCP-19, + Author = {{K. Pernal, O. V. Gritsenko, and R. van Meer}}, + Journal = {J. Chem. Phys.}, + Pages = {164122}, + Volume = {151}, + Year = {2019}} + +@article{Per-IJQC-93, + Author = {J. P. Perdew}, + Journal = {Int. J. Quantum. Chem.: Quantum Chem. Symp.}, + Pages = {93}, + Volume = {27}, + Year = {1993}} + +@incollection{Per-INC-91, + Address = {berlin}, + Author = {J. P. Perdew}, + Booktitle = {Electronic Structure of Solids '91}, + Editor = {P. Ziesche and H. Eschrig}, + Publisher = {Akademie Verlag}, + Year = {1991}} + +@article{Per-JCP-12, + Author = {K. Pernal}, + Journal = {J. Chem. Phys.}, + Pages = {184105}, + Title = {{Excitation energies from range-separated time-dependent density and density matrix functional theory}}, + Volume = {136}, + Year = {2012}} + +@article{PerLanSah-PRL-77, + Author = {J. P. Perdew and D. C. Langreth and V. Sahni}, + Journal = {Phys. Rev. Lett.}, + Pages = {1030}, + Volume = {{38}}, + Year = {1977}} + +@article{PerLev-PRL-83, + Author = {J. P. Perdew and M. Levy}, + Journal = {Phys. Rev. Lett.}, + Pages = {1884}, + Volume = {51}, + Year = {1983}} + +@article{PerMcMZun-PRA-81, + Author = {J. P. Perdew and E. R. McMullen and A. Zunger}, + Journal = {Phys. Rev. A}, + Pages = {2785}, + Volume = {{23}}, + Year = {1981}} + +@article{PerParLevBal-PRL-82, + Author = {J. P. Perdew and R. G. Parr and M. Levy and J. L. Balduz}, + Journal = {Phys. Rev. Lett.}, + Pages = {1691}, + Title = {Density-functional theory for fractional particle number: Derivative discontinuity of the energy}, + Volume = {49}, + Year = {1982}} + +@article{PerPerMorIll-JCC-07, + Author = {A. J. P\'erez-Jim\'enez and J. M. P\'erez-Jord\'a and I. De P. R. Moreira and F. Illas}, + Journal = {J. Comput. Chem.}, + Pages = {2559}, + Volume = {28}, + Year = {2007}} + +@article{PerPerSan-JCP-07, + Author = {A. J. P\'erez-Jim\'enez and J. M. P\'erez-Jord\'a and J. C. Sancho-Garc\'ia}, + Journal = {J. Chem. Phys.}, + Pages = {104102}, + Title = {Combining two-body density correlation functionals with multiconfigurational wave functions using natural orbitals and occupation numbers}, + Volume = {127}, + Year = {2007}} + +@article{Per-PRA-10, + Author = {Katarzyna Pernal}, + Journal = {Phys. Rev. A}, + Pages = {052511}, + Volume = {81}, + Year = {2010}} + +@article{Per-PRB-86, + Author = {J. P. Perdew}, + Journal = {Phys. Rev. B}, + Pages = {8822}, + Volume = {{33}}, + Year = {1986}} + +@article{Per-PRL-85, + Author = {J. P. Perdew}, + Journal = {Phys. Rev. Lett.}, + Pages = {1665}, + Volume = {{55}}, + Year = {1985}} + +@article{PerSavBur-PRA-95, + Author = {J. P. Perdew and A. Savin and K. Burke}, + Journal = {Phys. Rev. A}, + Pages = {4531}, + Title = {{Escaping the symmetry dilemma through a pair-density interpretation of spin-density functional theory}}, + Volume = {51}, + Year = {1995}} + +@article{PerSunRuzMezCso-PPCE-16, + Author = {J. P. Perdew and J. Sun and A. Ruzsinszky and P. D. Mezei and G. I. Csonka}, + Journal = {Period. Polytech. Chem. Eng.}, + Pages = {2}, + Volume = {60}, + Year = {2016}} + +@article{PerWan2-PRB-92, + Author = {J. P. Perdew and Y. Wang}, + Journal = {Phys. Rev. B}, + Pages = {12947}, + Volume = {{46}}, + Year = {1992}} + +@article{PerWan-PRB-86, + Author = {J. P. Perdew and Y. Wang}, + Journal = {Phys. Rev. B}, + Pages = {8800}, + Volume = {{33}}, + Year = {1986}} + +@article{PerWan-PRB-92, + Author = {J. P. Perdew and Y. Wang}, + Journal = {Phys. Rev. B}, + Pages = {13244}, + Volume = {45}, + Year = {1992}} + +@article{PerZun-PRB-81, + Author = {J. P. Perdew and A. Zunger}, + Journal = {Phys. Rev. B}, + Pages = {5048}, + Volume = {23}, + Year = {1981}} + +@article{PetDun-JCP-02, + Author = {{K. A. Peterson and T. H. Dunning, Jr}}, + Journal = {J. Chem. Phys.}, + Pages = {10548}, + Volume = {117}, + Year = {2002}} + +@article{PetDun-JPC-95, + Author = {{K. A. Peterson and T. H. Dunning Jr.}}, + Journal = {J. Phys. Chem.}, + Pages = {3898}, + Volume = {99}, + Year = {1995}} + +@article{PetGosGro-PRL-96, + Author = {M. Petersilka and U. J. Gossmann and E. K. U. Gross}, + Journal = {Phys. Rev. Lett.}, + Pages = {1212}, + Volume = {76}, + Year = {1996}} + +@article{PetGro-IJQC-96, + Author = {M. Petersilka and E. K. U. Gross}, + Journal = {Int. J. Quantum Chem. Symp.}, + Pages = {1393}, + Volume = {30}, + Year = {1996}} + +@article{PetHolChaNigUmr-PRL-12, + Author = {F. R. Petruzielo and A. A. Holmes and H. J. Changlani and M. P. Nightingale and C. J. Umrigar}, + Journal = {Phys. Rev. Lett.}, + Pages = {230201}, + Title = {Semistochastic Projector Monte Carlo Method}, + Volume = {109}, + Year = {2012}} + +@article{PetKenDun-JCP-93b, + Author = {{K. A. Peterson, R. A. Kendall and T. H. Dunning Jr.}}, + Journal = {J. Chem. Phys.}, + Pages = {9790}, + Volume = {99}, + Year = {1993}} + +@article{PetTouUmr-JCP-11, + Author = {F. R. Petruzielo and J. Toulouse and C. J. Umrigar}, + Journal = {J. Chem. Phys.}, + Pages = {064104}, + Title = {Basis set construction for molecular electronic structure theory: Natural orbital and Gauss--Slater basis for smooth pseudopotentials}, + Volume = {134}, + Year = {2011}} + +@article{PetTouUmr-JCP-12, + Author = {F. R. Petruzielo and J. Toulouse and C. J. Umrigar}, + Journal = {J. Chem. Phys.}, + Pages = {124116}, + Title = {{Approaching chemical accuracy with quantum Monte Carlo}}, + Volume = {136}, + Year = {2012}} + +@article{PevHea-JCP-13, + Author = {Roberto Peverati and Martin Head-Gordon}, + Journal = {J. Chem. Phys.}, + Pages = {024110}, + Title = {Orbital optimized double-hybrid density functionals}, + Volume = {139}, + Year = {2013}} + +@article{PevTru-JPCL-11, + Author = {Roberto Peverati and Donald G. Truhlar}, + Journal = {J. Phys. Chem. Lett.}, + Pages = {2810}, + Volume = {2}, + Year = {2011}} + +@article{PevTru-JPCL-12, + Author = {Roberto Peverati and Donald G. Truhlar}, + Journal = {J. Phys. Chem. Lett.}, + Pages = {117}, + Volume = {3}, + Year = {2012}} + +@article{PevTru-PTRSA-14, + Abstract = {Kohn{\textendash}Sham density functional theory is in principle an exact formulation of quantum mechanical electronic structure theory, but in practice we have to rely on approximate exchange{\textendash}correlation (xc) functionals. The objective of our work has been to design an xc functional with broad accuracy across as wide an expanse of chemistry and physics as possible, leading{\textemdash}as a long-range goal{\textemdash}to a functional with good accuracy for all problems, i.e. a universal functional. To guide our path towards that goal and to measure our progress, we have developed{\textemdash}building on earlier work of our group{\textemdash}a set of databases of reference data for a variety of energetic and structural properties in chemistry and physics. These databases include energies of molecular processes, such as atomization, complexation, proton addition and ionization; they also include molecular geometries and solid-state lattice constants, chemical reaction barrier heights, and cohesive energies and band gaps of solids. For this paper, we gather many of these databases into four comprehensive databases, two with 384 energetic data for chemistry and solid-state physics and another two with 68 structural data for chemistry and solid-state physics, and we test two wave function methods and 77 density functionals (12 Minnesota meta functionals and 65 others) in a consistent way across this same broad set of data. We especially highlight the Minnesota density functionals, but the results have broader implications in that one may see the successes and failures of many kinds of density functionals when they are all applied to the same data. Therefore, the results provide a status report on the quest for a universal functional.}, + Author = {Peverati, Roberto and Truhlar, Donald G.}, + Doi = {10.1098/rsta.2012.0476}, + Isbn = {1471-2962}, + Issn = {1364-503X}, + Journal = {Phil. Trans. R. Soc. A}, + Note = {preprint at http://arxiv.org/abs/1212.0944}, + Number = {2011}, + Pages = {20120476}, + Publisher = {The Royal Society}, + Title = {Quest for a universal density functional: the accuracy of density functionals across a broad spectrum of databases in chemistry and physics}, + Volume = {372}, + Year = {2014}, + Bdsk-Url-1 = {https://doi.org/10.1098/rsta.2012.0476}} + +@article{PhiKanZgi-JCP-15, + Author = {J. J. Phillips and A. A. Kananenka and D. Zgid}, + Journal = {J. Chem. Phys.}, + Pages = {194108}, + Volume = {142}, + Year = {2015}} + +@article{PicGos-MP-73, + Author = {B. T. Pickup and O. Goscinski}, + Journal = {Mol. Phys.