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@ -6,7 +6,7 @@
%Control: page (0) single
%Control: year (1) truncated
%Control: production of eprint (0) enabled
\begin{thebibliography}{57}%
\begin{thebibliography}{66}%
\makeatletter
\providecommand \@ifxundefined [1]{%
\@ifx{#1\undefined}
@ -50,6 +50,26 @@
\providecommand \BibitemShut [1]{\csname bibitem#1\endcsname}%
\let\auto@bib@innerbib\@empty
%</preamble>
\bibitem [{\citenamefont {Pople}(1999)}]{Pop-RMP-99}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~A.}\ \bibnamefont
{Pople}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Rev.
Mod. Phys.}\ }\textbf {\bibinfo {volume} {{71}}},\ \bibinfo {pages} {1267}
(\bibinfo {year} {1999})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Kohn}(1999)}]{Koh-RMP-99}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont
{Kohn}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Rev.
Mod. Phys.}\ }\textbf {\bibinfo {volume} {{71}}},\ \bibinfo {pages} {1253}
(\bibinfo {year} {1999})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Kohn}\ and\ \citenamefont
{Sham}(1965)}]{KohSha-PR-65}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont
{Kohn}}\ and\ \bibinfo {author} {\bibfnamefont {L.~J.}\ \bibnamefont
{Sham}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Phys.
Rev.}\ }\textbf {\bibinfo {volume} {140}},\ \bibinfo {pages} {A1133}
(\bibinfo {year} {1965})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Hylleraas}(1929)}]{Hyl-ZP-29}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.~A.}\ \bibnamefont
@ -78,6 +98,17 @@
{Kutzelnigg}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J.
Chem. Phys.}\ }\textbf {\bibinfo {volume} {101}},\ \bibinfo {pages} {7738}
(\bibinfo {year} {1994})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Helgaker}\ \emph {et~al.}(1997)\citenamefont
{Helgaker}, \citenamefont {Klopper}, \citenamefont {Koch},\ and\
\citenamefont {Noga}}]{HelKloKocNog-JCP-97}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
{Helgaker}}, \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Klopper}},
\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont {Koch}}, \ and\ \bibinfo
{author} {\bibfnamefont {J.}~\bibnamefont {Noga}},\ }\href@noop {} {\bibfield
{journal} {\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume}
{106}},\ \bibinfo {pages} {9639} (\bibinfo {year} {1997})}\BibitemShut
{NoStop}%
\bibitem [{\citenamefont {Halkier}\ \emph {et~al.}(1998)\citenamefont
{Halkier}, \citenamefont {Helgaker}, \citenamefont {J{\o}rgensen},
\citenamefont {Klopper}, \citenamefont {Koch}, \citenamefont {Olsen},\ and\
@ -276,14 +307,13 @@
}\textbf {\bibinfo {volume} {70}},\ \bibinfo {pages} {062505} (\bibinfo
{year} {2004})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Gori-Giorgi}\ and\ \citenamefont
{Savin}(2006{\natexlab{a}})}]{GorSav-PRA-06}%
{Savin}(2006)}]{GorSav-PRA-06}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
{Gori-Giorgi}}\ and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Savin}},\ }\href {\doibase 10.1103/PhysRevA.73.032506} {\bibfield {journal}
{\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {73}},\
\bibinfo {pages} {032506} (\bibinfo {year} {2006}{\natexlab{a}})}\BibitemShut
{NoStop}%
\bibinfo {pages} {032506} (\bibinfo {year} {2006})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Franck}\ \emph {et~al.}(2015)\citenamefont {Franck},
\citenamefont {Mussard}, \citenamefont {Luppi},\ and\ \citenamefont
{Toulouse}}]{FraMusLupTou-JCP-15}%
@ -360,6 +390,14 @@
{\bibinfo {volume} {142}},\ \bibinfo {pages} {154123} (\bibinfo {year}
{2015})},\ \bibinfo {note} {erratum: J. Chem. Phys. {\bf 142}, 219901
(2015)}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Kalai}\ and\ \citenamefont
{Toulouse}(2018)}]{KalTou-JCP-18}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont
{Kalai}}\ and\ \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
{Toulouse}},\ }\href {\doibase 10.1063/1.5025561} {\bibfield {journal}
{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {148}},\
\bibinfo {pages} {164105} (\bibinfo {year} {2018})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Leininger}\ \emph {et~al.}(1997)\citenamefont
{Leininger}, \citenamefont {Stoll}, \citenamefont {Werner},\ and\
\citenamefont {Savin}}]{LeiStoWerSav-CPL-97}%
@ -422,6 +460,14 @@
{pages} {084103} (\bibinfo {year} {2019})},\ \Eprint
{http://arxiv.org/abs/https://doi.org/10.1063/1.5082638}
{https://doi.org/10.1063/1.5082638} \BibitemShut {NoStop}%
\bibitem [{\citenamefont {Mussard}\ and\ \citenamefont
{Toulouse}(2017)}]{MusTou-MP-17}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {B.}~\bibnamefont
{Mussard}}\ and\ \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
{Toulouse}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Mol.
