working on equations
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Emmanuel Giner
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@ -51,12 +51,12 @@
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\newlabel{sec:basic}{{II\tmspace +\thinmuskip {.1667em}A}{3}{}{section*.5}{}}
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\newlabel{eq:levy}{{1}{3}{}{equation.2.1}{}}
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\newlabel{eq:levy_func}{{2}{3}{}{equation.2.2}{}}
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\newlabel{eq:def_levy_bas}{{3}{3}{}{equation.2.3}{}}
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\newlabel{eq:e0approx}{{5}{3}{}{equation.2.5}{}}
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\@writefile{toc}{\contentsline {subsection}{\numberline {B}Definition of an effective interaction within $\mathcal {B}$}{3}{section*.6}}
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\newlabel{sec:wee}{{II\tmspace +\thinmuskip {.1667em}B}{3}{}{section*.6}{}}
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\newlabel{eq:wbasis}{{6}{3}{}{equation.2.6}{}}
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\newlabel{eq:fbasis}{{8}{3}{}{equation.2.8}{}}
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\newlabel{eq:cbs_wbasis}{{10}{3}{}{equation.2.10}{}}
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\citation{GinPraFerAssSavTou-JCP-18}
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\citation{GinPraFerAssSavTou-JCP-18}
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\citation{TouGorSav-TCA-05}
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@ -69,6 +69,18 @@
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\citation{CarTruGag-JPCA-17}
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\bibdata{srDFT_SCNotes,srDFT_SC}
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\bibcite{Thom-PRL-10}{{1}{2010}{{Thom}}{{}}}
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\newlabel{eq:cbs_wbasis}{{10}{4}{}{equation.2.10}{}}
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\@writefile{toc}{\contentsline {subsection}{\numberline {C}Definition of a range-separation parameter varying in real space}{4}{section*.7}}
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\newlabel{sec:mur}{{II\tmspace +\thinmuskip {.1667em}C}{4}{}{section*.7}{}}
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\newlabel{eq:weelr}{{11}{4}{}{equation.2.11}{}}
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\newlabel{eq:def_mur}{{12}{4}{}{equation.2.12}{}}
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\newlabel{eq:cbs_mu}{{14}{4}{}{equation.2.14}{}}
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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}}
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\@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}}
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\newlabel{eq:def_ecmdpbebasis}{{15}{4}{}{equation.2.15}{}}
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\newlabel{eq:def_ecmdpbe}{{16}{4}{}{equation.2.16}{}}
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\newlabel{eq:lim_muinf}{{19}{4}{}{equation.2.19}{}}
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\newlabel{eq:lim_ebasis}{{20}{4}{}{equation.2.20}{}}
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\bibcite{ScoTho-JCP-17}{{2}{2017}{{Scott\ and\ Thom}}{{}}}
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\bibcite{SpeNeuVigFraTho-JCP-18}{{3}{2018}{{Spencer\ \emph {et~al.}}}{{Spencer, Neufeld, Vigor, Franklin,\ and\ Thom}}}
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\bibcite{DeuEmiShePie-PRL-17}{{4}{2017}{{Deustua, Shen,\ and\ Piecuch}}{{}}}
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@ -76,23 +88,6 @@
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\bibcite{DeuEmiYumShePie-JCP-19}{{6}{2019}{{Deustua\ \emph {et~al.}}}{{Deustua, Yuwono, Shen,\ and\ Piecuch}}}
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\bibcite{QiuHenZhaScu-JCP-17}{{7}{2017}{{Qiu\ \emph {et~al.}}}{{Qiu, Henderson, Zhao,\ and\ Scuseria}}}
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\bibcite{QiuHenZhaScu-JCP-18}{{8}{2018}{{Qiu\ \emph {et~al.}}}{{Qiu, Henderson, Zhao,\ and\ Scuseria}}}
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\@writefile{toc}{\contentsline {subsection}{\numberline {C}Definition of a range-separation parameter varying in real space}{4}{section*.7}}
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\newlabel{sec:mur}{{II\tmspace +\thinmuskip {.1667em}C}{4}{}{section*.7}{}}
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\newlabel{eq:weelr}{{11}{4}{}{equation.2.11}{}}
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\newlabel{eq:def_mur}{{12}{4}{}{equation.2.12}{}}
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\newlabel{eq:cbs_mu}{{14}{4}{}{equation.2.14}{}}
