From 8bdbfa5d11672d6429f339f4081831f51135f797 Mon Sep 17 00:00:00 2001 From: Emmanuel Giner Date: Sat, 12 Oct 2019 01:06:22 +0800 Subject: [PATCH] working on equations --- Manuscript/srDFT_SC.aux | 66 ++++++++++++++++++++++------------------- Manuscript/srDFT_SC.bbl | 11 ++++++- Manuscript/srDFT_SC.bib | 1 - Manuscript/srDFT_SC.blg | 64 +++++++++++++++++++-------------------- Manuscript/srDFT_SC.out | 4 +-- Manuscript/srDFT_SC.tex | 31 ++++++++++++++----- 6 files changed, 103 insertions(+), 74 deletions(-) diff --git a/Manuscript/srDFT_SC.aux b/Manuscript/srDFT_SC.aux index 796a401..6cd6bfe 100644 --- a/Manuscript/srDFT_SC.aux +++ b/Manuscript/srDFT_SC.aux @@ -66,6 +66,7 @@ \citation{TouColSav-PRA-04,GoriSav-PRA-06,PazMorGorBac-PRB-06} \citation{FerGinTou-JCP-18} \citation{GritMeePer-PRA-18} +\citation{CarTruGag-JPCA-17} \bibdata{srDFT_SCNotes,srDFT_SC} \bibcite{Thom-PRL-10}{{1}{2010}{{Thom}}{{}}} \bibcite{ScoTho-JCP-17}{{2}{2017}{{Scott\ and\ Thom}}{{}}} @@ -75,6 +76,23 @@ \bibcite{DeuEmiYumShePie-JCP-19}{{6}{2019}{{Deustua\ \emph {et~al.}}}{{Deustua, Yuwono, Shen,\ and\ Piecuch}}} \bibcite{QiuHenZhaScu-JCP-17}{{7}{2017}{{Qiu\ \emph {et~al.}}}{{Qiu, Henderson, Zhao,\ and\ Scuseria}}} \bibcite{QiuHenZhaScu-JCP-18}{{8}{2018}{{Qiu\ \emph {et~al.}}}{{Qiu, Henderson, Zhao,\ and\ Scuseria}}} +\@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 functionals $\mathaccentV {bar}916{E}^\mathcal {B}[{n}({\bf r})]$ }{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}{}} +\@writefile{toc}{\contentsline {subsubsection}{\numberline {2}Introduction of the effective spin-density}{4}{section*.10}} +\@writefile{toc}{\contentsline {subsubsection}{\numberline {3}Requirement for $\Psi _{}^{\mathcal {B}}$ for size extensivity}{4}{section*.11}} +\@writefile{toc}{\contentsline {section}{\numberline {III}Results}{4}{section*.12}} +\newlabel{sec:results}{{III}{4}{}{section*.12}{}} +\@writefile{toc}{\contentsline {section}{\numberline {IV}Conclusion}{4}{section*.13}} +\newlabel{sec:conclusion}{{IV}{4}{}{section*.13}{}} \bibcite{GomHenScu-JCP-19}{{9}{2019}{{Gomez, Henderson,\ and\ Scuseria}}{{}}} \bibcite{WerKno-JCP-88}{{10}{1988}{{Werner\ and\ Knowles}}{{}}} \bibcite{KnoWer-CPL-88}{{11}{1988}{{Knowles\ and\ Werner}}{{}}} @@ -92,19 +110,6 @@ \bibcite{BytRue-CP-09}{{23}{2009}{{Bytautas\ and\ Ruedenberg}}{{}}} \bibcite{GinSceCaf-CJC-13}{{24}{2013}{{Giner, Scemama,\ and\ Caffarel}}{{}}} \bibcite{CafGinScemRam-JCTC-14}{{25}{2014}{{Caffarel\ \emph {et~al.}}}{{Caffarel, Giner, Scemama,\ and\ Ram{\'\i }rez-Sol{\'\i }s}}} -\@writefile{toc}{\contentsline {subsection}{\numberline {C}Definition of an 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: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 of the approximations for functionals $\mathaccentV {bar}916{E}^\mathcal {B}[{n}({\bf r})]$}{4}{section*.9}} -\newlabel{eq:lim_muinf}{{19}{4}{}{equation.2.19}{}} -\@writefile{toc}{\contentsline {subsubsection}{\numberline {2}Introduction of the effective spin-density}{4}{section*.10}} -\@writefile{toc}{\contentsline {subsubsection}{\numberline {3}Requirement for $\Psi _{}^{\mathcal {B}}$ for size extensivity}{4}{section*.11}} -\@writefile{toc}{\contentsline {section}{\numberline {III}Results}{4}{section*.12}} -\newlabel{sec:results}{{III}{4}{}{section*.12}{}} -\@writefile{toc}{\contentsline {section}{\numberline {IV}Conclusion}{4}{section*.13}} -\newlabel{sec:conclusion}{{IV}{4}{}{section*.13}{}} \bibcite{GinSceCaf-JCP-15}{{26}{2015}{{Giner, Scemama,\ and\ Caffarel}}{{}}} \bibcite{CafAplGinScem-arxiv-16}{{27}{2016{}}{{Caffarel\ \emph {et~al.}}}{{Caffarel, Applencourt, Giner,\ and\ Scemama}}} \bibcite{CafAplGinSce-JCP-16}{{28}{2016{}}{{Caffarel\ \emph {et~al.