Phys.}\ }\textbf {\bibinfo {volume} {115}},\ \bibinfo {pages} {161} (\bibinfo
{year} {2017})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Giner}\ \emph {et~al.}(2018)\citenamefont {Giner},
\citenamefont {Pradines}, \citenamefont {Fert\'e}, \citenamefont {Assaraf},
\citenamefont {Savin},\ and\ \citenamefont
@ -465,6 +511,24 @@
{pages} {144118} (\bibinfo {year} {2019})},\ \Eprint
{http://arxiv.org/abs/https://doi.org/10.1063/1.5122976}
{https://doi.org/10.1063/1.5122976} \BibitemShut {NoStop}%
\bibitem [{\citenamefont {Levy}(1979)}]{Lev-PNAS-79}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
{Levy}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Proc.
Natl. Acad. Sci. U.S.A.}\ }\textbf {\bibinfo {volume} {76}},\ \bibinfo
{pages} {6062} (\bibinfo {year} {1979})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Lieb}(1983)}]{Lie-IJQC-83}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.~H.}\ \bibnamefont
{Lieb}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Int. J.
Quantum Chem.}\ }\textbf {\bibinfo {volume} {{24}}},\ \bibinfo {pages} {24}
(\bibinfo {year} {1983})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Kato}(1957)}]{Kat-CPAM-57}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
{Kato}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Comm.
Pure Appl. Math.}\ }\textbf {\bibinfo {volume} {10}},\ \bibinfo {pages} {151}
(\bibinfo {year} {1957})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Toulouse}, \citenamefont {Gori-Giorgi},\ and\
\citenamefont {Savin}(2005)}]{TouGorSav-TCA-05}%
\BibitemOpen
@ -483,15 +547,6 @@
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. Lett.}\
}\textbf {\bibinfo {volume} {77}},\ \bibinfo {pages} {3865} (\bibinfo {year}
{1996})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Gori-Giorgi}\ and\ \citenamefont
{Savin}(2006{\natexlab{b}})}]{GoriSav-PRA-06}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
{Gori-Giorgi}}\ and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Savin}},\ }\href {\doibase 10.1103/PhysRevA.73.032506} {\bibfield {journal}
{\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {73}},\
\bibinfo {pages} {032506} (\bibinfo {year} {2006}{\natexlab{b}})}\BibitemShut
{NoStop}%
\bibitem [{\citenamefont {Paziani}\ \emph {et~al.}(2006)\citenamefont
{Paziani}, \citenamefont {Moroni}, \citenamefont {Gori-Giorgi},\ and\
\citenamefont {Bachelet}}]{PazMorGorBac-PRB-06}%
@ -503,15 +558,6 @@
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. B}\
}\textbf {\bibinfo {volume} {73}},\ \bibinfo {pages} {155111} (\bibinfo
{year} {2006})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Gritsenko}, \citenamefont {van Meer},\ and\
\citenamefont {Pernal}(2018)}]{GritMeePer-PRA-18}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {O.~V.}\ \bibnamefont
{Gritsenko}}, \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {van Meer}},
\ and\ \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Pernal}},\ }\href
{\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
@ -521,6 +567,24 @@
{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 {Gritsenko}, \citenamefont {van Meer},\ and\
\citenamefont {Pernal}(2018)}]{GritMeePer-PRA-18}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {O.~V.}\ \bibnamefont
{Gritsenko}}, \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {van Meer}},
\ and\ \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Pernal}},\ }\href
{\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 {Becke}, \citenamefont {Savin},\ and\ \citenamefont
{Stoll}(1995)}]{BecSavSto-TCA-95}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.~D.}\ \bibnamefont