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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}}
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\@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}}
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\newlabel{eq:def_ecmdpbebasis}{{15}{4}{}{equation.2.15}{}}
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\newlabel{eq:def_ecmdpbe}{{16}{4}{}{equation.2.16}{}}
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\newlabel{eq:lim_muinf}{{19}{4}{}{equation.2.19}{}}
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\newlabel{eq:lim_ebasis}{{20}{4}{}{equation.2.20}{}}
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\@writefile{toc}{\contentsline {subsubsection}{\numberline {2}Introduction of the effective spin-density}{4}{section*.10}}
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\@writefile{toc}{\contentsline {subsubsection}{\numberline {3}Requirement for $\Psi _{}^{\mathcal {B}}$ for size extensivity}{4}{section*.11}}
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\@writefile{toc}{\contentsline {section}{\numberline {III}Results}{4}{section*.12}}
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\newlabel{sec:results}{{III}{4}{}{section*.12}{}}
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\@writefile{toc}{\contentsline {section}{\numberline {IV}Conclusion}{4}{section*.13}}
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\newlabel{sec:conclusion}{{IV}{4}{}{section*.13}{}}
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\bibcite{GomHenScu-JCP-19}{{9}{2019}{{Gomez, Henderson,\ and\ Scuseria}}{{}}}
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\bibcite{WerKno-JCP-88}{{10}{1988}{{Werner\ and\ Knowles}}{{}}}
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\bibcite{KnoWer-CPL-88}{{11}{1988}{{Knowles\ and\ Werner}}{{}}}
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@ -102,6 +97,16 @@
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\bibcite{EvaDauMal-ChemPhys-83}{{15}{1983}{{Evangelisti, Daudey,\ and\ Malrieu}}{{}}}
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\bibcite{Cim-JCP-1985}{{16}{1985}{{Cimiraglia}}{{}}}
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\bibcite{Cim-JCC-1987}{{17}{1987}{{Cimiraglia\ and\ Persico}}{{}}}
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\@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}}
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\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}{}}
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\@writefile{toc}{\contentsline {subsubsection}{\numberline {2}Introduction of the effective spin-density}{5}{section*.10}}
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\@writefile{toc}{\contentsline {subsubsection}{\numberline {3}Requirement for $\Psi _{}^{\mathcal {B}}$ for size extensivity}{5}{section*.11}}
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\@writefile{toc}{\contentsline {section}{\numberline {III}Results}{5}{section*.12}}
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\newlabel{sec:results}{{III}{5}{}{section*.12}{}}
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\@writefile{toc}{\contentsline {section}{\numberline {IV}Conclusion}{5}{section*.13}}
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\newlabel{sec:conclusion}{{IV}{5}{}{section*.13}{}}
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\@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}}
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\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}{}}
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\bibcite{IllRubRic-JCP-88}{{18}{1988}{{Illas, Rubio,\ and\ Ricart}}{{}}}
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\bibcite{PovRubIll-TCA-92}{{19}{1992}{{Povill, Rubio,\ and\ Illas}}{{}}}
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\bibcite{BunCarRam-JCP-06}{{20}{2006}{{Bunge\ and\ Carb{\'o}-Dorca}}{{}}}
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@ -118,10 +123,6 @@
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\bibcite{HolUmrSha-JCP-17}{{31}{2017}{{Holmes, Umrigar,\ and\ Sharma}}{{}}}
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\bibcite{ShaHolJeaAlaUmr-JCTC-17}{{32}{2017}{{Sharma\ \emph {et~al.}}}{{Sharma, Holmes, Jeanmairet, Alavi,\ and\ Umrigar}}}
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\bibcite{SchEva-JCTC-17}{{33}{2017}{{Schriber\ and\ Evangelista}}{{}}}
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\@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}}
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\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}{}}
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\@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}}