}, + Pages = {1013}, + Volume = {26}, + Year = {1973}} + +@book{PinNoz-BOOK-89, + Author = {D. Pines and P. Nozi\`eres}, + Publisher = {Addison-Wesley Publishing Company}, + Title = {The Theory of Quantum Liquids}, + Year = {1989}} + +@article{PisDov-IJQC-80, + Author = {C. Pisani and R. Dovesi}, + Journal = {Int. J. Quantum Chem.}, + Pages = {501}, + Volume = {17}, + Year = {1980}} + +@article{PisMasCasHalSchUsv-JCC-08, + Author = {C. Pisani and L. Maschio and S. Casassa and M. Halo and M. Sch\"utz and D. Usvyat}, + Journal = {J. Comput. Chem.}, + Pages = {2113}, + Volume = {29}, + Year = {2008}} + +@article{pittner:10275, + Author = {Jiri Pittner and Petr Nachtigall and Petr Carsky and Jozef Masik and Ivan Hubac}, + Journal = {The Journal of Chemical Physics}, + Keywords = {organic compounds; silicon compounds; coupled cluster calculations; configuration interactions}, + Number = {21}, + Pages = {10275-10282}, + Publisher = {AIP}, + Title = {Assessment of the single-root multireference Brillouin--Wigner coupled- cluster method: Test calculations on CH[sub 2], SiH[sub 2], and twisted ethylene}, + Url = {http://link.aip.org/link/?JCP/110/10275/1}, + Volume = {110}, + Year = {1999}, + Bdsk-Url-1 = {http://link.aip.org/link/?JCP/110/10275/1}} + +@article{PolColLeiStoWerSav-IJQC-03, + Author = {R. Pollet and F. Colonna and T. Leininger and H. Stoll and H.-J. Werner and A. Savin}, + Journal = {Int. J. Quantum. Chem.}, + Pages = {84}, + Title = {Exchange-Correlation Energies and Correlation Holes for Some Two- and Four-Electron Atoms along a Nonlinear Adiabatic Connection in Density Functional Theory}, + Volume = {{91}}, + Year = {2003}} + +@article{PolSavLeiSto-JCP-02, + Author = {R. Pollet and A. Savin and T. Leininger and H. Stoll}, + Journal = {J. Chem. Phys.}, + Pages = {1250}, + Title = {Combining multideterminantal wave functions with density functionals to handle near-degeneracy in atoms and molecules}, + Volume = {{116}}, + Year = {2002}} + +@phdthesis{Pol-THESIS-01, + Author = {R. Pollet}, + School = {Universit\'e Paris 6}, + Title = {D\'eveloppement et applications d'une m\'ethode g\'en\'eralisant la th\'eorie de la fonctionnelle de la densit\'e \`a plusieurs d\'eterminants de Slater}, + Type = {Th\`ese de doctorat}, + Year = {2001}} + +@article{Pop-RMP-99, + Author = {J. A. Pople}, + Journal = {Rev. Mod. Phys.}, + Pages = {1267}, + Title = {{Nobel Lecture: Quantum chemical models}}, + Volume = {{71}}, + Year = {1999}} + +@article{PorDer-PRA-02, + Author = {S. G. Porsev and A. Derevianko}, + Journal = {Phys. Rev. A}, + Pages = {020701(R)}, + Volume = {65}, + Year = {2002}} + +@article{PraWooPetDunWil-TCA-11, + Author = {{B. P. Prascher, D. E. Woon, K. A. Peterson, T. H. Dunning, Jr., and A. K. Wilson}}, + Journal = {Theor. Chem. Acc.}, + Pages = {69}, + Volume = {128}, + Year = {2011}} + +@article{PreBevFah-PRB-02, + Author = {David Prendergast and David Bevan and Stephen Fahy}, + Journal = {Phys. Rev. B}, + Pages = {155104}, + Volume = {66}, + Year = {2002}} + +@article{PreNolFilFahGre-JCP-01, + Author = {D. Prendergast and M. Nolan and C. Filippi and S. Fahy and J. C. Greer}, + Journal = {J. Chem. Phys.}, + Pages = {1626}, + Volume = {{115}}, + Year = {2001}} + +@book{PreTeuVetFla-BOOK-92, + Address = {Cambridge}, + Author = {W. Press and S. Teukolsky and W. Vetterling and B. Flannery}, + Publisher = {Cambridge University Press}, + Title = {Numerical Recipes}, + Year = {1992}} + +@article{Pul-MP-69, + Author = {P. Pulay}, + Journal = {Mol. Phys.}, + Pages = {197}, + Volume = {17}, + Year = {1969}} + +@article{PurZhaKra-JCP-09, + Author = {W. Purwanto and S. Zhang and H. Krakauer}, + Journal = {J. Chem. Phys.}, + Pages = {094107}, + Volume = {130}, + Year = {2009}} + +@misc{Qmc-PROG-XX, + Note = {QMCMOL, a quantum Monte Carlo program written by R. Assaraf, F. Colonna, X. Krokidis, P. Reinhardt and coworkers.}, + Url = {http://www.lct.jussieu.fr/pagesequipe/qmcmol/qmcmol/}, + Bdsk-Url-1 = {http://www.lct.jussieu.fr/pagesequipe/qmcmol/qmcmol/}} + +@misc{Quantumexpressoweb-PROG-XX, + Note = {\url{http://www.quantum-espresso.org}}, + Title = {QUANTUM ESPRESSO, an integrated suite of Open-Source computer codes for electronic-structure calculations and materials modeling at the nanoscale}} + +@article{RabBaeNeu-PRB-15, + Author = {E. Rabani and R. Baer and D. Neuhauser}, + Journal = {Phys. Rev. B}, + Pages = {235302}, + Volume = {91}, + Year = {2015}} + +@article{rabinovitch:1807, + Author = {B. S. Rabinovitch and John E. Douglas and F. S. Looney}, + Journal = {The Journal of Chemical Physics}, + Number = {11}, + Pages = {1807-1808}, + Publisher = {AIP}, + Title = {The Thermal Trans-Cis Isomerization of Dideuteroethylene}, + Url = {http://link.aip.org/link/?JCP/20/1807/2}, + Volume = {20}, + Year = {1952}, + Bdsk-Url-1 = {http://link.aip.org/link/?JCP/20/1807/2}} + +@book{Rai-BOOK-72, + Address = {Amsterdam-London}, + Author = {S. Raimes}, + Publisher = {North-Holland Publishing Company}, + Title = {Many-electron theory}, + Year = {1972}} + +@article{Raj-ACP-80, + Author = {A. K. Rajagopal}, + Journal = {Adv. Chem. Phys.}, + Pages = {59}, + Volume = {41}, + Year = {1980}} + +@article{RajCal-PRB-73, + Author = {A. K. Rajagopal and J. Callaway}, + Journal = {Phys. Rev. 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Astrophys. Astr.}, + Pages = {113}, + Volume = {17}, + Year = {1996}} + +@misc{RebSavTou-JJJ-XX, + Author = {E. Rebolini and A. Savin and J. Toulouse}, + Note = {unpublished}} + +@article{RebSavTou-MP-13, + Author = {E. Rebolini and A. Savin and J. Toulouse}, + Journal = {Mol. Phys.}, + Pages = {1219}, + Title = {Electronic excitations from a linear-response range-separated hybrid scheme}, + Volume = {111}, + Year = {2013}} + +@article{RebTeaHelSavTou-MP-18, + Author = {E. Rebolini and A. M. Teale and T. Helgaker and A. Savin and J. Toulouse}, + Doi = {doi.org/10.1080/00268976.2017.1422811}, + Journal = {Mol. Phys.}, + Pages = {1443-1451}, + Title = {Excitation energies from G\"orling--Levy perturbation theory along the range-separated adiabatic connection}, + Volume = {116}, + Year = {2018}, + Bdsk-Url-1 = {https://doi.org/10.1080/00268976.2017.1422811}} + +@phdthesis{Reb-THESIS-14, + Author = {E. Rebolini}, + School = {Universit\'e Pierre et Marie Curie}, + Title = {Range-separated density-functional theory for molecular excitation energies}, + Type = {{Th\`ese de doctorat}}, + Url = {https://tel.archives-ouvertes.fr/tel-01027522}, + Year = {2014}, + Bdsk-Url-1 = {https://tel.archives-ouvertes.fr/tel-01027522}} + +@phdthesis{Reb-THESIS-14_eng, + Author = {E. Rebolini}, + School = {Universit\'e Pierre et Marie Curie}, + Title = {Range-separated density-functional theory for molecular excitation energies}, + Type = {{PhD thesis}}, + Url = {https://tel.archives-ouvertes.fr/tel-01027522}, + Year = {2014}, + Bdsk-Url-1 = {https://tel.archives-ouvertes.fr/tel-01027522}} + +@article{RebTou-JCP-16, + Author = {E. Rebolini and J. Toulouse}, + Doi = {http://dx.doi.org/10.1063/1.4943003}, + Journal = {J. Chem. Phys.}, + Pages = {094107}, + Title = {Range-separated time-dependent density-functional theory with a frequency-dependent second-order Bethe-Salpeter correlation kernel}, + Volume = {144}, + Year = {2016}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4943003}} + +@misc{RebTou-JJJ-XX-note1, + Note = {The last two terms of the kernel in Eq. (31) of Ref.~\onlinecite{ZhaSteYan-JCP-13} contain non-antisymmetrized two-electron integrals. However, these terms can also be written with a factor of $1/2$ and antisymmetrized two-electron integrals, leading to our Eq.~(\ref{eq:eff kernel iajb}).}} + +@incollection{RebTouSav-INC-13, + Author = {E. Rebolini and J. Toulouse and A. Savin}, + Booktitle = {Electronic Structure and Reactivity}, + Editor = {S. K. Ghosh and P. K. Chattaraj}, + Note = {preprint at http://arxiv.org/abs/1304.1314}, + Pages = {367-390}, + Publisher = {CRC Press}, + Series = {Concepts and Methods in Modern Theoretical Chemistry Vol. 1}, + Title = {Electronic excitation energies of molecular systems from the Bethe-Salpeter equation: Example of the H$_2$ molecule}, + Year = {2013}} + +@misc{RebTouSav-JJJ-XX-note, + Note = {With the notations used here, the Hubbard model is obtained for $\Delta\varepsilon = 2 t$ and $J_{11}=J_{22}=J_{12}=J_{12}=K_{12}=U/2$ where $t$ is the hopping parameter and $U$ is the on-site Coulomb interaction.