{Becke}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Savin}}, \ and\
\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont {Stoll}},\ }\href@noop {}
{\bibfield {journal} {\bibinfo {journal} {Theoret. Chim. Acta}\ }\textbf
{\bibinfo {volume} {{91}}},\ \bibinfo {pages} {147} (\bibinfo {year}
{1995})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Perdew}, \citenamefont {Savin},\ and\ \citenamefont
{Burke}(1995)}]{PerSavBur-PRA-95}%
\BibitemOpen

326
Manuscript/srDFT_SC.tex
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@ -77,22 +77,22 @@
\newcommand{\emulda}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denr;\mu({\bf r};\wf{}{\Bas})\right)}
\newcommand{\emuldamodel}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denmodelr;\mu({\bf r};\wf{}{\Bas})\right)}
\newcommand{\emuldaval}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denval ({\bf r});\murval;\wf{}{\Bas})\right)}
\newcommand{\ecmd}[0]{\varepsilon^{\text{c,md}}_{\text{PBE}}}
\newcommand{\ecmd}[0]{\bar{\varepsilon}_{\text{c,md}}^{\text{sr},\text{PBE}}}
\newcommand{\psibasis}[0]{\Psi^{\basis}}
\newcommand{\BasFC}{\mathcal{A}}
%pbeuegxiHF
\newcommand{\pbeuegxihf}{\text{PBE-UEG-}\zeta\text{-HF}^\Bas}
\newcommand{\argpbeuegxihf}[0]{\den,\zeta,s,\ntwo_{\text{UEG}},\mu_{\text{HF}}^{\basis}}
\newcommand{\argrpbeuegxihf}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo_{\text{UEG}}(\br{}),\mu_{\text{HF}}^{\basis}(\br{})}
\newcommand{\argpbeuegxihf}[0]{\den,\zeta,s,\ntwo^{\text{UEG}},\mu^{\text{HF},\basis}}
\newcommand{\argrpbeuegxihf}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo^{\text{UEG}}(\br{}),\mu^{\text{HF},\basis}(\br{})}
%pbeuegxiCAS
\newcommand{\pbeuegxi}{\text{PBE-UEG-}\zeta\text{-CAS}^\Bas}
\newcommand{\argpbeuegxicas}[0]{\den,\zeta,s,\ntwo_{\text{UEG}},\mu_{\text{CAS}}^{\basis}}
\newcommand{\argrpbeuegxicas}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo_{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
\newcommand{\argpbeuegxicas}[0]{\den,\zeta,s,\ntwo^{\text{UEG}},\mu_{\text{CAS}}^{\basis}}
\newcommand{\argrpbeuegxicas}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo^{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
%pbeuegXiCAS
\newcommand{\pbeuegXi}{\text{PBE-UEG-}\tilde{\zeta}}
\newcommand{\argpbeuegXi}[0]{\den,\tilde{\zeta},s,\ntwo_{\text{UEG}},\mu_{\text{CAS}}^{\basis}}
\newcommand{\argrpbeuegXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),\ntwo_{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
\newcommand{\argpbeuegXi}[0]{\den,\tilde{\zeta},s,\ntwo^{\text{UEG}},\mu_{\text{CAS}}^{\basis}}
\newcommand{\argrpbeuegXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),\ntwo^{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
%pbeontxiCAS
\newcommand{\pbeontxi}{\text{PBE-ot-}\zeta}
\newcommand{\argpbeontxi}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
@ -109,12 +109,12 @@
%%%%%% arguments
\newcommand{\argepbe}[0]{\den,\zeta,s}
\newcommand{\argebasis}[0]{\den,\zeta,s,\ntwo,\mu_{\Psi^{\basis}}}
\newcommand{\argebasis}[0]{\den,\zeta,\ntwo,\mu}
\newcommand{\argecmd}[0]{\den,\zeta,s,\ntwo,\mu}
\newcommand{\argepbeueg}[0]{\den,\zeta,s,\ntwo_{\text{UEG}},\mu_{\Psi^{\basis}}}
\newcommand{\argepbeueg}[0]{\den,\zeta,s,\ntwo^{\text{UEG}},\mu_{\Psi^{\basis}}}
\newcommand{\argepbeontxicas}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
\newcommand{\argepbeuegXihf}[0]{\den,\tilde{\zeta},s,\ntwo_{\text{UEG}},\mu_{\Psi^{\basis}}}
\newcommand{\argrebasis}[0]{\denr,\zeta(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})}
\newcommand{\argepbeuegXihf}[0]{\den,\tilde{\zeta},s,\ntwo^{\text{UEG}},\mu_{\Psi^{\basis}}}
\newcommand{\argrebasis}[0]{\denr,\zeta(\br{}),s(\br{}),\ntwo(\br{}),\mu(\br{})}