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\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}{}}
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\bibcite{PerCle-JCP-17}{{34}{2017}{{Per\ and\ Cleland}}{{}}}
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\bibcite{OhtJun-JCP-17}{{35}{2017}{{Ohtsuka\ and\ ya~Hasegawa}}{{}}}
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\bibcite{Zim-JCP-17}{{36}{2017}{{Zimmerman}}{{}}}
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@ -131,6 +132,10 @@
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\bibcite{LooSceBloGarCafJac-JCTC-18}{{40}{2018}{{Loos\ \emph {et~al.}}}{{Loos, Scemama, Blondel, Garniron, Caffarel,\ and\ Jacquemin}}}
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\bibcite{GarSceGinCaffLoo-JCP-18}{{41}{2018}{{Garniron\ \emph {et~al.}}}{{Garniron, Scemama, Giner, Caffarel,\ and\ Loos}}}
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\bibcite{SceGarCafLoo-JCTC-18}{{42}{2018{}}{{Scemama\ \emph {et~al.}}}{{Scemama, Garniron, Caffarel,\ and\ Loos}}}
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\@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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\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}{}}
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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}}
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\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}{}}
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\bibcite{GarGinMalSce-JCP-16}{{43}{2017}{{Garniron\ \emph {et~al.}}}{{Garniron, Giner, Malrieu,\ and\ Scemama}}}
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\bibcite{LooBogSceCafJac-JCTC-19}{{44}{2019{}}{{Loos\ \emph {et~al.}}}{{Loos, Boggio-Pasqua, Scemama, Caffarel,\ and\ Jacquemin}}}
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\bibcite{Hyl-ZP-29}{{45}{1929}{{Hylleraas}}{{}}}
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@ -146,16 +151,18 @@
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\bibcite{GruHirOhnTen-JCP-17}{{55}{2017}{{Gr\"uneis\ \emph {et~al.}}}{{Gr\"uneis, Hirata, Ohnishi,\ and\ Ten-no}}}
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\bibcite{MaWer-WIREs-18}{{56}{2018}{{Ma\ and\ Werner}}{{}}}
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\bibcite{TewKloNeiHat-PCCP-07}{{57}{2007}{{Tew\ \emph {et~al.}}}{{Tew, Klopper, Neiss,\ and\ Hattig}}}
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\@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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\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}{}}
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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}}
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\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}{}}
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\bibcite{TouColSav-PRA-04}{{58}{2004}{{Toulouse, Colonna,\ and\ Savin}}{{}}}
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\bibcite{FraMusLupTou-JCP-15}{{59}{2015}{{Franck\ \emph {et~al.}}}{{Franck, Mussard, Luppi,\ and\ Toulouse}}}
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\bibcite{AngGerSavTou-PRA-05}{{60}{2005}{{\'Angy\'an\ \emph {et~al.}}}{{\'Angy\'an, Gerber, Savin,\ and\ Toulouse}}}
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\bibcite{GolWerSto-PCCP-05}{{61}{2005}{{Goll, Werner,\ and\ Stoll}}{{}}}
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\bibcite{TouGerJanSavAng-PRL-09}{{62}{2009}{{Toulouse\ \emph {et~al.}}}{{Toulouse, Gerber, Jansen, Savin,\ and\ \'Angy\'an}}}
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\bibcite{JanHenScu-JCP-09}{{63}{2009}{{Janesko, Henderson,\ and\ Scuseria}}{{}}}
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\@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}}
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\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}{}}
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\@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}}
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\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}{}}
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\@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}}
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\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}{}}
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\bibcite{TouZhuSavJanAng-JCP-11}{{64}{2011}{{Toulouse\ \emph {et~al.}}}{{Toulouse, Zhu, Savin, Jansen,\ and\ \'Angy\'an}}}
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\bibcite{MusReiAngTou-JCP-15}{{65}{2015}{{Mussard\ \emph {et~al.}}}{{Mussard, Reinhardt, \'Angy\'an,\ and\ Toulouse}}}
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\bibcite{LeiStoWerSav-CPL-97}{{66}{1997}{{Leininger\ \emph {et~al.}}}{{Leininger, Stoll, Werner,\ and\ Savin}}}
