From e1fc184d3d41001d7c1f054a484107390b5c177a Mon Sep 17 00:00:00 2001 From: Emmanuel Giner Date: Sun, 13 Oct 2019 19:00:39 +0800 Subject: [PATCH] working on equations ... --- Manuscript/srDFT_SC.aux | 73 +++++++++++++++---------- Manuscript/srDFT_SC.bbl | 13 ++++- Manuscript/srDFT_SC.blg | 67 +++++++++++------------ Manuscript/srDFT_SC.out | 18 ++++--- Manuscript/srDFT_SC.tex | 115 +++++++++++++++++++++++++++++++++------- 5 files changed, 200 insertions(+), 86 deletions(-) diff --git a/Manuscript/srDFT_SC.aux b/Manuscript/srDFT_SC.aux index 87cb759..b2f29cc 100644 --- a/Manuscript/srDFT_SC.aux +++ b/Manuscript/srDFT_SC.aux @@ -67,20 +67,26 @@ \citation{FerGinTou-JCP-18} \citation{GritMeePer-PRA-18} \citation{CarTruGag-JPCA-17} -\bibdata{srDFT_SCNotes,srDFT_SC} -\bibcite{Thom-PRL-10}{{1}{2010}{{Thom}}{{}}} \newlabel{eq:cbs_wbasis}{{10}{4}{}{equation.2.10}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {C}Definition of a range-separation parameter varying in real space}{4}{section*.7}} \newlabel{sec:mur}{{II\tmspace +\thinmuskip {.1667em}C}{4}{}{section*.7}{}} \newlabel{eq:weelr}{{11}{4}{}{equation.2.11}{}} \newlabel{eq:def_mur}{{12}{4}{}{equation.2.12}{}} \newlabel{eq:cbs_mu}{{14}{4}{}{equation.2.14}{}} -\@writefile{toc}{\contentsline {subsection}{\numberline {D}Approximation for $\mathaccentV {bar}916{E}^\mathcal {B}[{n}({\bf r})]$ : link with RSDFT}{4}{section*.8}} -\@writefile{toc}{\contentsline {subsubsection}{\numberline {1}Generic form and properties of the approximations for $\mathaccentV {bar}916{E}^\mathcal {B}[{n}({\bf r})]$ }{4}{section*.9}} +\@writefile{toc}{\contentsline {subsection}{\numberline {D}Generic form and properties of the approximations for $\mathaccentV {bar}916{E}^\mathcal {B}[{n}({\bf r})]$ }{4}{section*.8}} +\@writefile{toc}{\contentsline {subsubsection}{\numberline {1}Generic form of the approximated functionals}{4}{section*.9}} \newlabel{eq:def_ecmdpbebasis}{{15}{4}{}{equation.2.15}{}} \newlabel{eq:def_ecmdpbe}{{16}{4}{}{equation.2.16}{}} -\newlabel{eq:lim_muinf}{{19}{4}{}{equation.2.19}{}} -\newlabel{eq:lim_ebasis}{{20}{4}{}{equation.2.20}{}} +\newlabel{eq:def_beta}{{17}{4}{}{equation.2.17}{}} +\newlabel{eq:lim_mularge}{{19}{4}{}{equation.2.19}{}} +\newlabel{eq:lim_n2}{{20}{4}{}{equation.2.20}{}} +\newlabel{eq:lim_muinf}{{21}{4}{}{equation.2.21}{}} +\@writefile{toc}{\contentsline {subsubsection}{\numberline {2}Properties of approximated functionals}{4}{section*.10}} +\newlabel{eq:lim_ebasis}{{22}{4}{}{equation.2.22}{}} +\citation{GarBulHenScu-PCCP-15} +\citation{LooPraSceTouGin} +\bibdata{srDFT_SCNotes,srDFT_SC} +\bibcite{Thom-PRL-10}{{1}{2010}{{Thom}}{{}}} \bibcite{ScoTho-JCP-17}{{2}{2017}{{Scott\ and\ Thom}}{{}}} \bibcite{SpeNeuVigFraTho-JCP-18}{{3}{2018}{{Spencer\ \emph {et~al.}}}{{Spencer, Neufeld, Vigor, Franklin,\ and\ Thom}}} \bibcite{DeuEmiShePie-PRL-17}{{4}{2017}{{Deustua, Shen,\ and\ Piecuch}}{{}}} @@ -92,24 +98,34 @@ \bibcite{WerKno-JCP-88}{{10}{1988}{{Werner\ and\ Knowles}}{{}}} \bibcite{KnoWer-CPL-88}{{11}{1988}{{Knowles\ and\ Werner}}{{}}} \bibcite{BenErn-PhysRev-1969}{{12}{1969}{{Bender\ and\ Davidson}}{{}}} +\@writefile{toc}{\contentsline {subsection}{\numberline {E}Approximations for the strong correlation regime}{5}{section*.11}} +\@writefile{toc}{\contentsline {subsubsection}{\numberline {1}Requirements: separability of the energies and $S_z$ invariance}{5}{section*.12}} +\@writefile{toc}{\contentsline {subsubsection}{\numberline {2}Condition for the functional $\mathaccentV {bar}916{E}_{\text {PBE}}^\mathcal {B}[{n},\xi ,s,n^{(2)},\mu _{\Psi ^{\mathcal {B}}}]$ to obtain $S_z$ invariance}{5}{section*.13}} +\newlabel{eq:def_effspin}{{23}{5}{}{equation.2.23}{}} +\@writefile{toc}{\contentsline {subsubsection}{\numberline {3}Functionals for strong correlation}{5}{section*.14}} +\newlabel{eq:def_pbeueg}{{24}{5}{}{equation.2.24}{}} +\newlabel{eq:def_n2ueg}{{25}{5}{}{equation.2.25}{}} +\@writefile{toc}{\contentsline {subsection}{\numberline {F}Requirement on $\Psi ^{\mathcal {B}}$ for the extensivity of $\mu ({\bf r};\Psi _{}^{\mathcal {B}})$}{5}{section*.15}} +\@writefile{toc}{\contentsline {subsubsection}{\numberline {1}Introduction of the effective spin-density}{5}{section*.16}} +\@writefile{toc}{\contentsline {subsubsection}{\numberline {2}Requirement for $\Psi _{}^{\mathcal {B}}$ for size extensivity}{5}{section*.17}} +\@writefile{toc}{\contentsline {section}{\numberline {III}Results}{5}{section*.18}} +\newlabel{sec:results}{{III}{5}{}{section*.18}{}} +\@writefile{toc}{\contentsline {section}{\numberline {IV}Conclusion}{5}{section*.19}} +\newlabel{sec:conclusion}{{IV}{5}{}{section*.19}{}} \bibcite{WhiHac-JCP-1969}{{13}{1969}{{Whitten\ and\ Hackmeyer}}{{}}} \bibcite{HurMalRan-1973}{{14}{1973}{{Huron, Malrieu,\ and\ Rancurel}}{{}}} \bibcite{EvaDauMal-ChemPhys-83}{{15}{1983}{{Evangelisti, Daudey,\ and\ Malrieu}}{{}}} \bibcite{Cim-JCP-1985}{{16}{1985}{{Cimiraglia}}{{}}} \bibcite{Cim-JCC-1987}{{17}{1987}{{Cimiraglia\ and\ Persico}}{{}}} -\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces N$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{5}{figure.1}} -\newlabel{fig:N2_avdz}{{1}{5}{N$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.1}{}} -\@writefile{toc}{\contentsline {subsubsection}{\numberline {2}Introduction of the effective spin-density}{5}{section*.10}} -\@writefile{toc}{\contentsline {subsubsection}{\numberline {3}Requirement for $\Psi _{}^{\mathcal {B}}$ for size extensivity}{5}{section*.11}} -\@writefile{toc}{\contentsline {section}{\numberline {III}Results}{5}{section*.12}} -\newlabel{sec:results}{{III}{5}{}{section*.12}{}} -\@writefile{toc}{\contentsline {section}{\numberline {IV}Conclusion}{5}{section*.13}} -\newlabel{sec:conclusion}{{IV}{5}{}{section*.13}{}} -\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces N$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{5}{figure.2}} -\newlabel{fig:N2_avtz}{{2}{5}{N$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.2}{}} \bibcite{IllRubRic-JCP-88}{{18}{1988}{{Illas, Rubio,\ and\ Ricart}}{{}}} \bibcite{PovRubIll-TCA-92}{{19}{1992}{{Povill, Rubio,\ and\ Illas}}{{}}} \bibcite{BunCarRam-JCP-06}{{20}{2006}{{Bunge\ and\ Carb{\'o}-Dorca}}{{}}} +\@writefile{lot}{\contentsline {table}{\numberline {I}{\ignorespaces Total energies (in Hartree) for HF and $E$ in aug-cc-pvdz for the He atom, F$_2$ (with F-F=1.411 angstroms) and the super non interacting system He--F$_2$. }}{6}{table.1}} +\newlabel{conv_He_table}{{I}{6}{Total energies (in Hartree) for HF and $E$ in aug-cc-pvdz for the He atom, F$_2$ (with