}} + +@article{RebTouTeaHelSav-JCP-14, + Author = {E. Rebolini and J. Toulouse and A. M. Teale and T. Helgaker and A. Savin}, + Doi = {http://dx.doi.org/10.1063/1.4890652}, + Journal = {Journal of Chemical Physics}, + Pages = {044123}, + Title = {Excitation energies along a range-separated adiabatic connection}, + Volume = {141}, + Year = {2014}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4890652}} + +@misc{RebTouTeaHelSav-JJJ-XX-sup, + Note = {See supplementary material at http://xxxxxxxx for the fits of the total and excitation energies.}} + +@article{RebTouTeaHelSav-MP-15, + Author = {E. Rebolini and J. Toulouse and A. M. Teale and T. Helgaker and A. Savin}, + Doi = {http://dx.doi.org/10.1080/00268976.2015.1011248}, + Journal = {Mol. Phys.}, + Pages = {1740}, + Title = {Excited states from range-separated density-functional perturbation theory}, + Volume = {113}, + Year = {2015}, + Bdsk-Url-1 = {http://dx.doi.org/10.1080/00268976.2015.1011248}} + +@article{RebTouTeaHelSav-PRA-15, + Author = {E. Rebolini and J. Toulouse and A. M. Teale and T. Helgaker and A. Savin}, + Doi = {http://dx.doi.org/10.1103/PhysRevA.91.032519}, + Journal = {Phys. Rev. A}, + Pages = {032519}, + Title = {Calculating excitation energies by extrapolation along adiabatic connections}, + Volume = {91}, + Year = {2015}, + Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRevA.91.032519}} + +@article{RefJaiShaNeaKro-PRB-15, + Author = {Sivan Refaely-Abramson and Manish Jain and Sahar Sharifzadeh and Jeffrey B. Neaton and Leeor Kronik}, + Journal = {Phys. Rev. B}, + Pages = {081204(R)}, + Volume = {92}, + Year = {2015}} + +@article{RefShaGovAutNeaBaeKro-PRL-12, + Author = {S. Refaely-Abramson and Sahar Sharifzadeh and Niranjan Govind and Jochen Autschbach and Jeffrey B. Neaton and Roi Baer and Leeor Kronik}, + Journal = {Phys. Rev. Lett.}, + Pages = {226405}, + Volume = {109}, + Year = {2012}} + +@article{RegTha-JPB-84, + Author = {P. E. Regier and A. J. Thakkar}, + Journal = {J. Phys. 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Hoggan}, + Booktitle = {Advances in Quantum Monte Carlo}, + Editor = {{S. Tanaka, S. M. Rothstein, and W. A. Lester, Jr.}}, + Pages = {53-63}, + Publisher = {American Chemical Society}, + Series = {ACS Symposium Series Vol. 1094}, + Title = {Quantum Monte Carlo Facing the Hartree-Fock Symmetry Dilemma: The Case of Hydrogen Rings}, + Year = {2012}} + +@article{ReiTouSav-TCA-18, + Author = {Peter Reinhardt and Julien Toulouse and Andreas Savin}, + Doi = {10.1007/s00214-018-2370-5}, + Journal = {Theor. Chem. Acc.}, + Pages = {168}, + Title = {Range-separated density-functional theory applied to the beryllium dimer and trimer}, + Volume = {137}, + Year = {2018}, + Bdsk-Url-1 = {https://doi.org/10.1007/s00214-018-2370-5}} + +@article{RenRinBluWieTkaSanReuSch-NJP-12, + Author = {Xinguo Ren and Patrick Rinke and Volker Blum and Jurgen Wieferink and Alexandre Tkatchenko and Andrea Sanfilippo and Karsten Reuter and Matthias Scheffler}, + Journal = {New J. Phys.}, + Pages = {053020}, + Volume = {14}, + Year = {2012}} + +@article{RenRinJoaSch-JMS-12, + Affiliation = {Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany}, + Author = {Ren, Xinguo and Rinke, Patrick and Joas, Christian and Scheffler, Matthias}, + Journal = {J. Mater. Sci.}, + Keyword = {Chemistry and Materials Science}, + Optnote = {10.1007/s10853-012-6570-4}, + Opturl = {http://dx.doi.org/10.1007/s10853-012-6570-4}, + Owner = {gambort}, + Pages = {7447}, + Publisher = {Springer Netherlands}, + Timestamp = {2012.07.06}, + Title = {Random-phase approximation and its applications in computational chemistry and materials science}, + Volume = {47}, + Year = {2012}} + +@article{RenRinSch-PRB-09, + Author = {Ren, Xinguo and Rinke, Patrick and Scheffler, Matthias}, + Doi = {10.1103/PhysRevB.80.045402}, + Journal = {Phys. Rev. B}, + Month = {Jul}, + Number = {4}, + Numpages = {8}, + Pages = {045402}, + Publisher = {American Physical Society}, + Title = {Exploring the random phase approximation: Application to CO adsorbed on Cu(111)}, + Volume = {80}, + Year = {2009}, + Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevB.80.045402}} + +@article{RenTkaRinSch-PRL-11, + Author = {X. Ren and A. Tkatchenko and P. Rinke and M. Scheffler}, + Journal = {Phys. Rev. Lett.}, + Pages = {153003}, + Volume = {106}, + Year = {2011}} + +@article{ReyCepAldLes-JCP-82, + Author = {P. J. Reynolds and D. M. Ceperley and B. J. Alder and W. A. Lester}, + Journal = {J. Chem. Phys.}, + Pages = {5593}, + Title = {{Fixed-node quantum Monte Carlo for molecules}}, + Volume = {77}, + Year = {1982}} + +@article{ReyDupLes-JCP-85, + Author = {P. J. Reynolds and M. Dupuis and W. A. Lester}, + Journal = {J. Chem. Phys.}, + Pages = {1983}, + Volume = {82}, + Year = {1985}} + +@article{ReyOweLes-JCP-87, + Author = {P. J. Reynolds and R. K. 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Chem. A}, + Pages = {5314}, + Volume = {111}, + Year = {2007}} + +@article{RicAsh-PRB-94, + Author = {C. F. Richardson and N. W. Ashcroft}, + Journal = {Phys. Rev. B}, + Pages = {8170}, + Volume = {{50}}, + Year = {1994}} + +@article{RilAnd-MP-03, + Author = {K. E. Riley and J. B. Anderson}, + Journal = {Mol. Phys.}, + Pages = {3129}, + Volume = {101}, + Year = {2003}} + +@article{RocLuGal-JCP-10, + Author = {D. Rocca and D. Lu and G. Galli}, + Journal = {J. Chem. Phys.}, + Pages = {164109}, + Volume = {133}, + Year = {2010}} + +@article{RoeVes-IJQC-05, + Author = {I. R{\o}eggen and L. Veseth}, + Journal = {Int. J. Quantum Chem.}, + Pages = {201}, + Volume = {101}, + Year = {2005}} + +@article{RohGriBae-CPL-06, + Author = {D. Rohr and O. Gritsenko and E. J. Baerends}, + Journal = {Chem. Phys. Lett.}, + Pages = {336}, + Volume = {432}, + Year = {2006}} + +@article{RohHer-JCP-08, + Author = {M. A. Rohrdanz and J. M. Herbert}, + Journal = {J. Chem. 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Lett.}, + Pages = {1091}, + Volume = {57}, + Year = {1989}} + +@article{TalSha-PRA-76, + Author = {J. D. Talman and W. F. Shadwick}, + Journal = {Phys. Rev. A}, + Pages = {36}, + Volume = {14}, + Year = {1976}} + +@article{TanKobYamYosYam-JCP-89, + Author = {K. Tanaka and H. Kobayashi and S. Yamanaka and K. Yoshizawa and T. Yamaba}, + Journal = {J. Chem. Phys.}, + Pages = {3724}, + Volume = {91}, + Year = {1989}} + +@article{TanToe-JCP-03, + Author = {K. T. Tang and J. P. Toennies}, + Journal = {J. Chem. Phys.}, + Pages = {4976}, + Volume = {118}, + Year = {2003}} + +@article{TaoPerStaScu-PRL-03, + Author = {J. Tao and J. P. Perdew and V. N. Staroverov and G. E. Scuseria}, + Journal = {Phys. Rev. Lett.}, + Pages = {146401}, + Volume = {91}, + Year = {2003}} + +@article{TapTavRotFilCas-JCP-08, + Author = {E. Tapavicza and I. Tavernelli and U. Rothlisberger and C. Filippi and M. E. Casida}, + Journal = {J. Chem. Phys.}, + Pages = {124108}, + Volume = {129}, + Year = {2008}} + +@article{TarKarSerVuzMar-JPCA-08, + Author = {Alex Tarnopolsky and Amir Karton and Rotem Sertchook and Dana Vuzman and Jan M. L. Martin}, + Journal = {J. Phys. Chem. A}, + Pages = {3}, + Volume = {112}, + Year = {2008}} + +@article{TawTsuYanYanHir-JCP-04, + Author = {Y. Tawada and T. Tsuneda and S. Yanagisawa and T. Yanai and K. Hirao}, + Journal = {J. Chem. Phys.}, + Pages = {8425}, + Title = {A long-range-corrected time-dependent density functional theory}, + Volume = {{120}}, + Year = {2004}} + +@article{TayAngGalZhaGygHirSonRahLilPodBulHenScuTouPevTruSza-JCP-16, + Author = {Taylor, D. E. and \'Angy\'an, J. G. and Galli, G. and Zhang, C. and Gygi, F. and Hirao, K. and Song, J. W. and Rahul, K. and von Lilienfeld, O. A. and Podeszwa, R. and Bulik, I. W. and Henderson, T. M. and Scuseria, G. E. and Toulouse, J. and Peverati, R. and Truhlar, D. G. and Szalewicz, K.}, + Journal = {J. Chem. Phys.}, + Pages = {124105}, + Title = {Blind test of density-functional-based methods on intermolecular interaction energies}, + Volume = {145}, + Year = {2016}} + +@book{TDDFT-BOOK-06, + Address = {Berlin}, + Editor = {M. L. Marques and C. A. Ullrich and F. Nogueira and A. Rubio and K. Burke and E. K. U. Gross}, + Publisher = {Springer}, + Series = {Lecture Notes in Physics Vol. 706}, + Title = {Time-Dependent Density Functional Theory}, + Year = {2006}} + +@book{TDDFT-BOOK-12, + Address = {Berlin}, + Editor = {M. A. L. Marques and N. T. Maitra, F. M. S. Nogueira, E. K. U. Gross, A. Rubio}, + Publisher = {Springer}, + Series = {Lecture Notes in Physics Vol. 837}, + Title = {Fundamentals of Time-Dependent Density Functional Theory}, + Year = {2012}} + +@article{TeaCorHel-JCP-09, + Author = {A. M. Teale and S. Coriani and T. Helgaker}, + Journal = {J. Chem. Phys.