}}}{{Caffarel, Applencourt, Giner,\ and\ Scemama}}} @@ -113,6 +118,10 @@ \bibcite{HolUmrSha-JCP-17}{{31}{2017}{{Holmes, Umrigar,\ and\ Sharma}}{{}}} \bibcite{ShaHolJeaAlaUmr-JCTC-17}{{32}{2017}{{Sharma\ \emph {et~al.}}}{{Sharma, Holmes, Jeanmairet, Alavi,\ and\ Umrigar}}} \bibcite{SchEva-JCTC-17}{{33}{2017}{{Schriber\ and\ Evangelista}}{{}}} +\@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{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{PerCle-JCP-17}{{34}{2017}{{Per\ and\ Cleland}}{{}}} \bibcite{OhtJun-JCP-17}{{35}{2017}{{Ohtsuka\ and\ ya~Hasegawa}}{{}}} \bibcite{Zim-JCP-17}{{36}{2017}{{Zimmerman}}{{}}} @@ -126,10 +135,6 @@ \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}}{{}}} -\@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{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{KutKlo-JCP-91}{{47}{1991}{{Kutzelnigg\ and\ Klopper}}{{}}} \bibcite{NogKut-JCP-94}{{48}{1994}{{Noga\ and\ Kutzelnigg}}{{}}} \bibcite{HalHelJorKloKocOlsWil-CPL-98}{{49}{1998}{{Halkier\ \emph {et~al.}}}{{Halkier, Helgaker, J{\o }rgensen, Klopper, Koch, Olsen,\ and\ Wilson}}} @@ -141,6 +146,10 @@ \bibcite{GruHirOhnTen-JCP-17}{{55}{2017}{{Gr\"uneis\ \emph {et~al.}}}{{Gr\"uneis, Hirata, Ohnishi,\ and\ Ten-no}}} \bibcite{MaWer-WIREs-18}{{56}{2018}{{Ma\ and\ Werner}}{{}}} \bibcite{TewKloNeiHat-PCCP-07}{{57}{2007}{{Tew\ \emph {et~al.}}}{{Tew, Klopper, Neiss,\ and\ Hattig}}} +\@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{TouColSav-PRA-04}{{58}{2004}{{Toulouse, Colonna,\ and\ Savin}}{{}}} \bibcite{FraMusLupTou-JCP-15}{{59}{2015}{{Franck\ \emph {et~al.}}}{{Franck, Mussard, Luppi,\ and\ Toulouse}}} \bibcite{AngGerSavTou-PRA-05}{{60}{2005}{{\'Angy\'an\ \emph {et~al.}}}{{\'Angy\'an, Gerber, Savin,\ and\ Toulouse}}} @@ -152,28 +161,25 @@ \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}}{{}}} \bibcite{FroCimJen-PRA-10}{{68}{2010}{{Fromager, Cimiraglia,\ and\ Jensen}}{{}}} -\@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{HedKneKieJenRei-JCP-15}{{69}{2015}{{Hedeg{\r a}rd\ \emph {et~al.}}}{{Hedeg{\r a}rd, Knecht, Kielberg, Jensen,\ and\ Reiher}}} \bibcite{HedTouJen-JCP-18}{{70}{2018}{{Hedeg{\r a}rd, Toulouse,\ and\ Jensen}}{{}}} \bibcite{FerGinTou-JCP-18}{{71}{2019}{{Fert{\'e}, Giner,\ and\ Toulouse}}{{}}} \bibcite{GinPraFerAssSavTou-JCP-18}{{72}{2018}{{Giner\ \emph {et~al.}}}{{Giner, Pradines, Fert\'e, Assaraf, Savin,\ and\ Toulouse}}} \bibcite{LooPraSceTouGin-JCPL-19}{{73}{2019{}}{{Loos\ \emph {et~al.}}}{{Loos, Pradines, Scemama, Toulouse,\ and\ Giner}}} \bibcite{TouGorSav-TCA-05}{{74}{2005}{{Toulouse, Gori-Giorgi,\ and\ Savin}}{{}}} -\bibcite{PerBurErn-PRL-96}{{75}{1996}{{Perdew, Burke,\ and\ Ernzerhof}}{{}}} -\bibcite{GoriSav-PRA-06}{{76}{2006}{{Gori-Giorgi\ and\ Savin}}{{}}} -\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}}{{}}} -\bibstyle{aipnum4-1} -\citation{REVTEX41Control} -\citation{aip41Control} \@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}{}} -\newlabel{LastBibItem}{{78}{7}{}{section*.13}{}} -\newlabel{LastPage}{{}{7}{}{}{}} +\bibcite{PerBurErn-PRL-96}{{75}{1996}{{Perdew, Burke,\ and\ Ernzerhof}}{{}}} +\bibcite{GoriSav-PRA-06}{{76}{2006}{{Gori-Giorgi\ and\ Savin}}{{}}} +\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}}{{}}} +\bibstyle{aipnum4-1} +\citation{REVTEX41Control} +\citation{aip41Control} +\newlabel{LastBibItem}{{79}{8}{}{section*.13}{}} +\newlabel{LastPage}{{}{8}{}{}{}} diff --git a/Manuscript/srDFT_SC.bbl b/Manuscript/srDFT_SC.bbl index 8697b0c..f02a3ba 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}{78}% +\begin{thebibliography}{79}% \makeatletter \providecommand \@ifxundefined [1]{% \@ifx{#1\undefined} @@ -835,4 +835,13 @@ {\doibase 10.1103/PhysRevA.98.062510} {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {98}},\ \bibinfo {pages} {062510} (\bibinfo {year} {2018})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Carlson}, \citenamefont {Truhlar},\ and\ + \citenamefont {Gagliardi}(2017)}]{CarTruGag-JPCA-17}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {R.