\newcommand{\argrebasisab}[0]{\denr,\zeta(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})}
@ -129,17 +129,17 @@
% effective interaction
\newcommand{\twodm}[4]{\elemm{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}}
\newcommand{\murpsi}[0]{\mu({\bf r};\wf{}{\Bas})}
\newcommand{\ntwo}[0]{n^{(2)}}
\newcommand{\ntwohf}[0]{n^{(2),\text{HF}}}
\newcommand{\ntwophi}[0]{n^{(2)}_{\phi}}
\newcommand{\ntwoextrap}[0]{\mathring{n}^{(2)}_{\psibasis}}
\newcommand{\ntwoextrapcas}[0]{\mathring{n}^{(2)\,\basis}_{\text{CAS}}}
\newcommand{\murpsi}[0]{\mu_{\wf{}{\Bas}}({\bf r})}
\newcommand{\ntwo}[0]{n_{2}}
\newcommand{\ntwohf}[0]{n_2^{\text{HF}}}
\newcommand{\ntwophi}[0]{n_2^{{\phi}}}
\newcommand{\ntwoextrap}[0]{\mathring{n}_{2}^{\psibasis}}
\newcommand{\ntwoextrapcas}[0]{\mathring{n}_2^{\text{CAS},\basis}}
\newcommand{\mur}[0]{\mu({\bf r})}
\newcommand{\murr}[1]{\mu({\bf r}_{#1})}
\newcommand{\murval}[0]{\mu_{\text{val}}({\bf r})}
\newcommand{\murpsival}[0]{\mu_{\text{val}}({\bf r};\wf{}{\Bas})}
\newcommand{\murrval}[1]{\mu_{\text{val}}({\bf r}_{#1})}
\newcommand{\murpsival}[0]{\mu_{\wf{}{\Bas}}^{\text{val}}({\bf r})}
\newcommand{\murrval}[1]{\mu^{\text{val}}({\bf r}_{#1})}
\newcommand{\weeopmu}[0]{\hat{W}_{\text{ee}}^{\text{lr},\mu}}
@ -149,9 +149,9 @@
\newcommand{\fbasis}[0]{f_{\wf{}{\Bas}}(\bfr{1},\bfr{2})}
\newcommand{\fbasisval}[0]{f_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})}
\newcommand{\ontop}[2]{ n^{(2)}_{#1}({\bf #2}_1)}
\newcommand{\twodmrpsi}[0]{ \ntwo_{\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})}
\newcommand{\twodmrdiagpsi}[0]{ \ntwo_{\wf{}{\Bas}}(\rr{1}{2})}
\newcommand{\twodmrdiagpsival}[0]{ \ntwo_{\wf{}{\Bas},\,\text{val}}(\rr{1}{2})}
\newcommand{\twodmrpsi}[0]{ \ntwo^{\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})}
\newcommand{\twodmrdiagpsi}[0]{ n_{2,{\wf{}{\Bas}}}(\rr{1}{2})}
\newcommand{\twodmrdiagpsival}[0]{n_{2,\wf{}{\Bas}}^{\text{val}}(\rr{1}{2})}
\newcommand{\gammamnpq}[1]{\Gamma_{mn}^{pq}[#1]}
\newcommand{\gammamnkl}[0]{\Gamma_{mn}^{kl}}
\newcommand{\gammaklmn}[1]{\Gamma_{kl}^{mn}[#1]}
@ -239,10 +239,12 @@
\newcommand{\Cor}{\mathcal{C}}
% operators
\renewcommand{\d}{\text{d}}
\newcommand{\hT}{\Hat{T}}
\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}}
\newcommand{\f}[2]{f_{#1}^{#2}}
\newcommand{\Gam}[2]{\Gamma_{#1}^{#2}}
\newcommand{\isEquivTo}[1]{\underset{#1}{\sim}}
% coordinates
\newcommand{\br}[1]{{\mathbf{r}_{#1}}}
@ -251,19 +253,20 @@
\newcommand{\PBEspin}{PBEspin}
\newcommand{\PBEueg}{PBE-UEG-{$\tilde{\zeta}$}}
\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique (UMR 7616), Sorbonne Universit\'e, CNRS, Paris, France}
\newcommand{\ISCD}{Institut des Sciences du Calcul et des Donn\'ees, Sorbonne Universit\'e, Paris, France}
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, France}
\newcommand{\ISCD}{Institut des Sciences du Calcul et des Donn\'ees, Sorbonne Universit\'e, Paris, France}
\newcommand{\IUF}{Institut Universitaire de France, Paris, France}
\begin{document}
\title{A density-based basis set correction for strong correlation}
\title{A density-based basis-set correction for strong correlation}
\author{Emmanuel Giner}
\email{emmanuel.giner@lct.jussieu.fr}
\affiliation{\LCT}
\author{Bath\'elemy Pradines}
\author{Barth\'el\'emy Pradines}
\affiliation{\LCT}
\affiliation{\ISCD}
\author{Anthony Scemama}
@ -274,16 +277,13 @@
\author{Julien Toulouse}
\email{toulouse@lct.jussieu.fr}
\affiliation{\LCT}
\affiliation{\IUF}
\begin{abstract}
The present work proposes an application and extension to strongly correlated systems of the recently proposed basis set correction based on density functional theory (DFT).