}, + Pages = {104111}, + Title = {{The calculation of adiabatic-connection curves from full configuration-interaction densities: Two-electron systems}}, + Volume = {130}, + Year = {2009}} + +@article{TeaCorHel-JCP-10b, + Author = {Andrew M. Teale and S. Coriani and Trygve Helgaker}, + Journal = {J. Chem. Phys.}, + Pages = {164112}, + Volume = {133}, + Year = {2010}} + +@article{TeaCorHel-JCP-10, + Author = {Andrew M. Teale and S. Coriani and Trygve Helgaker}, + Journal = {J. Chem. Phys.}, + Pages = {164115}, + Volume = {132}, + Year = {2010}} + +@article{TeaDepToz-JCP-08, + Author = {{A. M. Teale, F. De Proft, and D. J. Tozer}}, + Journal = {J. Chem. Phys.}, + Pages = {044110}, + Volume = {129}, + Year = {2008}} + +@article{TecGomEksVis-PCCP-11, + Author = {P. Tecmer and A. S. P. Gomes and U. Ekstr\"om and L. Visscher}, + Journal = {Phys. Chem. Chem. Phys.}, + Pages = {6249}, + Volume = {13}, + Year = {2011}} + +@article{TelKvaSagEksTeaHel-PRA-12, + Author = {E. I. Tellgren and S. Kvaal and E. Sagvolden and U. Ekstr\"om and A. M. Teale and T. Helgaker}, + Journal = {Phys. Rev. A}, + Pages = {062506}, + Volume = {86}, + Year = {2012}} + +@article{TelSosRozChu-PRA-13, + Author = {D. A. Telnov and K. E. Sosnova and E. Rozenbaum and S.-I. Chu}, + Journal = {Phys. Rev. A}, + Pages = {053406}, + Volume = {87}, + Year = {2013}} + +@article{Tha-CPL-03, + Author = {A. J. Thakkar}, + Journal = {Chem. Phys. Lett.}, + Pages = {80}, + Volume = {381}, + Year = {2003}} + +@incollection{Tha-INC-87, + Address = {Dordrecht}, + Author = {A. J. Thakkar}, + Booktitle = {Density Matrices and Density Functionals}, + Editor = {R. M. Erdahl and V. H. Smith}, + Pages = {553-581}, + Publisher = {Reidel}, + Year = {1987}} + +@article{ThaSmi-CPL-76, + Author = {A. J. Thakkar and V. H. Smith}, + Journal = {Chem. Phys. Lett.}, + Pages = {476}, + Volume = {42}, + Year = {1976}} + +@article{ThaSmi-JCP-77, + Author = {A. J. Thakkar and V. H. Smith}, + Journal = {J. Chem. Phys.}, + Pages = {1191}, + Volume = {67}, + Year = {1977}} + +@article{ThaTriSmi-IJQC-84, + Author = {A. J. Thakkar and A. N. Tripathi and V. H. Smith}, + Journal = {Int. J. Quantum. Chem.}, + Pages = {157}, + Volume = {26}, + Year = {1984}} + +@article{ThaTriSmi-PRA-84, + Author = {A. J. Thakkar and A. N. Tripathi and V. H. Smith}, + Journal = {Phys. Rev. A}, + Pages = {1108}, + Volume = {29}, + Year = {1984}} + +@book{Tho-BOOK-61, + Address = {New York}, + Author = {D. J. Thouless}, + Publisher = {Academic Press}, + Title = {The Quantum Mechanics of Many Body Systems}, + Year = {1961}} + +@article{Tho-NAT-25, + Author = {W. Thomas}, + Journal = {Naturwissenschaften}, + Pages = {627}, + Volume = {13}, + Year = {1925}} + +@article{Tho-NP-61, + Author = {D. J. Thouless}, + Journal = {Nucl. Phys.}, + Pages = {78}, + Volume = {22}, + Year = {1961}} + +@article{ThyFleJen-JCP-08, + Author = {J. Thyssen and T. Fleig and H. J. Aa. Jensen}, + Journal = {J. Chem. Phys.}, + Pages = {034109}, + Volume = {129}, + Year = {2008}} + +@article{TiaChe-PRB-06, + Author = {M. L. Tiago and J. R. Chelikowsky}, + Journal = {Phys. Rev. B}, + Pages = {205334}, + Volume = {73}, + Year = {2006}} + +@article{TiaChe-SSC-05, + Author = {M. L. Tiago and J. R. Chelikowsky}, + Journal = {Solid State Commun.}, + Pages = {333}, + Volume = {136}, + Year = {2005}} + +@article{TiaKenHooReb-JCP-08, + Author = {M. L. Tiago and P. R. C. Kent and R. Q. Hood and F. A. Reboredo}, + Journal = {J. Chem. Phys.}, + Pages = {084311}, + Volume = {129}, + Year = {2008}} + +@article{TkaSch-PRL-09, + Author = {A. Tkatchenko and M. Scheffler}, + Journal = {Phys. Rev. Lett.}, + Pages = {073005}, + Title = {{Accurate molecular van der Waals interactions from ground-state electron density and free-atom reference data}}, + Volume = {102}, + Year = {2009}} + +@article{TofSteDec-PRA-06, + Author = {D. Toffoli and M. Stener and P. Decleva}, + Journal = {Phys. Rev. A}, + Pages = {042704}, + Volume = {73}, + Year = {2006}} + +@article{ToiWoo-PRB-70, + Author = {F. Toigo and T. O. Woodruff}, + Journal = {Phys. Rev. B}, + Pages = {3958}, + Volume = {{2}}, + Year = {1970}} + +@article{ToiWoo-PRB-71, + Author = {F. Toigo and T. O. Woodruff}, + Journal = {Phys. Rev. B}, + Pages = {371}, + Volume = {{4}}, + Year = {1971}} + +@incollection{Tos-INC-99, + Address = {University of Oxford}, + Author = {M. P. Tosi}, + Booktitle = {Electron Correlation in the Solid State}, + Editor = {N. H. March}, + Publisher = {World Scientific Publishing Company}, + Title = {Many-Body Effects in Jellium}, + Year = {1999}} + +@incollection{TouAssUmr-INC-16, + Author = {J. Toulouse and R. Assaraf and C. J. Umrigar}, + Booktitle = {Electron Correlation in Molecules - ab initio Beyond Gaussian Quantum Chemistry}, + Doi = {10.1016/bs.aiq.2015.07.003}, + Editor = {P. E. Hoggan and T. Ozdogan}, + Pages = {285-314}, + Publisher = {Academic Press}, + Series = {Advances in Quantum Chemistry Vol. 73}, + Title = {Introduction to the Variational and Diffusion Monte Carlo Methods}, + Year = {2016}, + Bdsk-Url-1 = {https://doi.org/10.1016/bs.aiq.2015.07.003}} + +@article{TouAssUmr-JCP-07, + Author = {J. Toulouse and R. Assaraf and C. J. Umrigar}, + Journal = {J. Chem. Phys.}, + Pages = {244112}, + Title = {{Zero-variance zero-bias quantum Monte Carlo estimators of the spherically and system-averaged pair density}}, + Volume = {126}, + Year = {2007}} + +@incollection{TouCafReiHogUmr-INC-12, + Address = {Dordrecht Heidelberg London New York}, + Author = {J. Toulouse and M. Caffarel and P. Reinhardt and P. E. Hoggan and C. J. Umrigar}, + Booktitle = {Advances in the Theory of Quantum Systems in Chemistry and Physics}, + Editor = {P. E. Hoggan and J. Maruani and P. Piecuch and G. Delgado-Barrio and E. J. Br\"andas}, + Pages = {345-353}, + Publisher = {Springer}, + Series = {Progress in Theoretical Chemistry and Physics Vol. 22}, + Title = {Quantum Monte Carlo calculations of electronic excitation energies: The case of the singlet $n\to\pi^*$ (CO) transition in acrolein}, + Year = {2012}} + +@article{TouColSav-JCP-05, + Author = {J. Toulouse and F. Colonna and A. Savin}, + Journal = {J. Chem. Phys.}, + Pages = {014110}, + Title = {Short-range exchange and correlation energy density functionals: Beyond the local-density approximation}, + Volume = {122}, + Year = {2005}} + +@misc{TouColSav-JJJ-XXa, + Author = {J. Toulouse and F. Colonna and A. Savin}, + Title = {in preparation}} + +@article{TouColSav-MP-05, + Author = {Julien Toulouse and Francois Colonna and Andreas Savin}, + Journal = {Mol. Phys.}, + Pages = {2725}, + Title = {Exchange-correlation potentials and local energies per particle along non-linear adiabatic connections}, + Volume = {103}, + Year = {2005}} + +@article{TouColSav-PRA-04, + Author = {J. Toulouse and F. Colonna and A. Savin}, + Journal = {Phys. Rev. A}, + Keywords = {density functional theory; wave functions; exchange interactions (electron); electron correlations}, + Number = {6}, + Pages = {062505}, + Publisher = {APS}, + Title = {Long-range--short-range separation of the electron-electron interaction in density-functional theory}, + Volume = {70}, + Year = {2004}} + +@misc{TouGerJanSavAng-JJJ-XX-note, + Note = {For Ne$_{2}$ at equilibrium distance with aug-cc-pVQZ basis, the counterpoise correction on the interaction energy is 16 $\mu$H for RSH+RPAx and 62 $\mu$H for standard MP2.}} + +@article{TouGerJanSavAng-PRL-09, + Author = {J. Toulouse and I. C. Gerber and G. Jansen and A. Savin and J. G. \'Angy\'an}, + Journal = {Phys. Rev. Lett.}, + Pages = {096404}, + Title = {Adiabatic-connection fluctuation-dissipation density-functional theory based on range separation}, + Volume = {102}, + Year = {2009}} + +@article{TouGorSav-IJQC-06, + Author = {J. Toulouse and P. Gori-Giorgi and A. Savin}, + Journal = {Int. J. Quantum Chem.}, + Pages = {2026}, + Title = {Scaling relations, virial theorem and energy densities for long-range and short-range density functionals}, + Volume = {106}, + Year = {2006}} + +@article{TouGorSav-TCA-05, + Author = {J. Toulouse and P. Gori-Giorgi and A. Savin}, + Journal = {Theor. Chem. Acc.}, + Pages = {305}, + Volume = {114}, + Year = {2005}} + +@misc{Tou-JJJ-XX, + Author = {S. Dubillard and E. Eliav and L. Visscher and R. Bast and T. Saue}, + Note = {unpublished}} + +@article{Tou-PRB-05, + Author = {Julien Toulouse}, + Journal = {Phys. Rev. B}, + Pages = {035117}, + Title = {Simple model of the static exchange-correlation kernel of a uniform electron gas with long-range electron-electron interaction}, + Volume = {72}, + Year = {2005}} + +@article{TouRebGouDobSeaAng-JCP-13, + Author = {Julien Toulouse and Elisa Rebolini and Tim Gould and John F. Dobson and Prasenjit Seal and J\'anos G. \'Angy\'an}, + Journal = {J. Chem. Phys.