F-F=1.411 angstroms) and the super non interacting system He--F$_2$}{table.1}{}} +\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces N$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{6}{figure.1}} +\newlabel{fig:N2_avdz}{{1}{6}{N$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.1}{}} +\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces N$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{6}{figure.2}} +\newlabel{fig:N2_avtz}{{2}{6}{N$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.2}{}} \bibcite{AbrSheDav-CPL-05}{{21}{2005}{{Abrams\ and\ Sherrill}}{{}}} \bibcite{MusEngels-JCC-06}{{22}{2006}{{Musch\ and\ Engels}}{{}}} \bibcite{BytRue-CP-09}{{23}{2009}{{Bytautas\ and\ Ruedenberg}}{{}}} @@ -132,11 +148,11 @@ \bibcite{LooSceBloGarCafJac-JCTC-18}{{40}{2018}{{Loos\ \emph {et~al.}}}{{Loos, Scemama, Blondel, Garniron, Caffarel,\ and\ Jacquemin}}} \bibcite{GarSceGinCaffLoo-JCP-18}{{41}{2018}{{Garniron\ \emph {et~al.}}}{{Garniron, Scemama, Giner, Caffarel,\ and\ Loos}}} \bibcite{SceGarCafLoo-JCTC-18}{{42}{2018{}}{{Scemama\ \emph {et~al.}}}{{Scemama, Garniron, Caffarel,\ and\ Loos}}} -\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces F$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{6}{figure.3}} -\newlabel{fig:F2_avdz}{{3}{6}{F$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.3}{}} -\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces F$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{6}{figure.4}} -\newlabel{fig:F2_avtz}{{4}{6}{F$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.4}{}} \bibcite{GarGinMalSce-JCP-16}{{43}{2017}{{Garniron\ \emph {et~al.}}}{{Garniron, Giner, Malrieu,\ and\ Scemama}}} +\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces F$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{7}{figure.3}} +\newlabel{fig:F2_avdz}{{3}{7}{F$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.3}{}} +\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces F$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{7}{figure.4}} +\newlabel{fig:F2_avtz}{{4}{7}{F$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.4}{}} \bibcite{LooBogSceCafJac-JCTC-19}{{44}{2019{}}{{Loos\ \emph {et~al.}}}{{Loos, Boggio-Pasqua, Scemama, Caffarel,\ and\ Jacquemin}}} \bibcite{Hyl-ZP-29}{{45}{1929}{{Hylleraas}}{{}}} \bibcite{Kut-TCA-85}{{46}{1985}{{Kutzelnigg}}{{}}} @@ -157,13 +173,13 @@ \bibcite{GolWerSto-PCCP-05}{{61}{2005}{{Goll, Werner,\ and\ Stoll}}{{}}} \bibcite{TouGerJanSavAng-PRL-09}{{62}{2009}{{Toulouse\ \emph {et~al.}}}{{Toulouse, Gerber, Jansen, Savin,\ and\ \'Angy\'an}}} \bibcite{JanHenScu-JCP-09}{{63}{2009}{{Janesko, Henderson,\ and\ Scuseria}}{{}}} -\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces H$_{10}$, cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{7}{figure.5}} -\newlabel{fig:H10_vdz}{{5}{7}{H$_{10}$, cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.5}{}} -\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces H$_{10}$, cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{7}{figure.6}} -\newlabel{fig:H10_vtz}{{6}{7}{H$_{10}$, cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.6}{}} -\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces H$_{10}$, cc-pvqz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{7}{figure.7}} -\newlabel{fig:H10_vqz}{{7}{7}{H$_{10}$, cc-pvqz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.7}{}} \bibcite{TouZhuSavJanAng-JCP-11}{{64}{2011}{{Toulouse\ \emph {et~al.