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@ -167,12 +174,6 @@
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\bibcite{GinPraFerAssSavTou-JCP-18}{{72}{2018}{{Giner\ \emph {et~al.}}}{{Giner, Pradines, Fert\'e, Assaraf, Savin,\ and\ Toulouse}}}
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\bibcite{LooPraSceTouGin-JCPL-19}{{73}{2019{}}{{Loos\ \emph {et~al.}}}{{Loos, Pradines, Scemama, Toulouse,\ and\ Giner}}}
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\bibcite{TouGorSav-TCA-05}{{74}{2005}{{Toulouse, Gori-Giorgi,\ and\ Savin}}{{}}}
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\@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}}
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\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}{}}
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\@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}}
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\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}{}}
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\@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}}
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\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}{}}
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\bibcite{PerBurErn-PRL-96}{{75}{1996}{{Perdew, Burke,\ and\ Ernzerhof}}{{}}}
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\bibcite{GoriSav-PRA-06}{{76}{2006}{{Gori-Giorgi\ and\ Savin}}{{}}}
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\bibcite{PazMorGorBac-PRB-06}{{77}{2006}{{Paziani\ \emph {et~al.}}}{{Paziani, Moroni, Gori-Giorgi,\ and\ Bachelet}}}
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@ -6,7 +6,7 @@
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\BOOKMARK [2][-]{section*.6}{Definition of an effective interaction within B}{section*.4}% 6
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\BOOKMARK [2][-]{section*.7}{Definition of a range-separation parameter varying in real space}{section*.4}% 7
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\BOOKMARK [2][-]{section*.8}{Approximation for B[n\(r\)] : link with RSDFT}{section*.4}% 8
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\BOOKMARK [3][-]{section*.9}{Generic form and properties of the approximations for functionals B[n\(r\)] }{section*.8}% 9
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\BOOKMARK [3][-]{section*.9}{Generic form and properties of the approximations for B[n\(r\)] }{section*.8}% 9
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\BOOKMARK [3][-]{section*.10}{Introduction of the effective spin-density}{section*.8}% 10
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\BOOKMARK [3][-]{section*.11}{Requirement for B for size extensivity}{section*.8}% 11
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\BOOKMARK [1][-]{section*.12}{Results}{section*.2}% 12
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@ -292,11 +292,11 @@ Then in section \ref{sec:results} we discuss the potential energy surfaces (PES)
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\section{Theory}
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\label{sec:theory}
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%%%%%%%%%%%%%%%%%%%%%%%%
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The theoretical framework of the basis set correction has been derived in details in \cite{GinPraFerAssSavTou-JCP-18}, so we recall briefly the main equations involved for the present study.
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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.
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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}.
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\subsection{Basic formal equations}
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\label{sec:basic}
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The exact ground state energy $E_0$ of a $N-$electron system can be obtained by the Levy-Lieb constrained search formalism which is an elegant mathematical framework connecting WFT and DFT
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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
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\begin{equation}
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\label{eq:levy}
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E_0 = \min_{\denr} \bigg\{ F[\denr] + (v_{\text{ne}} (\br{}) |\denr) \bigg\},
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@ -307,35 +307,37 @@ where $(v_{ne}(\br{})|\denr)$ is the nuclei-electron interaction for a given den
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F[\denr] = \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop +\weeop }{\Psi}.
