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Emmanuel Giner 2019-10-12 01:06:22 +08:00
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Manuscript/srDFT_SC.aux
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@ -66,6 +66,7 @@
\citation{TouColSav-PRA-04,GoriSav-PRA-06,PazMorGorBac-PRB-06} \citation{TouColSav-PRA-04,GoriSav-PRA-06,PazMorGorBac-PRB-06}
\citation{FerGinTou-JCP-18} \citation{FerGinTou-JCP-18}
\citation{GritMeePer-PRA-18} \citation{GritMeePer-PRA-18}
\citation{CarTruGag-JPCA-17}
\bibdata{srDFT_SCNotes,srDFT_SC} \bibdata{srDFT_SCNotes,srDFT_SC}
\bibcite{Thom-PRL-10}{{1}{2010}{{Thom}}{{}}} \bibcite{Thom-PRL-10}{{1}{2010}{{Thom}}{{}}}
\bibcite{ScoTho-JCP-17}{{2}{2017}{{Scott\ and\ 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{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-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}}} \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}}
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\@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}{}}
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\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}}
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\@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{GomHenScu-JCP-19}{{9}{2019}{{Gomez, Henderson,\ and\ Scuseria}}{{}}}
\bibcite{WerKno-JCP-88}{{10}{1988}{{Werner\ and\ Knowles}}{{}}} \bibcite{WerKno-JCP-88}{{10}{1988}{{Werner\ and\ Knowles}}{{}}}
\bibcite{KnoWer-CPL-88}{{11}{1988}{{Knowles\ and\ Werner}}{{}}} \bibcite{KnoWer-CPL-88}{{11}{1988}{{Knowles\ and\ Werner}}{{}}}
@ -92,19 +110,6 @@
\bibcite{BytRue-CP-09}{{23}{2009}{{Bytautas\ and\ Ruedenberg}}{{}}} \bibcite{BytRue-CP-09}{{23}{2009}{{Bytautas\ and\ Ruedenberg}}{{}}}
\bibcite{GinSceCaf-CJC-13}{{24}{2013}{{Giner, Scemama,\ and\ Caffarel}}{{}}} \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}}} \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}}
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\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{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{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}}} \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{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{ShaHolJeaAlaUmr-JCTC-17}{{32}{2017}{{Sharma\ \emph {et~al.}}}{{Sharma, Holmes, Jeanmairet, Alavi,\ and\ Umrigar}}}
\bibcite{SchEva-JCTC-17}{{33}{2017}{{Schriber\ and\ Evangelista}}{{}}} \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{PerCle-JCP-17}{{34}{2017}{{Per\ and\ Cleland}}{{}}}
\bibcite{OhtJun-JCP-17}{{35}{2017}{{Ohtsuka\ and\ ya~Hasegawa}}{{}}} \bibcite{OhtJun-JCP-17}{{35}{2017}{{Ohtsuka\ and\ ya~Hasegawa}}{{}}}
\bibcite{Zim-JCP-17}{{36}{2017}{{Zimmerman}}{{}}} \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{LooBogSceCafJac-JCTC-19}{{44}{2019{}}{{Loos\ \emph {et~al.}}}{{Loos, Boggio-Pasqua, Scemama, Caffarel,\ and\ Jacquemin}}}
\bibcite{Hyl-ZP-29}{{45}{1929}{{Hylleraas}}{{}}} \bibcite{Hyl-ZP-29}{{45}{1929}{{Hylleraas}}{{}}}
\bibcite{Kut-TCA-85}{{46}{1985}{{Kutzelnigg}}{{}}} \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{KutKlo-JCP-91}{{47}{1991}{{Kutzelnigg\ and\ Klopper}}{{}}}
\bibcite{NogKut-JCP-94}{{48}{1994}{{Noga\ and\ Kutzelnigg}}{{}}} \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}}} \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{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{MaWer-WIREs-18}{{56}{2018}{{Ma\ and\ Werner}}{{}}}
\bibcite{TewKloNeiHat-PCCP-07}{{57}{2007}{{Tew\ \emph {et~al.}}}{{Tew, Klopper, Neiss,\ and\ Hattig}}} \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}}
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\@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{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{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}}} \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{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{FroTouJen-JCP-07}{{67}{2007}{{Fromager, Toulouse,\ and\ Jensen}}{{}}}
\bibcite{FroCimJen-PRA-10}{{68}{2010}{{Fromager, Cimiraglia,\ and\ Jensen}}{{}}} \bibcite{FroCimJen-PRA-10}{{68}{2010}{{Fromager, Cimiraglia,\ and\ Jensen}}{{}}}
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\bibcite{HedKneKieJenRei-JCP-15}{{69}{2015}{{Hedeg{\r a}rd\ \emph {et~al.}}}{{Hedeg{\r a}rd, Knecht, Kielberg, Jensen,\ and\ Reiher}}} \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{HedTouJen-JCP-18}{{70}{2018}{{Hedeg{\r a}rd, Toulouse,\ and\ Jensen}}{{}}}
\bibcite{FerGinTou-JCP-18}{{71}{2019}{{Fert{\'e}, Giner,\ and\ Toulouse}}{{}}} \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{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{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{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}}{{}}}
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@ -6,7 +6,7 @@
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{journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {98}},\ \bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {98}},\ \bibinfo
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@ -12657,4 +12657,3 @@ eprint = {https://doi.org/10.1021/acs.jpclett.9b01176}
doi = {10.1103/PhysRevA.98.062510}, doi = {10.1103/PhysRevA.98.062510},
url = {https://link.aps.org/doi/10.1103/PhysRevA.98.062510} url = {https://link.aps.org/doi/10.1103/PhysRevA.98.062510}