}, + Pages = {194106}, + Title = {Assessment of range-separated time-dependent density-functional theory for calculating $C_6$ dispersion coefficients}, + Volume = {138}, + Year = {2013}} + +@article{TouSavFla-IJQC-04, + Author = {J. Toulouse and A. Savin and H.-J. Flad}, + Journal = {Int. J. Quantum Chem.}, + Pages = {1047}, + Title = {Short-range exchange-correlation energy of a uniform electron gas with modified electron-electron interaction}, + Volume = {100}, + Year = {2004}} + +@misc{TouSav-JJJ-XX, + Author = {J. Toulouse and A. Savin}, + Title = {in preparation}} + +@article{TouSav-JMS-06, + Author = {Julien Toulouse and Andreas Savin}, + Journal = {J. Mol. Struct. (Theochem)}, + Pages = {147}, + Title = {Local density approximation for long-range or for short-range energy functionals?}, + Volume = {762}, + Year = {2006}} + +@article{TouShaBreAda-JCP-11, + Author = {J. Toulouse and K. Sharkas and E. Br\'emond and C. Adamo}, + Journal = {J. Chem. Phys.}, + Pages = {101102}, + Title = {Rationale for a new class of double-hybrid approximations in density-functional theory}, + Volume = {135}, + Year = {2011}} + +@phdthesis{Tou-THESIS-05, + Author = {J. Toulouse}, + Note = {tel.archives-ouvertes.fr/tel-00550772}, + School = {Universit\'e Pierre et Marie Curie (Paris 6)}, + Year = {2005}} + +@article{TouUmr-JCP-07, + Author = {J. Toulouse and C. J. Umrigar}, + Journal = {J. Chem. Phys.}, + Pages = {084102}, + Title = {{Optimization of quantum Monte Carlo wave functions by energy minimization}}, + Volume = {126}, + Year = {2007}} + +@article{TouUmr-JCP-08, + Author = {J. Toulouse and C. J. Umrigar}, + Journal = {J. Chem. Phys.}, + Pages = {174101}, + Title = {{Full optimization of Jastrow-Slater wave functions with application to the first-row atoms and homonuclear diatomic molecules}}, + Volume = {128}, + Year = {2008}} + +@misc{TouUmr-JJJ-XX, + Author = {J. Toulouse and C. J. Umrigar}, + Note = {unpublished}} + +@misc{TouZhuAngSav-JJJ-XX-note2, + Note = {In the context of density-functional theory RPA is usually derived from the Kohn-Sham reference, while in the context of many-body perturbation theory (see appendices) RPA is usually derived from the HF reference. Therefore, both PBE+RPA and HF+RPA are theoretically justified.}} + +@misc{TouZhuAngSav-JJJ-XX-note3, + Note = {The inverse of a 4-point function $\chi (1,2;1',2')$ is defined according to $\int d1'd2' \chi (1,2;1',2') \chi^{-1} (2',1';4,3) = \delta(1,3) \delta (2,4)$.}} + +@misc{TouZhuAngSav-JJJ-XX-note, + Note = {The short-range self-energy correction $\Delta \Sigma^\sr_\l$ is wrongly missing in Eq.~(11) of Ref.~\onlinecite{TouGerJanSavAng-PRL-09}. However, in practice, this term vanishes in the RPA or RPAx approximation so that the results of Ref.~\onlinecite{TouGerJanSavAng-PRL-09} are correct.}} + +@article{TouZhuAngSav-PRA-10, + Author = {Julien Toulouse and Wuming Zhu and J\'anos G. \'Angy\'an and Andreas Savin}, + Journal = {Phys. Rev. A}, + Pages = {032502}, + Title = {Range-separated density-functional theory with the random-phase approximation: Detailed formalism and illustrative applications}, + Volume = {82}, + Year = {2010}} + +@article{TouZhuSavJanAng-JCP-11, + Author = {J. Toulouse and W. Zhu and A. Savin and G. Jansen and J. G. \'Angy\'an}, + Journal = {J. Chem. Phys.}, + Pages = {084119}, + Title = {Closed-shell ring coupled cluster doubles theory with range separation applied on weak intermolecular interactions}, + Volume = {135}, + Year = {2011}} + +@misc{TouZhuSavJanAng-JJJ-XX, + Author = {J. Toulouse and W. Zhu and A. Savin and G. Jansen and J. G. \'Angyan}, + Note = {unpublished}} + +@misc{TouZhuSavJanAng-JJJ-XX-note, + Note = {In Ref.~\onlinecite{Hes-JCP-11}, AC-RPA, NRPA1, NRPA3, and NRPA4 refer to what we call here RPAx-I, RPAx-II, RPA-SO2, and RPA-SO1, respectively. In addition, NRPA2 corresponds to the variant of Fukuda {\it et al.}, i.e. $2E_{c,\RPAxII}-E_{c,\text{MP2}}$.}} + +@article{TowZupCau-CPC-96, + Author = {M. D. Towler and A. Zupan and M. Caus\'a}, + Journal = {Comp. Phys. Comm.}, + Pages = {181}, + Volume = {98}, + Year = {1996}} + +@article{TozHan-JCP-98, + Author = {D. J. Tozer and N. C. Handy}, + Journal = {J. Chem. Phys.}, + Pages = {10180}, + Volume = {109}, + Year = {1998}} + +@article{Tri-JPB-80, + Author = {H. P. Trivedi}, + Journal = {J. Phys. B}, + Pages = {839}, + Volume = {13}, + Year = {1980}} + +@article{Tru-CPL-98, + Author = {D. G. Truhlar}, + Journal = {Chem. Phys. Lett.}, + Pages = {45}, + Volume = {294}, + Year = {1998}} + +@article{TsuScu-JCP-11, + Author = {T. Tsuchimochi and G. E. Scuseria}, + Journal = {J. Chem. Phys.}, + Pages = {064101}, + Volume = {134}, + Year = {2011}} + +@article{TsuScuSav-JCP-10, + Author = {T. Tsuchimochi and G. E. Scuseria and A. Savin}, + Journal = {J. Chem. Phys.}, + Pages = {024111}, + Volume = {132}, + Year = {2010}} + +@article{TsuSonSuzHir-JCP-10, + Author = {Takao Tsuneda and Jong-Won Song and Satoshi Suzuki and Kimihiko Hirao}, + Journal = {J. Chem. Phys.}, + Pages = {174101}, + Volume = {133}, + Year = {2010}} + +@article{TsuSuzHir-JCP-99, + Author = {Takao Tsuneda and Toshihisa Suzumura and Kimihiko Hirao}, + Journal = {J. Chem. Phys.}, + Pages = {5656}, + Volume = {111}, + Year = {1999}} + +@article{UgaBoy-IJQC-85, + Author = {J. M. Ugalde and R. J. Boyd}, + Journal = {Int. J. Quantum. Chem.}, + Pages = {439}, + Volume = {27}, + Year = {1985}} + +@book{Ull-BOOK-11, + Author = {Carsten A. Ullrich}, + Publisher = {OUP Oxford}, + Title = {Time-Dependent Density-Functional Theory: Concepts and Applications}, + Year = {2011}} + +@article{UllKoh-PRL-01, + Author = {C. A. Ullrich and W. Kohn}, + Journal = {Phys. Rev. Lett.}, + Pages = {093001}, + Volume = {87}, + Year = {2001}} + +@article{UmrFil-PRL-05, + Author = {C. J. Umrigar and C. Filippi}, + Journal = {Phys. Rev. Lett.}, + Pages = {150201}, + Title = {{Energy and variance optimization of many-body wave functions}}, + Volume = {94}, + Year = {2005}} + +@misc{UmrFil-PRL-05-note, + Note = {The convergence in 2 iterations, mentioned near the end of Ref.~\onlinecite{UmrFil-PRL-05} was obtained using the correlated sampling adjustment of $a_\diag$ and the TU Hessian.}} + +@incollection{UmrGon-INC-93, + Address = {Singapore}, + Author = {C. J. Umrigar and X. Gonze}, + Booktitle = {High Performance Computing and its Application to the Physical Sciences, Proceedings of the Mardi Gras '93 Conference}, + Editor = {D. A. Browne {\it et al.}}, + Publisher = {World Scientific}, + Year = {1993}} + +@article{UmrGon-PRA-94, + Author = {C. J. Umrigar and X. Gonze}, + Journal = {Phys. Rev. A}, + Pages = {3827}, + Volume = {50}, + Year = {1994}} + +@article{Umr-IJQC-89, + Author = {C. J. Umrigar}, + Journal = {Int. J. Quantum Chem.}, + Pages = {217}, + Volume = {{23}}, + Year = {1989}} + +@incollection{Umr-INC-99, + Address = {Dordrecht}, + Author = {C. J. Umrigar}, + Booktitle = {Quantum Monte Carlo Methods in Physics and Chemistry}, + Editor = {M. P. Nightingale and C. J. Umrigar}, + Pages = {129}, + Publisher = {Kluwer}, + Series = {NATO ASI Ser. C 525}, + Year = {1999}} + +@article{UmrNigRun-JCP-93, + Author = {C. J. Umrigar and M. P. Nightingale and K. J. Runge}, + Journal = {J. Chem. Phys.}, + Pages = {2865}, + Title = {{A diffusion Monte Carlo algorithm with very small time-step errors}}, + Volume = {99}, + Year = {1993}} + +@article{Umr-PRL-93, + Author = {C. J. Umrigar}, + Journal = {Phys. Rev. Lett.}, + Pages = {408}, + Title = {{Accelerated Metropolis method}}, + Volume = {71}, + Year = {1993}} + +@incollection{UmrSavGon-INC-98, + Author = {Umrigar, C.J. and Savin, A. and Gonze, Xavier}, + Booktitle = {Electronic Density Functional Theory}, + Doi = {10.1007/978-1-4899-0316-7_12}, + Editor = {Dobson, John F. and Vignale, Giovanni and Das, Mukunda P.}, + Pages = {167-176}, + Publisher = {Springer US}, + Title = {Are Unoccupied Kohn-Sham Eigenvalues Related to Excitation Energies?}, + Year = {1998}, + Bdsk-Url-1 = {https://doi.org/10.1007/978-1-4899-0316-7_12}} + +@article{UmrTouFilSorHen-PRL-07, + Author = {C. J. Umrigar and J. Toulouse and C. Filippi and S. Sorella and R. G. Hennig}, + Journal = {Phys. Rev. Lett.