~K.}\ \bibnamefont + {Carlson}}, \bibinfo {author} {\bibfnamefont {D.~G.}\ \bibnamefont + {Truhlar}}, \ and\ \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont + {Gagliardi}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J. + Phys. Chem. A}\ }\textbf {\bibinfo {volume} {121}},\ \bibinfo {pages} {5540} + (\bibinfo {year} {2017})}\BibitemShut {NoStop}% \end{thebibliography}% diff --git a/Manuscript/srDFT_SC.bib b/Manuscript/srDFT_SC.bib index 51f8ebf..8ee1f4a 100644 --- a/Manuscript/srDFT_SC.bib +++ b/Manuscript/srDFT_SC.bib @@ -12657,4 +12657,3 @@ eprint = {https://doi.org/10.1021/acs.jpclett.9b01176} doi = {10.1103/PhysRevA.98.062510}, url = {https://link.aps.org/doi/10.1103/PhysRevA.98.062510} } - diff --git a/Manuscript/srDFT_SC.blg b/Manuscript/srDFT_SC.blg index 06c3d14..8827a50 100644 --- a/Manuscript/srDFT_SC.blg +++ b/Manuscript/srDFT_SC.blg @@ -27,45 +27,45 @@ Control: page (0) single Control: year (1) truncated Control: production of eprint (0) enabled Warning--missing journal in CafAplGinScem-arxiv-16 -You've used 80 entries, +You've used 81 entries, 5918 wiz_defined-function locations, - 2180 strings with 31028 characters, -and the built_in function-call counts, 82782 in all, are: -= -- 5304 -> -- 2712 -< -- 506 -+ -- 842 -- -- 690 -* -- 12769 -:= -- 8522 -add.period$ -- 79 -call.type$ -- 80 -change.case$ -- 315 -chr.to.int$ -- 77 -cite$ -- 81 -duplicate$ -- 7366 -empty$ -- 5871 -format.name$ -- 1395 -if$ -- 16416 + 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 int.to.chr$ -- 4 -int.to.str$ -- 87 -missing$ -- 973 -newline$ -- 286 -num.names$ -- 234 -pop$ -- 3151 +int.to.str$ -- 88 +missing$ -- 985 +newline$ -- 289 +num.names$ -- 237 +pop$ -- 3194 preamble$ -- 1 -purify$ -- 390 +purify$ -- 395 quote$ -- 0 -skip$ -- 2923 +skip$ -- 2958 stack$ -- 0 -substring$ -- 2163 -swap$ -- 7163 -text.length$ -- 252 +substring$ -- 2189 +swap$ -- 7251 +text.length$ -- 255 text.prefix$ -- 0 top$ -- 10 -type$ -- 1122 +type$ -- 1136 warning$ -- 2 -while$ -- 310 +while$ -- 314 width$ -- 0 -write$ -- 686 +write$ -- 694 (There were 5 warnings) diff --git a/Manuscript/srDFT_SC.out b/Manuscript/srDFT_SC.out index d61d5e5..9ae1f2c 100644 --- a/Manuscript/srDFT_SC.out +++ b/Manuscript/srDFT_SC.out @@ -4,9 +4,9 @@ \BOOKMARK [1][-]{section*.4}{Theory}{section*.2}% 4 \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 an range-separation parameter varying in real space}{section*.4}% 7 +\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 of the approximations for functionals B[n\(r\)]}{section*.8}% 9 +\BOOKMARK [3][-]{section*.9}{Generic form and properties of the approximations for functionals 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 diff --git a/Manuscript/srDFT_SC.tex b/Manuscript/srDFT_SC.tex index d1d22c3..33884ec 100644 --- a/Manuscript/srDFT_SC.tex +++ b/Manuscript/srDFT_SC.tex @@ -368,7 +368,7 @@ As it was shown in \cite{GinPraFerAssSavTou-JCP-18}, the effective interaction $ \lim_{\Bas \rightarrow \text{CBS}} \wbasis = \frac{1}{|\br{1}-\br{2}|}. \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 an range-separation parameter varying in real space} +\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} @@ -377,6 +377,7 @@ As the effective