} }

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@ -27,45 +27,45 @@ Control: page (0) single
Control: year (1) truncated Control: year (1) truncated
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Warning--missing journal in CafAplGinScem-arxiv-16 Warning--missing journal in CafAplGinScem-arxiv-16
You've used 80 entries, You've used 81 entries,
5918 wiz_defined-function locations, 5918 wiz_defined-function locations,
2180 strings with 31028 characters, 2185 strings with 31117 characters,
and the built_in function-call counts, 82782 in all, are: and the built_in function-call counts, 83825 in all, are:
= -- 5304 = -- 5372
> -- 2712 > -- 2749
< -- 506 < -- 512
+ -- 842 + -- 853
- -- 690 - -- 699
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:= -- 8522 := -- 8630
add.period$ -- 79 add.period$ -- 80
call.type$ -- 80 call.type$ -- 81
change.case$ -- 315 change.case$ -- 319
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format.name$ -- 1395 format.name$ -- 1413
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int.to.chr$ -- 4 int.to.chr$ -- 4
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Manuscript/srDFT_SC.out
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@ -4,9 +4,9 @@
\BOOKMARK [1][-]{section*.4}{Theory}{section*.2}% 4 \BOOKMARK [1][-]{section*.4}{Theory}{section*.2}% 4
\BOOKMARK [2][-]{section*.5}{Basic formal equations}{section*.4}% 5 \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*.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 [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*.10}{Introduction of the effective spin-density}{section*.8}% 10
\BOOKMARK [3][-]{section*.11}{Requirement for B for size extensivity}{section*.8}% 11 \BOOKMARK [3][-]{section*.11}{Requirement for B for size extensivity}{section*.8}% 11
\BOOKMARK [1][-]{section*.12}{Results}{section*.2}% 12 \BOOKMARK [1][-]{section*.12}{Results}{section*.2}% 12

31
Manuscript/srDFT_SC.tex
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@ -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}|}. \lim_{\Bas \rightarrow \text{CBS}} \wbasis = \frac{1}{|\br{1}-\br{2}|}.
\end{equation} \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. 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} \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 $\mu$ 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} \begin{equation}
@ -377,6 +377,7 @@ As the effective interaction within a basis set $\wbasis$ is non divergent, one
\end{equation} \end{equation}
As originally proposed in \cite{GinPraFerAssSavTou-JCP-18}, we introduce a range-separation parameter $\murpsi$ varying in real space As originally proposed in \cite{GinPraFerAssSavTou-JCP-18}, we introduce a range-separation parameter $\murpsi$ varying in real space
\begin{equation} \begin{equation}
\label{eq:def_mur}
\murpsi = \frac{\sqrt{\pi}}{2} \wbasiscoal \murpsi = \frac{\sqrt{\pi}}{2} \wbasiscoal
\end{equation} \end{equation}
such that 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. which is fundamental to guarantee the good behaviour of the theory at the CBS limit.
\subsection{Approximation for $\efuncden{\denr}$ : link with RSDFT} \subsection{Approximation for $\efuncden{\denr}$ : link with RSDFT}
\subsubsection{Generic form of the approximations for functionals $\efuncden{\denr}$} \subsubsection{Generic form and properties 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}. 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}.
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 $\psibasis$. 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 $\efuncden{\denr}$: Therefore, we take the following form for the approximation of $\efuncden{\denr}$:
\begin{equation} \begin{equation}
\begin{aligned} \begin{aligned}
\label{eq:def_ecmdpbebasis}
\efuncdenpbe{\argebasis} = &\int d\br{} \,\denr \\ & \ecmd(\argrebasis) \efuncdenpbe{\argebasis} = &\int d\br{} \,\denr \\ & \ecmd(\argrebasis)
\end{aligned} \end{aligned}
\end{equation} \end{equation}
where $\ecmd(\argebasis)$ is the correlation energy density defined as where $\ecmd(\argebasis)$ is the correlation energy density defined as
\begin{equation} \begin{equation}
\label{eq:def_ecmdpbe}
\ecmd(\argebasis) = \frac{\varepsilon_{\text{c,PBE}}(\argepbe)}{1+ \mu^3 \beta(\argepbe)} \ecmd(\argebasis) = \frac{\varepsilon_{\text{c,PBE}}(\argepbe)}{1+ \mu^3 \beta(\argepbe)}
\end{equation} \end{equation}
with 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}, \beta(\argepbe) = \frac{3}{2\sqrt{\pi}(1 - \sqrt{2})}\frac{\varepsilon_{\text{c,PBE}}(\argepbe)}{n^{2}/\den},
\end{equation} \end{equation}
and where $\varepsilon_{\text{c,PBE}}(\argepbe)$ is the usual PBE correlation density\cite{PerBurErn-PRL-96}. 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} \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} \end{equation}
which can be qualified as the weak correlation regime, and
\begin{equation} \begin{equation}
\label{eq:lim_muinf} \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} \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{Introduction of the effective spin-density}
\subsubsection{Requirement for $\wf{}{\Bas}$ for size extensivity} \subsubsection{Requirement for $\wf{}{\Bas}$ for size extensivity}
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