}, + Pages = {110201}, + Title = {Alleviation of the Fermion-sign problem by optimization of many-body wave functions}, + Volume = {98}, + Year = {2007}} + +@misc{Umr-UNP-XXb, + Author = {C. J. Umrigar}, + Note = {unpublished}} + +@misc{Umr-UNP-XX, + Author = {C. J. Umrigar}, + Note = {unpublished}} + +@misc{Umr-UNP-XX_fr, + Author = {C. J. Umrigar}, + Note = {non publi{\'e}}} + +@incollection{UmrWilWil-INC-88, + Address = {Berlin}, + Author = {C. J. Umrigar and K. G. Wilson and J. W. Wilkins}, + Booktitle = {Computer Simulation Studies in Condensed Matter Physics: Recent Developments}, + Editor = {D. P. Landau and K. K. Mon and H. B. Sch\"uttler}, + Publisher = {Springer}, + Year = {1988}} + +@article{UmrWilWil-PRL-88, + Author = {C. J. Umrigar and K. G. Wilson and J. W. Wilkins}, + Journal = {Phys. Rev. Lett.}, + Pages = {1719}, + Title = {{Optimized trial wave functions for quantum Monte Carlo calculations}}, + Volume = {60}, + Year = {1988}} + +@article{UtsIch-PRB-80a, + Author = {K. Utsumi and S. Ichimaru}, + Journal = {Phys. Rev. B}, + Pages = {1522}, + Volume = {{22}}, + Year = {1980}} + +@article{UtsIch-PRB-80, + Author = {K. Utsumi and S. Ichimaru}, + Journal = {Phys. Rev. B}, + Pages = {5203}, + Volume = {{22}}, + Year = {1980}} + +@article{Val-JCP-80a, + Author = {S. M. Valone}, + Journal = {J. Chem. Phys.}, + Pages = {4653}, + Volume = {73}, + Year = {1980}} + +@phdthesis{Van-THESIS-18, + Author = {Marco Vanzini}, + School = {{\'E}cole Polytechnique, Universit{\'e} Paris-Saclay}, + Title = {Auxiliary systems for observables: dynamical local connector approximation for electron addition and removal spectra}, + Year = {2018}} + +@article{VasSin-PRB-72, + Author = {P. Vashista and K.S. Singwi}, + Journal = {Phys. Rev. B}, + Pages = {875}, + Volume = {{6}}, + Year = {1972}} + +@article{VeiCle-JCP-69, + Author = {A. Veillard and E. Clementi}, + Journal = {J. Chem. Phys.}, + Pages = {2415}, + Volume = {{49}}, + Year = {1969}} + +@article{VitDelGor-JCP-05, + Author = {{V. Vitale, F. Della Sala, and A. G\"orling}}, + Journal = {J. Chem. Phys.}, + Pages = {244102}, + Volume = {122}, + Year = {2005}} + +@phdthesis{Vog-THESIS-03, + Author = {M. Vogt}, + School = {University of Cambridge}, + Title = {Spectral Moments in the Homogeneous Electron Gas}, + Year = {2003}} + +@article{VogZimNee-PRB-04, + Author = {M. Vogt and R. Zimmermann and R. J. Needs}, + Journal = {Phys. Rev. B}, + Pages = {045113}, + Volume = {{69}}, + Year = {2004}} + +@article{VosWilNus-CJP-80, + Author = {S. J. Vosko and L. Wilk and M. Nusair}, + Journal = {Can. J. Phys.}, + Pages = {1200}, + Volume = {{58}}, + Year = {1980}} + +@article{VrbDepRot-JCP-88, + Author = {J. Vrbik and M. F. DePasquale and S. M. Rothstein}, + Journal = {J. Chem. Phys.}, + Pages = {3784}, + Volume = {88}, + Year = {1988}} + +@article{VutCamGurHutparPatPetSpaDoyGabDem-JPB-10, + Author = {A. C. Vutha and W. C. Campbell and Y. V. Gurevich and N. R. Hutzler and M. Parsons and D. Patterson and E. Petrik and B. Spaun and J. M. Doyle and G. Gabrielse and D. DeMille}, + Journal = {J. Phys. B}, + Pages = {074007}, + Volume = {43}, + Year = {2010}} + +@article{VydScu-JCP-06, + Author = {O. A. Vydrov and G. E. Scuseria}, + Journal = {J. Chem. Phys.}, + Pages = {234109}, + Volume = {125}, + Year = {2006}} + +@article{VydScuPer-JCP-07, + Author = {Vydrov, O. A. and Scuseria, G. E. and Perdew, J. P.}, + Journal = {J. Chem. Phys.}, + Pages = {154109}, + Title = {Tests of functionals for systems with fractional electron number}, + Volume = {126}, + Year = {2007}} + +@article{VydVoo-JCP-09, + Author = {O. A. Vydrov and T. van Voorhis}, + Journal = {J. Chem. Phys.}, + Pages = {104105}, + Volume = {130}, + Year = {2009}} + +@article{VydVoo-JCP-10b, + Author = {O. A. Vydrov and T. V. Voorhis}, + Journal = {J. Chem. Phys.}, + Pages = {244103}, + Title = {{Nonlocal van der Waals density functional: The simpler the better}}, + Volume = {133}, + Year = {2010}} + +@article{VydVoo-PRL-09, + Author = {O. A. Vydrov and T. van Voorhis}, + Journal = {Phys. Rev. Lett.}, + Pages = {063004}, + Volume = {103}, + Year = {2009}} + +@article{Wal-TFS-45, + Author = {A. D. Walsh}, + Journal = {Trans. Faraday Soc.}, + Pages = {498}, + Volume = {41}, + Year = {1945}} + +@article{WanGauWul-JCP-08, + Author = {{F. Wang, J. Gauss, and C. van W\"ullen}}, + Journal = {J. Chem. Phys.}, + Pages = {064113}, + Volume = {129}, + Year = {2008}} + +@article{WanPer-PRB-91, + Author = {Y. Wang and J.P. Perdew}, + Journal = {Phys. Rev. B}, + Pages = {13298}, + Volume = {{44}}, + Year = {1991}} + +@article{WanSch-JCP-96, + Author = {S. G. Wang and W. H. E. Schwarz}, + Journal = {J. Chem. Phys.}, + Pages = {4641}, + Volume = {{105}}, + Year = {1996}} + +@article{WanSchSmi-PRA-00, + Author = {J. Wang and H. L. Schmider and V. H. Smith}, + Journal = {Phys. Rev. A}, + Pages = {016501}, + Volume = {62}, + Year = {2000}} + +@article{WanSmi-IJQC-94, + Author = {J. Wang and V. H. Smith}, + Journal = {Int. J. Quantum. Chem.}, + Pages = {147}, + Volume = {49}, + Year = {1994}} + +@article{WanTriSmi-JCP-92, + Author = {J. Wang and A. N. Tripathi and V. H. Smith}, + Journal = {J. Chem. Phys.}, + Pages = {9188}, + Volume = {97}, + Year = {1992}} + +@article{WanTriSmi-JCP-94, + Author = {J. Wang and A. N. Tripathi and V. H. Smith}, + Journal = {J. Chem. Phys.}, + Pages = {4842}, + Volume = {101}, + Year = {1994}} + +@article{WanZie-JCP-04, + Author = {F. Wang and T. Ziegler}, + Journal = {J. Chem. Phys.}, + Pages = {12191}, + Volume = {121}, + Year = {2004}} + +@article{WasBur-PRL-05, + Author = {A. Wasserman and K. Burke}, + Journal = {Phys. Rev. Lett.}, + Pages = {163006}, + Volume = {95}, + Year = {2005}} + +@article{WasMaiBur-PRL-03, + Author = {A. Wasserman and N. T. Maitra and K. Burke}, + Journal = {Phys. Rev. Lett.}, + Pages = {263001}, + Volume = {91}, + Year = {2003}} + +@article{WatKamYamUdaMul-MP-04, + Author = {Noboru Watanabe and Yohei Kamata and Kota Yamauchi and Yasuo Udagawa and Thomas M\"uller}, + Journal = {Mol. Phys.}, + Pages = {649}, + Volume = {102}, + Year = {2004}} + +@article{WeiDelGor-JCP-08, + Author = {M. Weimer and F. 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Phys.}, + Number = {13}, + Pages = {134108}, + Title = {Relations between coordinate and potential scaling in the high-density limit}, + Volume = {122}, + Year = {2005}} + +@article{Wig-PR-48, + Author = {E. Wigner}, + Journal = {Phys. Rev.}, + Pages = {1002}, + Volume = {73}, + Year = {1948}} + +@article{WilMouDun-JMS-96, + Author = {A. K. Wilson and T. v. Mourik and T. H. Dunning}, + Journal = {J. Mol. Struct.}, + Pages = {339}, + Volume = {388}, + Year = {1996}} + +@article{WilWooPetDun-JCP-99, + Author = {A.K. Wilson and D.E. Woon and K.A. Peterson and T.H.Dunning}, + Journal = {J. Chem. Phys.}, + Pages = {7667}, + Volume = {110}, + Year = {1999}} + +@article{WodJanRagCor-JPCA-08, + Author = {{M. D. Wodrich, D. F. Jana, P. v. R. Schleyer, and C. Corminboeuf}}, + Journal = {J. Phys. Chem. A}, + Pages = {11495}, + Volume = {112}, + Year = {2008}} + +@article{Wol-PRL-89, + Author = {Wolff, U.}, + Doi = {10.1103/PhysRevLett.62.361}, + Issue = {4}, + Journal = {Phys. Rev. Lett.}, + Numpages = {0}, + Pages = {361--364}, + Publisher = {American Physical Society}, + Title = {{Collective Monte Carlo updating for spin systems}}, + Url = {http://link.aps.org/doi/10.1103/PhysRevLett.62.361}, + Volume = {62}, + Year = {1989}, + Bdsk-Url-1 = {http://link.aps.org/doi/10.1103/PhysRevLett.62.361}, + Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.62.361}} + +@article{wolinski:3647, + Author = {Krzysztof Wolinski and Peter Pulay}, + Journal = {The Journal of Chemical Physics}, + Keywords = {PERTURBATION THEORY; WAVE FUNCTIONS; ELECTRONIC STRUCTURE; DIPOLE MOMENTS; AFFINITY; IONIZATION POTENTIAL; POTENTIAL ENERGY SURFACES; HYDROFLUORIC ACID; FLUORINE; ETHYLENE; OZONE; FORMALDEHYDE; ELECTRON CORRELATION}, + Number = {7}, + Pages = {3647-3659}, + Publisher = {AIP}, + Title = {Generalized M[o-slash]ller--Plesset perturbation theory: Second order results for two-configuration, open-shell excited singlet, and doublet wave functions}, + Url = {http://link.aip.org/link/?JCP/90/3647/1}, + Volume = {90}, + Year = {1989}, + Bdsk-Url-1 = {http://link.aip.org/link/?JCP/90/3647/1}} + +@article{WooDun-JCP-93, + Author = {D.E. Woon and T.H. Dunning}, + Journal = {J. Chem. Phys.}, + Pages = {1358}, + Volume = {98}, + Year = {1993}} + +@article{WooDun-JCP-94, + Author = {D.E. Woon and T.H. Dunning}, + Journal = {J. Chem. Phys.}, + Pages = {2975}, + Volume = {100}, + Year = {1994}} + +@article{WooDun-JCP-95, + Author = {D.E. Woon and T.H. Dunning}, + Journal = {J. Chem. Phys.