We study the potential energy surfaces (PES) of the H$_{10}$, C$_2$, N$_2$, O$_2$ and F$_2$ molecules up to full dissociation limit in increasing basis sets at near full configuration interaction (FCI) level with and without the present basis set correction.
Such basis set correction relies on a mapping between range-separated DFT (RSDFT) and wave function calculations in a finite basis set through the definition of an effective non-divergent interaction mimicking the coulomb operator projected in a finite basis set. From that mapping, RSDFT-types functionals are used to recover the dominant part of the short-range correlation effects missing in a finite basis set.
The scope of the present work is to develop new approximations for the complementary functionals which are suited to describe strong correlation regimes and which fulfill two very desirable properties: $S_z$ invariance and size extensivity.
In that context, we investigate the dependence of the functionals on different flavours of on-top pair densities and spin-polarizations. An important result is that the explicit dependence on the on-top pair density allows one to completely remove the dependence on any form of spin-polarization without any significant loss of accuracy.
In the general context of multi-configurational DFT, such findings show that one can avoid the effective spin polarization whose mathematical definition is rather \textit{ad hoc} and which can become complex valued.
Quantitatively, we show that the basis set correction allows chemical accuracy on atomization energies in a triple-zeta quality for most of the systems studied. Also, the present basis set correction provides smooth curves all along the PES.
We extend to strongly correlated systems the recently introduced basis-set correction based on density-functional theory (DFT) [E. Giner \textit{et al.}, J. Chem. Phys. \textbf{149}, 194301 (2018)]. This basis-set correction relies on a mapping between wave-function calculations in a finite basis set and range-separated DFT (RSDFT) through the definition of an effective non-divergent interaction corresponding to the Coulomb electron-electron interaction projected in the finite basis set, allowing one to use RSDFT-type complementary functionals to recover the dominant part of the short-range correlation effects missing in a finite basis set. Using as test cases the potential energy curves of the H$_{10}$, C$_2$, N$_2$, O$_2$, and F$_2$ molecules up to the dissociation limit, we systematically explore different approximations for the complementary functionals which are suited to describe strong-correlation regimes and which fulfill two very desirable properties: $S_z$ invariance and size consistency. Specifically, we investigate the dependence of the functionals on different flavours of on-top pair densities and spin polarizations. An important result is that the explicit dependence on the on-top pair density allows one to completely remove the dependence on any form of spin polarization without any significant loss of accuracy.
In the general context of multiconfigurational DFT, this finding shows that one can avoid the effective spin polarization whose mathematical definition is rather \textit{ad hoc} and which can become complex valued. Quantitatively, we show that the basis-set correction reaches chemical accuracy on atomization energies with triple-zeta quality basis sets for most of the systems studied. Also, the present basis-set correction provides smooth curves along the whole potential energy curves.
%We study the potential energy surfaces (PES) of the H$_{10}$, C$_2$, N$_2$, O$_2$, and F$_2$ molecules up to the dissociation limit using increasing basis sets at near full configuration interaction (FCI) level with and without the present basis-set correction.
\end{abstract}
\maketitle
@ -291,35 +291,11 @@ In the general context of multi-configurational DFT, such findings show that one
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%
The main goal of quantum chemistry is to propose reliable theoretical tools to explore the rich area of chemistry.
The accurate computation of the electronic structure of molecular systems plays a central role in the development of methods in quantum chemistry,
but despite intense developments, no definitive solution to that problem have been found.
The theoretical challenge to be overcome falls back in the category of the quantum many-body problem due the intrinsic quantum nature
of the electrons and the coulomb repulsion between them, inducing the so-called electronic correlation problem.
Tackling this problem translates into solving the Schroedinger equation for a $N$~-~electron system, and two roads have emerged to approximate the solution to this formidably complex mathematical problem: the wave function theory (WFT) and density functional theory (DFT).
Although both WFT and DFT spring from the same equation, their formalisms are very different as the former deals with the complex
$N$~-~body wave function whereas the latter handles the much simpler one~-~body density.
In its Kohn-Sham (KS) formulation, the computational cost of DFT is very appealing as it can be recast in a mean-field procedure.