}}}{{Toulouse, Zhu, Savin, Jansen,\ and\ \'Angy\'an}}} +\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces H$_{10}$, cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{8}{figure.5}} +\newlabel{fig:H10_vdz}{{5}{8}{H$_{10}$, cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.5}{}} +\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces H$_{10}$, cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{8}{figure.6}} +\newlabel{fig:H10_vtz}{{6}{8}{H$_{10}$, cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.6}{}} +\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces H$_{10}$, cc-pvqz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{8}{figure.7}} +\newlabel{fig:H10_vqz}{{7}{8}{H$_{10}$, cc-pvqz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.7}{}} \bibcite{MusReiAngTou-JCP-15}{{65}{2015}{{Mussard\ \emph {et~al.}}}{{Mussard, Reinhardt, \'Angy\'an,\ and\ Toulouse}}} \bibcite{LeiStoWerSav-CPL-97}{{66}{1997}{{Leininger\ \emph {et~al.}}}{{Leininger, Stoll, Werner,\ and\ Savin}}} \bibcite{FroTouJen-JCP-07}{{67}{2007}{{Fromager, Toulouse,\ and\ Jensen}}{{}}} @@ -179,8 +195,9 @@ \bibcite{PazMorGorBac-PRB-06}{{77}{2006}{{Paziani\ \emph {et~al.}}}{{Paziani, Moroni, Gori-Giorgi,\ and\ Bachelet}}} \bibcite{GritMeePer-PRA-18}{{78}{2018}{{Gritsenko, van Meer,\ and\ Pernal}}{{}}} \bibcite{CarTruGag-JPCA-17}{{79}{2017}{{Carlson, Truhlar,\ and\ Gagliardi}}{{}}} +\bibcite{GarBulHenScu-PCCP-15}{{80}{2015}{{Garza\ \emph {et~al.}}}{{Garza, Bulik, Henderson,\ and\ Scuseria}}} \bibstyle{aipnum4-1} \citation{REVTEX41Control} \citation{aip41Control} -\newlabel{LastBibItem}{{79}{8}{}{section*.13}{}} -\newlabel{LastPage}{{}{8}{}{}{}} +\newlabel{LastBibItem}{{80}{9}{}{section*.19}{}} +\newlabel{LastPage}{{}{9}{}{}{}} diff --git a/Manuscript/srDFT_SC.bbl b/Manuscript/srDFT_SC.bbl index f02a3ba..2f61993 100644 --- a/Manuscript/srDFT_SC.bbl +++ b/Manuscript/srDFT_SC.bbl @@ -6,7 +6,7 @@ %Control: page (0) single %Control: year (1) truncated %Control: production of eprint (0) enabled -\begin{thebibliography}{79}% +\begin{thebibliography}{80}% \makeatletter \providecommand \@ifxundefined [1]{% \@ifx{#1\undefined} @@ -844,4 +844,15 @@ {Gagliardi}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J. Phys. Chem. A}\ }\textbf {\bibinfo {volume} {121}},\ \bibinfo {pages} {5540} (\bibinfo {year} {2017})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Garza}\ \emph {et~al.}(2015)\citenamefont {Garza}, + \citenamefont {Bulik}, \citenamefont {Henderson},\ and\ \citenamefont + {Scuseria}}]{GarBulHenScu-PCCP-15}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {A.~J.}\ \bibnamefont + {Garza}}, \bibinfo {author} {\bibfnamefont {I.~W.}\ \bibnamefont {Bulik}}, + \bibinfo {author} {\bibfnamefont {T.~M.}\ \bibnamefont {Henderson}}, \ and\ + \bibinfo {author} {\bibfnamefont {G.~E.}\ \bibnamefont {Scuseria}},\ + }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Phys. Chem. Chem. + Phys.