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\end{equation}
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The minimizing density $n_0$ of equation \eqref{eq:levy} is the exact ground state density.
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As 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 $\setdenbasis$.
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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 $\setdenbasis$.
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Following equation (7) of \cite{GinPraFerAssSavTou-JCP-18}, we split $F[\denr]$ as
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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.
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An important step proposed originally by some of the present authors in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}
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was to propose to split the minimization in the definition of $F[\denr]$ using $\wf{}{\Bas}$ which are wave functions developed in $\basis$
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\begin{equation}
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F[\denr] = \min_{\wf{}{\Bas} \rightarrow \denr} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\denr}
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\label{eq:def_levy_bas}
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F[\denr] = \min_{\wf{}{\Bas} \rightarrow \denr} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\denr},
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\end{equation}
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where $\wf{}{\Bas}$ refer to $N-$electron wave functions expanded in $\Bas$, and
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where $\efuncden{\denr}$ is the density functional complementary to the basis set $\Bas$ defined as
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which leads to the following definition of $\efuncden{\denr}$ which is the the density functional complementary to the basis set $\Bas$
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\begin{equation}
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\begin{aligned}
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\efuncden{\denr} =& \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop +\weeop }{\Psi} \\
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&- \min_{\Psi^{\Bas} \rightarrow \denr} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}}.
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\end{aligned}
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\end{equation}
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The functional $\efuncden{\denr}$ must therefore recover all physical effects not included in the basis set $\Bas$.
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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.
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Assuming that the FCI density $\denFCI$ in $\Bas$ is a good approximation of the exact density, one obtains the following approximation for the exact ground state density (see equations 12-15 of \cite{GinPraFerAssSavTou-JCP-18})
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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 \cite{GinPraFerAssSavTou-JCP-18} and further emphasized in \cite{LooPraSceTouGin-JCPL-19,excited}, 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.
|
||||
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,excited}, 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, the lack of cusp in any wave function $\wf{}{\Bas}$ could also originate from an effective non-divergent electron-electron interaction. In other words, the incompleteness of a finite basis set can be understood as the removal of the divergence at the electron coalescence point.
|
||||
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 \cite{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.
|
||||
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}
|
||||
@ -362,10 +364,10 @@ 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 \cite{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, that is
|
||||
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}|}.
|
||||
\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}
|
||||
@ -373,36 +375,36 @@ The condition of equation \eqref{eq:cbs_wbasis} is fundamental as it guarantees
|
||||
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;\br{1},\br{2}) = \frac{\text{erf}\big(\mu \,|\br{1}-\br{2}| \big)}{|\br{1}-\br{2}|}.
|
||||
w_{ee}^{\lr}(\mu;r_{12}) = \frac{\text{erf}\big(\mu \,r_{12} \big)}{r_{12}}.
|
||||
\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 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;\br{ },\br{ }) = \wbasiscoal.
|
||||
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,
|
||||
\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{Approximation for $\efuncden{\denr}$ : link with RSDFT}
|
||||
\subsubsection{Generic form and properties of the approximations for functionals $\efuncden{\denr}$ }
|
||||
\subsubsection{Generic form and properties of the approximations for $\efuncden{\denr}$ }
|
||||
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 $\efuncden{\denr}$:
|
||||
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 $\murpsi$.
|
||||
The generic form for the approximations to $\efuncden{\denr}$ proposed here reads
|
||||
\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
|
||||
where $\ecmd(\argebasis)$ 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)}
|
||||
@ -411,19 +413,19 @@ with
|
||||
\begin{equation}
|
||||
\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}.
|
||||
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
|
||||
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
|
||||
\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} + o(\frac{1}{\mu^5})
|
||||
\lim_{\mu \rightarrow \infty} \ecmd(\argebasis) = \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.~\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 $\psibasis$.
|
||||
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 $\psibasis$.
|
||||
|
||||
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 $\mu_{\Psi^{\basis}}$:
|
||||
|
||||
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