Therefore, although constant efforts are performed to reduce the computational cost of WFT, DFT remains still the workhorse of quantum chemistry.
The general goal of quantum chemistry is to provide reliable theoretical tools to explore the rich area of chemistry. More specifically, developments in quantum chemistry primarily aim at accurately computing the electronic structure of molecular systems, but despite intense developments, no definitive solution to this problem has been found. The theoretical challenge to tackle belongs to the quantum many-body problem, due the intrinsic quantum nature of the electrons and the Coulomb repulsion between them. This so-called electronic correlation problem corresponds to finding a solution to the Schr\"odinger equation for a $N$-electron system, and two main roads have emerged to approximate this solution: wave-function theory (WFT)~\cite{Pop-RMP-99} and density-functional theory (DFT)~\cite{Koh-RMP-99}. Although both WFT and DFT spring from the same Schr\"odinger equation, they rely on very different formalisms, as the former deals with the complicated $N$-electron wave function whereas the latter focuses on the much simpler one-electron density. In its Kohn-Sham (KS) formulation~\cite{KohSha-PR-65}, the computational cost of DFT is very appealing since it is a simple mean-field procedure. Therefore, although continued efforts have been done to reduce the computational cost of WFT, DFT still remains the workhorse of quantum chemistry.
The complexity of a reliable theoretical description of a given chemical system can be roughly
categorized by the strength of the electronic correlation appearing in its electronic structure.
The so-called weakly correlated systems, such as closed-shell organic molecules near their equilibrium geometry, are typically dominated by correlation effects which do not affect the qualitative mean-field picture of the system. These weak correlation effects can be either short-range when electron are near the electron coalescence point, or long-range
with dispersion forces. The theoretical description of weakly correlated systems is one of the more concrete achievement
of quantum chemistry, and the main remaining issue for these systems is to push the limit in terms of the size of the chemical systems that can be treated.
The case of the so-called strongly correlated systems, which are ubiquitous in chemistry, is much more problematic as they exhibits
a much more exotic electronic structure.
Transition metals containing systems, low-spin open shell systems, covalent bond breaking or excited states
have all in common that they cannot be even qualitatively described by a single electronic configuration.
It is now clear that the usual semi-local approximations in KS-DFT fail in giving an accurate description of these situations and WFT has become
the standard for the treatment of strongly correlated systems.
The difficulty of obtaining a reliable theoretical description of a given chemical system can be roughly categorized by the strength of the electronic correlation appearing in its electronic structure. The so-called weakly correlated systems, such as closed-shell organic molecules near their equilibrium geometries, are typically dominated by correlation effects which do not affect the qualitative mean-field picture of the system. These weak correlation effects can be either short range(near the electron-electron coalescence point) or long range (London dispersion interactions). The theoretical description of weakly correlated systems is one of the most concrete achievement of quantum chemistry, and the main remaining issue for these systems is to push the limit of the size of the chemical systems that can be treated. The case of the so-called strongly correlated systems, which are ubiquitous in chemistry, is much more problematic as they exhibit a much more complex electronic structure. For example, transition metal complexes, low-spin open-shell systems, covalent bond breaking situations have all in common that they cannot be even qualitatively described by a single electronic configuration. It is now clear that the usual semilocal density-functional approximations of KS DFT fail to accurately describe these situations and WFT is king for the treatment of strongly correlated systems.
In practice WFT uses a finite one-particle basis set (here referred as $\basis$) to project the Schroedinger equation whose exact solution becomes clear: the full configuration interaction (FCI) which consists in a linear algebra problem whose dimension scales exponentially with the system size.
Because of the exponential growth of the FCI, many approximations have appeared and in that regard the complexity of the strong correlation problem is, at least, two-fold:
i) the qualitative description of the wave function is determined by a primary set of electronic configurations (whose size can scale exponentially in many cases) among which near degeneracies and/or strong interactions appear in the Hamiltonian matrix,
ii) the quantitative description of the systems must take into account weak correlation effects which requires to take into account many
other electronic configurations with typically much smaller weights in the wave function.
Fulfilling these two objectives is a rather complicated task for a given approximated approach, specially if one adds the requirement of satisfying formal properties, such $S_z$ invariance or additivity of the computed energy in the case of non interacting fragments.