}\ }\textbf {\bibinfo {volume} {17}},\ \bibinfo {pages} {22412} + (\bibinfo {year} {2015})}\BibitemShut {NoStop}% \end{thebibliography}% diff --git a/Manuscript/srDFT_SC.blg b/Manuscript/srDFT_SC.blg index 8827a50..b14d44f 100644 --- a/Manuscript/srDFT_SC.blg +++ b/Manuscript/srDFT_SC.blg @@ -14,6 +14,7 @@ Database file #2: srDFT_SC.bib Warning--I didn't find a database entry for "exicted" Warning--I didn't find a database entry for "excited" Warning--I didn't find a database entry for "kato" +Warning--I didn't find a database entry for "LooPraSceTouGin" 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 @@ -27,45 +28,45 @@ Control: page (0) single Control: year (1) truncated Control: production of eprint (0) enabled Warning--missing journal in CafAplGinScem-arxiv-16 -You've used 81 entries, +You've used 82 entries, 5918 wiz_defined-function locations, - 2185 strings with 31117 characters, -and the built_in function-call counts, 83825 in all, are: -= -- 5372 -> -- 2749 -< -- 512 -+ -- 853 -- -- 699 -* -- 12928 -:= -- 8630 -add.period$ -- 80 -call.type$ -- 81 -change.case$ -- 319 -chr.to.int$ -- 78 -cite$ -- 82 -duplicate$ -- 7458 -empty$ -- 5945 -format.name$ -- 1413 -if$ -- 16624 + 2192 strings with 31260 characters, +and the built_in function-call counts, 84952 in all, are: += -- 5441 +> -- 2788 +< -- 519 ++ -- 865 +- -- 709 +* -- 13106 +:= -- 8744 +add.period$ -- 81 +call.type$ -- 82 +change.case$ -- 323 +chr.to.int$ -- 79 +cite$ -- 83 +duplicate$ -- 7558 +empty$ -- 6026 +format.name$ -- 1434 +if$ -- 16850 int.to.chr$ -- 4 -int.to.str$ -- 88 -missing$ -- 985 -newline$ -- 289 -num.names$ -- 237 -pop$ -- 3194 +int.to.str$ -- 89 +missing$ -- 998 +newline$ -- 292 +num.names$ -- 240 +pop$ -- 3236 preamble$ -- 1 -purify$ -- 395 +purify$ -- 400 quote$ -- 0 -skip$ -- 2958 +skip$ -- 2995 stack$ -- 0 -substring$ -- 2189 -swap$ -- 7251 -text.length$ -- 255 +substring$ -- 2218 +swap$ -- 7349 +text.length$ -- 259 text.prefix$ -- 0 top$ -- 10 -type$ -- 1136 +type$ -- 1151 warning$ -- 2 -while$ -- 314 +while$ -- 318 width$ -- 0 -write$ -- 694 -(There were 5 warnings) +write$ -- 702 +(There were 6 warnings) diff --git a/Manuscript/srDFT_SC.out b/Manuscript/srDFT_SC.out index 91d198e..c18a19b 100644 --- a/Manuscript/srDFT_SC.out +++ b/Manuscript/srDFT_SC.out @@ -5,9 +5,15 @@ \BOOKMARK [2][-]{section*.5}{Basic formal equations}{section*.4}% 5 \BOOKMARK [2][-]{section*.6}{Definition of an effective interaction within B}{section*.4}% 6 \BOOKMARK [2][-]{section*.7}{Definition of a range-separation parameter varying in real space}{section*.4}% 7 -\BOOKMARK [2][-]{section*.8}{Approximation for B[n\(r\)] : link with RSDFT}{section*.4}% 8 -\BOOKMARK [3][-]{section*.9}{Generic form and properties of the approximations for B[n\(r\)] }{section*.8}% 9 -\BOOKMARK [3][-]{section*.10}{Introduction of the effective spin-density}{section*.8}% 10 -\BOOKMARK [3][-]{section*.11}{Requirement for B for size extensivity}{section*.8}% 11 -\BOOKMARK [1][-]{section*.12}{Results}{section*.2}% 12 -\BOOKMARK [1][-]{section*.13}{Conclusion}{section*.2}% 13 +\BOOKMARK [2][-]{section*.8}{Generic form and properties of the approximations for B[n\(r\)] }{section*.4}% 8 +\BOOKMARK [3][-]{section*.9}{Generic form of the approximated functionals}{section*.8}% 9 +\BOOKMARK [3][-]{section*.10}{Properties of approximated functionals}{section*.8}% 10 +\BOOKMARK [2][-]{section*.11}{Approximations for the strong correlation