}, + Pages = {4572}, + Volume = {103}, + Year = {1995}} + +@article{WouNec-EPJD-14, + Author = {{S. Wouters and D. Van Neck}}, + Journal = {Eur. Phys. J. D}, + Pages = {272}, + Title = {The density matrix renormalization group for ab initio quantum chemistry}, + Volume = {68}, + Year = {2014}} + +@article{WuAyeYan-JCP-03, + Author = {Qin Wu and Paul W. Ayers and Weitao Yang}, + Journal = {J. Chem. Phys.}, + Keywords = {density functional theory; variational techniques; electron affinity; negative ions}, + Number = {6}, + Pages = {2978-2990}, + Publisher = {AIP}, + Title = {Density-functional theory calculations with correct long-range potentials}, + Url = {http://link.aip.org/link/?JCP/119/2978/1}, + Volume = {119}, + Year = {2003}, + Bdsk-Url-1 = {http://link.aip.org/link/?JCP/119/2978/1}} + +@article{WuCheVoo-JCP-07, + Author = {{Q. Wu, C.-L. Cheng, and T. Van Voorhis}}, + Journal = {J. Chem. Phys.}, + Pages = {164119}, + Title = {Configuration interaction based on constrained density functional theory: A multireference method}, + Volume = {127}, + Year = {2007}} + +@article{WuKadVoo-JCP-09, + Author = {{Q. Wu, B. Kaduk, and T. Van Voorhis}}, + Journal = {J. Chem. Phys.}, + Pages = {034109}, + Volume = {130}, + Year = {2009}} + +@incollection{Wul-INC-10, + Address = {Netherlands}, + Author = {{C. van W\"ullen}}, + Booktitle = {Relativistic Methods for Chemists, Challenges and Advances in Computational Chemistry and Physics 10}, + Editor = {M. Barysz and Y. Ishikawa}, + Pages = {191-214}, + Publisher = {Springer}, + Title = {RELATIVISTIC DENSITY FUNCTIONAL THEORY}, + Year = {2010}} + +@article{WuSha-CPL-99, + Author = {W. Wu and S. Shaik}, + Journal = {Chem. Phys. Lett.}, + Pages = {37}, + Volume = {301}, + Year = {1999}} + +@article{WuShaHib-xx, + Author = {P. Su and J. Wu. and J. Gu and W. Wu and S. Shaik and P. C. Hiberty}, + Journal = {J. Chem. Theory Comput.}, + Pages = {121}, + Volume = {7}, + Year = {2011}} + +@article{WuSonCaoZhaSha-JPCA-02, + Author = {W. Wu and L. Song and Z. Cao and Q. Zhang and S. Shaik}, + Journal = {J. Phys. Chem. A}, + Pages = {2721}, + Volume = {106}, + Year = {2002}} + +@article{WuVarNayLotSco-JCP-01, + Author = {X. Wu and M. C. Vargas and S. Nayak and V. Lotrich and G. Scoles}, + Journal = {J. Chem. Phys.}, + Pages = {8748}, + Volume = {115}, + Year = {2001}} + +@article{WuWuMo-IJQC-98, + Author = {W. Wu and A. Wu and Y. Mo and M. Lin and Q. Zhang}, + Journal = {Int. J. Quantum Chem.}, + Pages = {287}, + Volume = {67}, + Year = {1998}} + +@article{WuYan-JCP-02, + Author = {Q. Wu and W. Yang}, + Journal = {J. Chem. Phys.}, + Pages = {515}, + Volume = {116}, + Year = {2002}} + +@article{WuYan-JCP-03, + Author = {Qin Wu and Weitao Yang}, + Journal = {J. Chem. Phys.}, + Number = {6}, + Pages = {2498-2509}, + Title = {A direct optimization method for calculating density functionals and exchange--correlation potentials from electron densities}, + Volume = {118}, + Year = {2003}} + +@article{xantheas:8054, + Author = {Sotiris S. Xantheas and Gregory J. Atchity and Stephen T. Elbert and Klaus Ruedenberg}, + Journal = {The Journal of Chemical Physics}, + Keywords = {OZONE; ELECTRONIC STRUCTURE; POTENTIAL ENERGY; SELF–CONSISTENT FIELD; CONFIGURATION INTERACTION; POTENTIAL ENERGY SURFACES; QUANTUM CHEMISTRY}, + Number = {12}, + Pages = {8054-8069}, + Publisher = {AIP}, + Title = {Potential energy surfaces of ozone. I}, + Url = {http://link.aip.org/link/?JCP/94/8054/1}, + Volume = {94}, + Year = {1991}, + Bdsk-Url-1 = {http://link.aip.org/link/?JCP/94/8054/1}} + +@article{YamKocTen-JCP-07, + Author = {D. Yamaki and H. Koch and S. Ten-no}, + Journal = {J. Chem. Phys.}, + Pages = {144104}, + Volume = {127}, + Year = {2007}} + +@article{YamNakUkaTakYam-IJQC-06, + Author = {S. Yamanaka and K. Nakata and T. Ukai and T. Takada and K. Yamaguchi}, + Journal = {Int. J. Quantum Chem.}, + Pages = {3312}, + Title = {Multireference Density Functional Theory With Orbital-Dependent Correlation Corrections}, + Volume = {106}, + Year = {2006}} + +@article{YanAyeWu-PRL-04, + Author = {Weitao Yang and Paul W. Ayers and Qin Wu}, + Journal = {Phys. Rev. Lett.}, + Pages = {146404}, + Volume = {92}, + Year = {2003}} + +@article{YanCohMor-JCP-12, + Author = {Weitao Yang and Aron J. Cohen and Paula Mori-S\'anchez}, + Journal = {J. Chem. Phys.}, + Pages = {204111}, + Volume = {136}, + Year = {2012}} + +@article{YanFaaBur-JCP-09, + Author = {{Z.-h. Yang, M. van Faassen, and K. Burke}}, + Journal = {J. Chem. Phys.}, + Pages = {114308}, + Volume = {131}, + Year = {2009}} + +@article{YanHarHan-MP-05, + Author = {T. Yanai and R. J. Harrison and N. C. Handy}, + Journal = {Mol. Phys.}, + Pages = {413}, + Volume = {103}, + Year = {2005}} + +@article{Yan-JCP-98, + Author = {W. Yang}, + Journal = {J. Chem. Phys.}, + Pages = {10107}, + Title = {Generalized adiabatic connection in density functional theory}, + Volume = {{109}}, + Year = {1998}} + +@article{YanMorCoh-JCP-13, + Author = {W. Yang and P. Mori-S{\'a}nchez and A. J. Cohen}, + Journal = {J. Chem. Phys.}, + Pages = {104114}, + Volume = {139}, + Year = {2013}} + +@article{YanPerKur-PRB-00, + Author = {Yan, Z. and Perdew, J. P. and Kurth, S.}, + Journal = {Phys. Rev. B.}, + Pages = {16430-16439}, + Title = {Density functional for short-range correlation: Accuracy of the random-phase approximation for isoelectronic energy changes}, + Volume = {61}, + Year = {2000}} + +@article{YanTewHan-CPL-04, + Author = {Takeshi Yanai and David P. Tew and Nicholas C. Handy}, + Journal = {Chem. Phys. Lett.}, + Pages = {51}, + Volume = {393}, + Year = {2004}} + +@article{YanWu-PRL-02, + Author = {Weitao Yang and Qin Wu}, + Journal = {Phys. Rev. Lett.}, + Pages = {143002}, + Title = {Direct Method for Optimized Effective Potentials in Density-Functional Theory}, + Volume = {89}, + Year = {2002}} + +@article{YanZhaAye-PRL-00, + Author = {Weitao Yang and Yingkai Zhang and Paul W. Ayers}, + Journal = {Phys. Rev. Lett.}, + Pages = {5172}, + Volume = {84}, + Year = {2000}} + +@article{Yar-CPL-81, + Author = {D. R. Yarkony}, + Journal = {Chem. Phys. Lett.}, + Pages = {634}, + Volume = {77}, + Year = {1981}} + +@article{Yas-PRL-02, + Author = {Koji Yasuda}, + Eid = {053001}, + Journal = {Physical Review Letters}, + Keywords = {density functional theory; electron density; electron correlations; Schrodinger equation; scaling phenomena; electron gas}, + Number = {5}, + Numpages = {4}, + Pages = {053001}, + Publisher = {APS}, + Title = {Local Approximation of the Correlation Energy Functional in the Density Matrix Functional Theory}, + Url = {http://link.aps.org/abstract/PRL/v88/e053001}, + Volume = {88}, + Year = {2002}, + Bdsk-Url-1 = {http://link.aps.org/abstract/PRL/v88/e053001}} + +@article{YeaJor-JCP-79, + Author = {D. L. Yeager and P. J{\o}rgensen}, + Journal = {J. Chem. Phys.}, + Pages = {755}, + Volume = {{71}}, + Year = {1979}} + +@article{YinSuCheShaWu-JCTC-12, + Author = {F. Ying and P. Su and Z. Chen and S. Shaik and W. Wu}, + Journal = {J. Chem. Theory Comput.}, + Pages = {1608}, + Volume = {8}, + Year = {2012}} + +@article{Yu-IJQC-13, + Author = {F. Yu}, + Journal = {Int. J. Quantum Chem.}, + Pages = {2355}, + Volume = {113}, + Year = {2013}} + +@article{ZahLeaGor-JCP-13, + Author = {F. Zahariev and S. S. Leang and M. S. Gordon}, + Journal = {J. Chem. Phys.}, + Pages = {244108}, + Volume = {138}, + Year = {2013}} + +@book{ZalPapMezLes-BOOK-11, + Address = {Netherlands}, + Editor = {R. Zalesny and M. G. Papadopoulos and P. G. Mezey and J. Leszczynski}, + Publisher = {Springer}, + Series = {Challenges and Advances in Computational Chemistry and Physics Vol. 13}, + Title = {Linear-Scaling Techniques in Computational Chemistry and Physics}, + Year = {2011}} + +@article{ZanSov-PRA-80, + Author = {A. Zangwill and P. Soven}, + Journal = {Phys. Rev. A}, + Pages = {1561}, + Volume = {21}, + Year = {1980}} + +@article{ZapLupTou-JCP-19, + Author = {F. Zapata and E. Luppi and J. Toulouse}, + Doi = {10.1063/1.5096037}, + Journal = {J. Chem. Phys.}, + Pages = {234104}, + Title = {Linear-response range-separated density-functional theory for atomic photoexcitation and photoionization spectra}, + Volume = {150}, + Year = {2019}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.5096037}} + +@misc{ZapLupTou-JJJ-XX-note, + Note = {Contrary to our Fig. \ref{Hespectra}a, the TDLDA spectrum of the He atom shown in Fig. 6 of Ref.~\onlinecite{WasMaiBur-PRL-03} has a larger maximum than the LDA spectrum. This discrepancy is due to the fact that the TDLDA spectrum shown in Ref.