In practice WFT uses a finite one-particle basis set (here denoted as $\basis$) to project the Schr\"odinger equation. The exact solution within the basis set is then provided by full configuration interaction (FCI) which consists in a linear-algebra problem with a dimension scaling exponentially with the system size. Due to this exponential growth of the FCI computational cost, introducing approximations is necessary, with at least two difficulties for strongly correlated systems: i) the qualitative description of the wave function is determined by a primary set of electronic configurations (whose size can scale exponentially in many cases) among which near degeneracies and/or strong interactions appear in the Hamiltonian matrix; ii) the quantitative description of the system requires to also account for weak correlation effects which involve many other electronic configurations with typically much smaller weights in the wave function. Addressing these two objectives is a rather complicated task for a given approximate WFT method, especially if one adds the requirement of satisfying formal properties, such as spin-multiplet degeneracy and size consistency.
%To tackle this complicated problem, many methods have been proposed and an exhaustive review of the zoology of methods for strong correlation goes beyond the scope and purpose of this article.
@ -339,74 +315,68 @@ Fulfilling these two objectives is a rather complicated task for a given approxi
%Among the SCI algorithms, the CI perturbatively selected iteratively (CIPSI) can be considered as a pioneer. The main idea of the CIPSI and other related SCI algorithms is to iteratively select the most important Slater determinants thanks to perturbation theory in order to build a MRCI zeroth-order wave function which automatically concentrate the strongly interacting part of the wave function. On top of this MRCI zeroth-order wave function, a rather simple MRPT approach is used to recover the missing weak correlation and the process is iterated until reaching a given convergence criterion. It is important to notice that in the SCI algorithms, neither the SCI or the MRPT are size extensive \textit{per se}, but the extensivity property is almost recovered by approaching the FCI limit.
%When the SCI are affordable, their clear advantage are that they provide near FCI wave functions and energies, whatever the level of knowledge of the user on the specific physical/chemical problem considered. The drawback of SCI is certainly their \textit{intrinsic} exponential scaling due to their linear parametrisation. Nevertheless, such an exponential scaling is lowered by the smart selection of the zeroth-order wave function together with the MRPT calculation.
Besides the difficulties of accurately describing the electronic structure within a given basis set, a crucial component of the limitations of applicability of WFT concerns the slow convergence of the energies and properties with respect to the quality of the basis set.
As initially shown by the seminal work of Hylleraas\cite{Hyl-ZP-29} and further developed by Kutzelnigg \textit{et. al.}\cite{Kut-TCA-85,KutKlo-JCP-91, NogKut-JCP-94}, the main convergence problem originates from the divergence of the coulomb interaction at the electron coalescence point, which induces a discontinuity in the first-derivative of the exact wave function (the so-called electron-electron cusp).
Describing such a discontinuity with an incomplete basis set is impossible and as a consequence, the convergence of the computed energies and properties can be strongly affected. To attenuate this problem, extrapolation techniques has been developed, either based on the Hylleraas's expansion of the coulomb operator\cite{HalHelJorKloKocOlsWil-CPL-98}, or more recently based on perturbative arguments\cite{IrmHulGru-arxiv-19}. A more rigorous approach to tackle the basis set convergence problem has been proposed by the so-called R12 and F12 methods\cite{Ten-TCA-12,TenNog-WIREs-12,HatKloKohTew-CR-12, KonBisVal-CR-12, GruHirOhnTen-JCP-17, MaWer-WIREs-18} which introduce a function explicitly depending on the interelectronic coordinates ensuring the correct cusp condition in the wave function, and the resulting correlation energies converge much faster than the usual WFT. For instance, using the explicitly correlated version of coupled cluster with single, double and perturbative triple substitution (CCSD(T)) in a triple-$\zeta$ quality basis set is equivalent to a quintuple-$\zeta$ quality of the usual CCSD(T) method\cite{TewKloNeiHat-PCCP-07}, although inherent computational overhead are introduced by the auxiliary basis sets needed to resolve the rather complex three- and four-electron integrals involved in the F12 theory. In addition to the computational cost, a possible drawback of F12 theory is its rather complex formalism which requires a quite involved mathematical development to adapt to a new theory. In the context of strong correlation, several multi-reference methods have been extended to explicitly correlation (see for instance Ref. \cite{Ten-CPL-07,ShiWer-JCP-10,TorKniWer-JCP-11,DemStanMatTenPitNog-PCCP-12,GuoSivValNee-JCP-17}), including approaches based on so-called universal F12 which are potentially applicable to any electronic structure approaches\cite{TorVal-JCP-09,KonVal-JCP-11,HauMaoMukKlo-CPL-12,BooCleAlaTew-JCP-12}.