regime}{section*.4}% 11 +\BOOKMARK [3][-]{section*.12}{Requirements: separability of the energies and Sz invariance}{section*.11}% 12 +\BOOKMARK [3][-]{section*.13}{Condition for the functional PBEB[n,,s,n\(2\),B] to obtain Sz invariance}{section*.11}% 13 +\BOOKMARK [3][-]{section*.14}{Functionals for strong correlation}{section*.11}% 14 +\BOOKMARK [2][-]{section*.15}{Requirement on B for the extensivity of \(r;B\)}{section*.4}% 15 +\BOOKMARK [3][-]{section*.16}{Introduction of the effective spin-density}{section*.15}% 16 +\BOOKMARK [3][-]{section*.17}{Requirement for B for size extensivity}{section*.15}% 17 +\BOOKMARK [1][-]{section*.18}{Results}{section*.2}% 18 +\BOOKMARK [1][-]{section*.19}{Conclusion}{section*.2}% 19 diff --git a/Manuscript/srDFT_SC.tex b/Manuscript/srDFT_SC.tex index 3923565..a9c1a4c 100644 --- a/Manuscript/srDFT_SC.tex +++ b/Manuscript/srDFT_SC.tex @@ -69,8 +69,11 @@ \newcommand{\efuncdenpbe}[1]{\bar{E}_{\text{PBE}}^\Bas[#1]} \newcommand{\ecompmodel}[0]{\bar{E}^\Bas[\denmodel]} \newcommand{\argepbe}[0]{\den,\xi,s} -\newcommand{\argebasis}[0]{\den,\xi,s,n^{2},\mu_{\Psi^{\basis}}} -\newcommand{\argrebasis}[0]{\denr,\xi(\br{}),s,n^{2}(\br{}),\mu_{\Psi^{\basis}}(\br{})} +\newcommand{\argebasis}[0]{\den,\xi,s,\ntwo,\mu_{\Psi^{\basis}}} +\newcommand{\argecmd}[0]{\den,\xi,s,\ntwo,\mu} +\newcommand{\argepbeueg}[0]{\den,\xi,s,\ntwo_{\text{UEG}},\mu_{\Psi^{\basis}}} +\newcommand{\argepbeuegspin}[0]{\den,\Xi,s,\ntwo_{\text{UEG}},\mu_{\Psi^{\basis}}} +\newcommand{\argrebasis}[0]{\denr,\xi(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})} \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]} @@ -102,6 +105,7 @@ % 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{\mur}[0]{\mu({\bf r})} \newcommand{\murr}[1]{\mu({\bf r}_{#1})} \newcommand{\murval}[0]{\mu_{\text{val}}({\bf r})} @@ -215,6 +219,7 @@ \newcommand{\br}[1]{{\mathbf{r}_{#1}}} \newcommand{\bx}[1]{\mathbf{x}_{#1}} \newcommand{\dbr}[1]{d\br{#1}} +\newcommand{\PBEspin}{PBEspin} \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} @@ -393,10 +398,10 @@ Because of the very definition of $\wbasis$, one has the following properties at \end{equation} which is fundamental to guarantee the good behaviour of the theory at the CBS limit. -\subsection{Approximation for $\efuncden{\denr}$ : link with RSDFT} -\subsubsection{Generic form and properties of the approximations for $\efuncden{\denr}$ } +\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 $\xi(\br{}) = n_{\alpha}(\br{}) - n_{\beta}(\br{})$, reduced density gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$ and the on-top pair density $n^{2}(\br{})$. In the present work, unless explicitly stated the quantities $\denr$, $\xi(\br{})$, $s(\br{})$ and $n^{2}(\br{})$ will be computed from the wave function $\psibasis$ used to define $\murpsi$. +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 $\xi(\br{}) = n_{\alpha}(\br{}) - n_{\beta}(\br{})$, reduced density gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$ and the on-top pair density $n^{2}(\br{})$. In the present work, the quantities $\denr$, $\xi(\br{})$, $s(\br{})$ and $n^{2}(\br{})$ are be computed from the same wave function $\psibasis$ used to define $\murpsi$. The generic form for the approximations to $\efuncden{\denr}$ proposed here reads \begin{equation} \begin{aligned} @@ -404,37 +409,111 @@ The generic form for the approximations to $\efuncden{\denr}$ proposed here read \efuncdenpbe{\argebasis} = &\int d\br{} \,\denr \\ & \ecmd(\argrebasis) \end{aligned} \end{equation} -where $\ecmd(\argebasis)$ is the ECMD correlation energy density defined as +where $\ecmd(\argecmd)$ is the ECMD correlation energy density defined as \begin{equation} \label{eq:def_ecmdpbe} - \ecmd(\argebasis) = \frac{\varepsilon_{\text{c,PBE}}(\argepbe)}{1+ \mu^3 \beta(\argepbe)} + \ecmd(\argecmd) = \frac{\varepsilon_{\text{c,PBE}}(\argepbe)}{1+ \mu^3 \beta(\argepbe)} \end{equation} with \begin{equation} + \label{eq:def_beta} \beta(\argepbe) = \frac{3}{2\sqrt{\pi}(1 - \sqrt{2})}\frac{\varepsilon_{\text{c,PBE}}(\argepbe)}{n^{2}/\den}, \end{equation} and where $\varepsilon_{\text{c,PBE}}(\argepbe)$ is the usual PBE correlation energy density\cite{PerBurErn-PRL-96}. -The actual functional form of $\ecmd(\argebasis)$ 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 +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(\argebasis) = \varepsilon_{\text{c,PBE}}(\argepbe), + \lim_{\mu \rightarrow 0} \ecmd(\argecmd) = \varepsilon_{\text{c,PBE}}(\argepbe), \end{equation} -which can be qualified as the weak correlation regime, and +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} n^{2} + 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 $n^{2}$ 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 vanishing spin 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(\argebasis) = \frac{1}{\mu^3} n^{2} + o(\frac{1}{\mu^5}), + \lim_{\mu \rightarrow \infty} \ecmd(\argecmd) = 0. \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 $n^{2}$ 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.~\cite{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}. -For equation \eqref{eq:lim_muinf} to be exact, the \textit{exact} on-top pair density $n^{2}$ of the physical system is needed, which is of course rarely affordable and therefore, in the present work, it will be approximated by that computed by an approximated wave function $\psibasis$. +%For equation \eqref{eq:lim_muinf} to be exact, the \textit{exact} on-top pair density $n^{2}$ of the physical system is needed, which is of course rarely affordable and therefore, in the present work, it will be approximated by that computed by an approximated wave function $\psibasis$. -Within the definition of \eqref{eq:def_mur} and \eqref{eq:def_ecmdpbebasis}, the approximated complementary basis set functionals $\efuncdenpbe{\argebasis}$ satisfies two important properties. -Because of the properties \eqref{eq:cbs_mu} and \eqref{eq:lim_muinf}, $\efuncdenpbe{\argebasis}$ vanishes when reaching the complete basis set limit, whatever the wave function $\psibasis$ used to define the range separation parameter $\mu_{\Psi^{\basis}}$: +\subsubsection{Properties of approximated functionals} +Within the definition of \eqref{eq:def_mur} and \eqref{eq:def_ecmdpbebasis}, the 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{\argebasis} = 0\quad \forall\, \psibasis. + \lim_{\basis \rightarrow \text{CBS}} \efuncdenpbe{\argecmd} = 0\quad \forall\, \psibasis, \end{equation} which guarantees an unaltered limit when reaching