interaction within a basis set $\wbasis$ is non divergent, one \end{equation} As originally proposed in \cite{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 @@ -391,17 +392,19 @@ Because of the very definition of $\wbasis$, one has the following properties at 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 of the approximations for functionals $\efuncden{\denr}$} -As originally proposed 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)\cite{TouGorSav-TCA-05}. -Here, we extend the recent work\cite{LooPraSceTouGin-JCPL-19} and 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 gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$ and the on-top pair density $n^{2}(\br{})$ taken from a given wave function $\psibasis$. +\subsubsection{Generic form and properties of the approximations for functionals $\efuncden{\denr}$ } +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 gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$ and the on-top pair density $n^{2}(\br{})$ taken from a given wave function $\psibasis$. Therefore, we take the following form for the approximation of $\efuncden{\denr}$: \begin{equation} \begin{aligned} + \label{eq:def_ecmdpbebasis} \efuncdenpbe{\argebasis} = &\int d\br{} \,\denr \\ & \ecmd(\argrebasis) \end{aligned} \end{equation} where $\ecmd(\argebasis)$ is the correlation energy density defined as \begin{equation} + \label{eq:def_ecmdpbe} \ecmd(\argebasis) = \frac{\varepsilon_{\text{c,PBE}}(\argepbe)}{1+ \mu^3 \beta(\argepbe)} \end{equation} with @@ -409,15 +412,27 @@ with \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 density\cite{PerBurErn-PRL-96}. -As initially proposed by some of the authors~\cite{FerGinTou-JCP-18}, such a correlation energy density admits the two following limits +The function $\ecmd(\argebasis)$ have been originally proposed by some of the authors~\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(\argebasis) = \varepsilon_{\text{c,PBE}}(\argepbe) \end{equation} +which can be qualified as the weak correlation regime, and \begin{equation} \label{eq:lim_muinf} - \lim_{\mu \rightarrow \infty} \ecmd(\argebasis) = \frac{1}{\mu^3} n^{2}. + \lim_{\mu \rightarrow \infty} \ecmd(\argebasis) = \frac{1}{\mu^3} n^{2} + o(\frac{1}{\mu^5}) \end{equation} -As it was previously shown\cite{TouColSav-PRA-04, GoriSav-PRA-06,PazMorGorBac-PRB-06}, the condition \eqref{eq:lim_muinf} is exact for the ECMD in the limit of large $\mu$ provided that $n^{2}$ is the \textit{exact} on-top pair density of the system. Therefore, in the present work we will approximate the \textit{exact} on-top pair density of the system by that computed by an approximated wave function $\psibasis$. In the context of RSDFT, some of the present authors have illustrated in Ref.~\cite{FerGinTou-JCP-18} that the on-top pair density plays a crucial role when reaching strong correlation limit, which have been also found in a related context by Pernal and co-workers\cite{GritMeePer-PRA-18}. +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 strong correlation limit. 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}. +Of course, the \textit{exact} on-top pair density of a system is rarely affordable and therefore, in the present work, we will approximate it 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 $\mu_{\Psi^{\basis}}$: +\begin{equation} + \label{eq:lim_ebasis} + \lim_{\basis \rightarrow \text{CBS}} \efuncdenpbe{\argebasis} = 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. \subsubsection{Introduction of the effective spin-density} \subsubsection{Requirement for $\wf{}{\Bas}$ for size extensivity} %%%%%%%%%%%%%%%%%%%%%%%%