~\onlinecite{WasMaiBur-PRL-03} comes in fact from Ref.~\onlinecite{SteDecGor-JCP-01} where it was calculated by replacing the LDA 1s orbital energy by the opposite of the exact ionization energy. We have checked that this results not only in an energy shift of the spectrum but also to larger oscillator strengths. The true TDLDA spectrum of the He atom is thus the one shown in the present Fig. \ref{Hespectra}a.}} + +@article{ZarKoh-PRB-76, + Author = {E. Zaremba and W. Kohn}, + Journal = {Phys. Rev. B}, + Pages = {2270}, + Volume = {13}, + Year = {1976}} + +@article{ZecGorMorBac-PRB-04, + Author = {Lorenzo Zecca and Paola Gori-Giorgi and Saverio Moroni and Giovanni B. Bachelet}, + Journal = {Phys. Rev. B}, + Keywords = {density functional theory; Monte Carlo methods; ground states; electron gas; electron density; electron correlations}, + Number = {20}, + Pages = {205127}, + Publisher = {APS}, + Title = {Local density functional for the short-range part of the electron-electron interaction}, + Volume = {70}, + Year = {2004}} + +@article{ZeiMea-MP-77, + Author = {G. D. Zeiss and W. J. Meath}, + Journal = {Mol. Phys.}, + Pages = {1155}, + Volume = {33}, + Year = {1977}} + +@article{ZhaBur-PRA-04, + Author = {F. Zhang and K. Burke}, + Journal = {Phys. Rev. A}, + Pages = {052510}, + Volume = {69}, + Year = {2004}} + +@article{ZhaLuoXu-JCP-10b, + Author = {I. Y. Zhang and Y. Luo and X. Xu}, + Journal = {J. Chem. Phys.}, + Pages = {104105}, + Volume = {133}, + Year = {2010}} + +@article{ZhaLuoXu-JCP-10, + Author = {I. Y. Zhang and Y. Luo and X. Xu}, + Journal = {J. Chem. Phys.}, + Pages = {194105}, + Volume = {132}, + Year = {2010}} + +@article{ZhaLynTru-JPCA-04, + Author = {Y. Zhao and B. J. Lynch and D. G. Truhlar}, + Journal = {J. Phys. Chem. A}, + Pages = {4786}, + Volume = {108}, + Year = {2004}} + +@article{ZhaLynTru-PCCP-05, + Author = {Y. Zhao and B. J. Lynch and D. G. Truhlar}, + Journal = {Phys. Chem. Chem. Phys.}, + Pages = {43}, + Volume = {7}, + Year = {2005}} + +@article{ZhaPar-PRA-92, + Author = {Q. Zhao and R. G. Parr}, + Journal = {Phys. Rev. A.}, + Pages = {2337}, + Volume = {{46}}, + Year = {1992}} + +@article{ZhaSchTru-JCTC-06, + Author = {Y. Zhao and N. E. Schultz and D. G. Truhlar}, + Journal = {J. Chem. Theory Comput.}, + Pages = {364}, + Volume = {2}, + Year = {2006}} + +@article{ZhaSteYan-JCP-13, + Abstract = {We present a dynamical second-order kernel for the Bethe-Salpeter equation to calculate electronic excitation energies. The derivation takes explicitly the functional derivative of the exact second-order self energy with respect to the one-particle Green's function. It includes naturally a frequency dependence, going beyond the adiabatic approximation. Perturbative calculations under the Tamm-Dancoff approximation, using the configuration interaction singles (CIS) eigenvectors, reveal an appreciable improvement over CIS, time-dependent Hartree-Fock, and adiabatic time-dependent density functional theory results. The perturbative results also compare well with equation-of-motion coupled-cluster and experimental results.}, + Author = {Zhang, Du and Steinmann, Stephan N and Yang, Weitao}, + Journal = {J. Chem. Phys.}, + Month = oct, + Number = {15}, + Pages = {154109}, + Pmid = {24160502}, + Publisher = {American Institute of Physics}, + Title = {{Dynamical second-order Bethe-Salpeter equation kernel: A method for electronic excitation beyond the adiabatic approximation.}}, + Volume = {139}, + Year = {2013}} + +@article{ZhaSuBreAdaXu-JCP-12, + Author = {I. Y. Zhang and N. Q. Su and E. A. G. Br\'emond and C. Adamo}, + Journal = {J. Chem. Phys.}, + Pages = {174103}, + Volume = {136}, + Year = {2012}} + +@article{ZhaTru-JCTC-07, + Author = {Y. Zhao and D. G. Truhlar}, + Journal = {J. Chem. Theory Comput.}, + Pages = {289}, + Title = {Density Functionals for Noncovalent Interaction Energies of Biological Importance}, + Volume = {3}, + Year = {2007}} + +@article{ZhaTru-TCA-08, + Author = {Y. Zhao and D. G. Truhlar}, + Journal = {Theor. Chem. Acc.}, + Pages = {215}, + Volume = {120}, + Year = {2008}} + +@article{ZhaWuXu-CC-10, + Author = {I. Y. Zhang and J. Wu and X. Xu}, + Journal = {Chem. Commun.}, + Pages = {3057}, + Volume = {46}, + Year = {2010}} + +@article{ZhaXuGod-PNAS-09, + Author = {{Y. Zhang, X. Xu and W. A. Goddard III}}, + Journal = {Proc. Natl. Acad. Sci. U.S.A.}, + Pages = {4963}, + Volume = {106}, + Year = {2009}} + +@article{ZhaXuGod-PNAS-09_fr, + Author = {{Y. Zhang, X. Xu et W. A. Goddard III}}, + Journal = {Proc. Natl. Acad. Sci. U.S.A.}, + Pages = {4963}, + Volume = {106}, + Year = {2009}} + +@article{ZhaXu-IRPC-11, + Author = {I. Y. Zhang and X. Xu}, + Journal = {Int. Rev. Phys. Chem.}, + Pages = {115}, + Volume = {30}, + Year = {2011}} + +@article{ZhaXu-JPCL-13, + Author = {I. Y. Zhang and X. Xu}, + Journal = {J. Phys. Chem. Lett.}, + Pages = {1669}, + Volume = {4}, + Year = {2013}} + +@article{ZhaXuJunGod-PNAS-11, + Author = {{I. Y. Zhang, X. Xu, Y. Jung, and W. A. Goddard III}}, + Journal = {Proc. Natl. Acad. Sci. U.S.A.}, + Pages = {19896}, + Volume = {108}, + Year = {2011}} + +@article{ZhaYan-JCP-98, + Author = {Y. Zhang and W. Yang}, + Journal = {J. Chem. Phys.}, + Pages = {2604}, + Volume = {109}, + Year = {1998}} + +@article{ZheZhaTru-JCTC-07, + Author = {J. Zheng and Y. Zhao and D. G. Truhlar}, + Journal = {J. Chem. Theory Comput.}, + Pages = {569}, + Volume = {3}, + Year = {2007}} + +@article{ZheZhaTru-JCTC-09, + Author = {J. Zheng and Y. Zhao and D. G. Truhlar}, + Journal = {J. Chem. Theory Comput.}, + Pages = {808}, + Volume = {5}, + Year = {2009}} + +@article{ZhoChu-PRA-09, + Author = {Z. Zhou and S.-I. Chu}, + Journal = {Phys. Rev. A}, + Pages = {053412}, + Volume = {79}, + Year = {2009}} + +@article{ZhuTouSavAng-JCP-10, + Author = {Wuming Zhu and Julien Toulouse and Andreas Savin and J\'anos G. \'Angy\'an}, + Journal = {J. Chem. Phys.}, + Pages = {244108}, + Title = {Range-separated density-functional theory with random phase approximation applied to noncovalent intermolecular interactions}, + Volume = {132}, + Year = {2010}} + +@misc{ZhuTouSavAng-JJJ-XX, + Author = {W. Zhu and J. Toulouse and A. Savin and J. G. \'Angy\'an}, + Note = {unpublished}} + +@article{ZieCio-PA-05, + Author = {P. Ziesche and J. Cioslowski}, + Journal = {Physica A}, + Year = {to appear}} + +@article{Zie-JPC-80, + Author = {P. Ziesche}, + Journal = {J. Phys. C}, + Pages = {3625}, + Volume = {13}, + Year = {1980}} + +@article{ZieRauBae-TCA-77, + Author = {T. Ziegler and A. Rauk and E. J. Baerends}, + Journal = {Theor. Chem. Acc.}, + Pages = {261}, + Volume = {43}, + Year = {1977}} + +@article{ZieSetKryAutWan-JCP-09, + Author = {T. Ziegler and M. Seth and M. Krykunov and J. Autschbach and F. Wang}, + Journal = {J. Chem. Phys.}, + Pages = {154102}, + Volume = {130}, + Year = {2009}} + +@article{ZimParKou-JCP-04, + Author = {U. Zimmerli and M. Parrinello and P. Koumoutsakos}, + Journal = {J. Chem. Phys.}, + Pages = {2693}, + Volume = {120}, + Year = {2004}} + +@article{ZimTouZhaMusUmr-JCP-09, + Author = {P. M. Zimmerman and J. Toulouse and Z. Zhang and C. B. Musgrave and C. J. Umrigar}, + Journal = {J. Chem. Phys.}, + Pages = {124103}, + Title = {Excited states of methylene from quantum Monte Carlo}, + Volume = {131}, + Year = {2009}} + +@article{ZucPodMosJezSza-JCP-08, + Author = {P. S. Zuchowski and R. Podeszwa and R. Moszynski and B. Jeziorski and K. Szalewicz}, + Journal = {J. Chem. Phys.}, + Pages = {084101}, + Title = {{Symmetry-adapted perturbation theory utilizing density functional description of monomers for high-spin open-shell complexes}}, + Volume = {129}, + Year = {2008}} diff --git a/Revised_Manuscript/data/C2/DFT_avdzE_relat-eps-converted-to.pdf b/Revised_Manuscript/data/C2/DFT_avdzE_relat-eps-converted-to.pdf new file mode 100644 index 0000000000000000000000000000000000000000..66a8e4e2f3a20e43bfd23558222455330acdc6c7 GIT binary patch literal 13914 zcmb_@2Ut_f_OCRNfCz#hC=hxlB-GH9-g^fD1BBiQMLLQi9Rx%H>4-Fuj(~#nj`ZF^ zdQ*CRJ08!uo^$X2|K9t)yYppd_L?(v?_x!@BCMClO_MK%&56p643)&s zV%E<>#d^R_kb7Z^Ltk2)1Fq~wUZE+0Uo-3O2(-N3*|A)7pHQ`NwnpFc1zPem4wyK z8V*Jk=W5~YL_tvso;#Af6p0b3TU<5g*ML}!OXkHoL z_i5aQ$_^Q4atYyb<17jx9a%(8-`>g6fi0hZbbj>xzDu8Y#57lQ1y|4XbL!E|G-@nq zv%v`btilvCKNbf2TR7T&tD&RDEH7@W^GZ!s$t5-vd{@)Sst7ZWGuc?R;Ns__GuJ(Q z)f;DaZug=a>w3jrR)xaRPJIPayUNtNO~Osb2T49x!aW)VNJNyp6bZ z)5Ww)XiuT%gIG(Ssqe$6#|&2=S1(wIx4hmwsp4XOM5&0e@$_!f^SjQIT+GklND4)G 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