the CBS limit. -Also, because of eq. \eqref{eq:lim_muinf}, $\efuncdenpbe{\argebasis}$ 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. +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{Approximations for 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 any quantity related to $S_z$, which is the spin density $s(\b{})$ in the case of the definition $\ecmd(\argecmd)$. +The spin density is involved in the usual PBE correlation functional $\varepsilon_{\text{c,PBE}}(\argepbe)$ which contributes to the definition of $\ecmd(\argecmd)$ (see equation \eqref{eq:def_ecmdpbe}). A possible way to eliminate the $S_z$ dependency would be then to simply set $\xi(\br{})=0$, but this would lower the accuracy of the usual PBE correlation functional $\varepsilon_{\text{c,PBE}}(\argepbe)$. Therefore, we use the proposal by Scuseria and co-workers\cite{GarBulHenScu-PCCP-15} which introduce an effective spin density depending on the on-top pair density and the total density +\begin{equation} + \label{eq:def_effspin} + \Xi(n,\ntwo) = + \begin{cases} + \sqrt{ n^2 - 4 \ntwo }. & \text{if $n^2 - 4 \ntwo > 0$,} \\ + 0 & \text{otherwise.} + \end{cases} +\end{equation} +The definition of equation \eqref{eq:def_effspin} is exact when $n$ and $\ntwo$ are obtained from a single Slater determinant. +With this definition, the $\Xi(n,\ntwo)$ depends only on $S_z$ invariants quantities, which naturally makes it $S_z$ invariant. +We also propose + +\subsubsection{Functionals for strong correlation} +The first one, referred as the \PBEspin functional, is a natural extension with effective spin density of the previously introduced PBE\cite{LooPraSceTouGin}, +\begin{equation} + \label{eq:def_pbeueg} + \begin{aligned} + \efuncdenpbe{\argepbeuegspin} = &\int d\br{} \,\denr \\ & \ecmd(\argepbeuegspin) + \end{aligned} +\end{equation} +where $\ntwo_{\text{UEG}}$ is the on-top pair density of the UEG defined as +\begin{equation} + \label{eq:def_n2ueg} + \ntwo_{\text{UEG}}(n,\xi) = n^2(1-\xi)g_0(n). +\end{equation} +Therefore, such a functional used the on-top pair density of the UEG computed with the total density and effective spin density which depends on the on-top pair density. + +\subsection{Requirement on $\psibasis$ for the extensivity of $\murpsi$} +\begin{table*} +\caption{Total energies (in Hartree) for HF and $E$ in aug-cc-pvdz for the He atom, F$_2$ (with F-F=1.411 angstroms) and the super non interacting system He--F$_2$. } +\begin{tabular}{lcc} +%\hline + System & HF & $E$ \\ +\hline +F$_2$ & -2.85570466771188 & -0.0112667838948910 \\ +He & -198.698792752661 & -0.1596345827582842 \\ +He $\ldots$ F$_2$ & -201.554497420371 & -0.1709013666531826 \\ + \hline +Error to additivity & 1.2 $\times 10^{-12}$ & 7 $\times 10^{-15}$ \\ +\end{tabular} +\label{conv_He_table} +\end{table*} + + +In the case of the present basis set correction, as $\murpsi$ is a local quantity, one of the necessary but not sufficient requirements for extensivity is that $\murpsi$ must be the same on the system $A$ that in the subsystem $A$ of the super system $A+B$ in the limit of non interacting fragments. + + + \subsubsection{Introduction of the effective spin-density} \subsubsection{Requirement for $\wf{}{\Bas}$ for size extensivity} %%%%%%%%%%%%%%%%%%%%%%%%