Beside the difficulties of accurately describing the molecular electronic structure within a given basis set, a crucial limitation of WFT methods is the slow convergence of the energies and properties with respect to the size of the basis set. As initially shown by the seminal work of Hylleraas\cite{Hyl-ZP-29} and further developed by Kutzelnigg and coworkers~\textit{et al.}\cite{Kut-TCA-85,KutKlo-JCP-91, NogKut-JCP-94}, the main convergence problem originates from the divergence of the Coulomb electron-electron interaction at the coalescence point, which induces a discontinuity in the first derivative of the exact wave function (the so-called electron-electron cusp). Describing such a discontinuity with an incomplete one-electron basis set is impossible and, as a consequence, the convergence of the computed energies and properties can be strongly affected. To attenuate this problem, extrapolation techniques have been developed, either based on a partial-wave expansion analysis~\cite{HelKloKocNog-JCP-97,HalHelJorKloKocOlsWil-CPL-98}, or more recently based on perturbative arguments\cite{IrmHulGru-arxiv-19}. A more rigorous approach to tackle the basis-set convergence problem is provided by the so-called R12 and F12 explicitly correlated methods\cite{Ten-TCA-12,TenNog-WIREs-12,HatKloKohTew-CR-12, KonBisVal-CR-12, GruHirOhnTen-JCP-17, MaWer-WIREs-18} which introduce a geminal function explicitly depending on the interelectronic distances ensuring the correct cusp condition in the wave function, and lead to a much faster convergence of the correlation energies than usual WFT methods. For instance, using the explicitly correlated version of coupled cluster with singles, doubles, and perturbative triples (CCSD(T)) in a triple-$\zeta$ quality basis set is equivalent to using a quintuple-$\zeta$ quality basis set with the usual CCSD(T) method\cite{TewKloNeiHat-PCCP-07}, although a computational overhead is introduced by the auxiliary basis set needed to compute the three- and four-electron integrals involved in F12 theory. In addition to the computational cost, a possible drawback of F12 theory is its rather complex formalism which requires non-trivial developments for adapting it to a new method. For strongly correlated systems, several multi-reference methods have been extended to explicitly correlation (see for instance Ref.~\onlinecite{Ten-CPL-07,ShiWer-JCP-10,TorKniWer-JCP-11,DemStanMatTenPitNog-PCCP-12,GuoSivValNee-JCP-17}), including approaches based on the so-called universal F12 theory which are potentially applicable to any electronic-structure computational methods~\cite{TorVal-JCP-09,KonVal-JCP-11,HauMaoMukKlo-CPL-12,BooCleAlaTew-JCP-12}.
An alternative point of view to improve the convergence towards the CBS limit is to leave the short-range correlation effects to DFT and to use WFT to deal only with the long-range and/or strong-correlation effects. A rigorous approach to mix DFT and WFT is the range-separated DFT (RSDFT) formalism (see Ref.~\onlinecite{TouColSav-PRA-04} and references therein) which rely on a splitting of the coulomb interaction in terms of the interelectronic distance thanks to a range-separation parameter $\mu$. The advantage of such approach is at least two-folds: i) the DFT part deals only with the short-range part of the coulomb interaction, and therefore the usual semi-local approximations to the unknown exchange-correlation functional are more suited to that correlation regime, ii) as the WFT part deals with a smooth non divergent interaction, the exact wave function has no cusp\cite{GorSav-PRA-06} and therefore the basis set convergence is much faster\cite{FraMusLupTou-JCP-15}.
Therefore, a number of approximate RS-DFT schemes have been developed within single-reference \cite{AngGerSavTou-PRA-05, GolWerSto-PCCP-05, TouGerJanSavAng-PRL-09,JanHenScu-JCP-09, TouZhuSavJanAng-JCP-11, MusReiAngTou-JCP-15} or multi-reference \cite{LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, HedTouJen-JCP-18, FerGinTou-JCP-18} WFT approaches.
Nevertheless, there are still some open issues in RSDFT, such remaining self-interaction errors or
the dependence of the quality of the results on the value of the range separation $\mu$ which can be seen as an empirical parameter.
An alternative way to improve the convergence towards the complete-basis-set (CBS) limit is to treat the short-range correlation effects within DFT and to use WFT methods to deal only with the long-range and/or strong-correlation effects. A rigorous approach achieving this mixing of DFT and WFT is range-separated DFT (RSDFT) (see Ref.~\onlinecite{TouColSav-PRA-04} and references therein) which relies on a splitting of the Coulomb electron-electron interaction in terms of the interelectronic distance thanks to a range-separation parameter $\mu$. The advantage of this approach is at least two-fold: i) the DFT part deals primarily with the short-range part of the Coulomb interaction, and consequently the usual semilocal density-functional approximations are more accurate than for standard KS DFT; ii) the WFT part deals only with a smooth non-divergent interaction, and consequently the wave function has no electron-electron cusp\cite{GorSav-PRA-06} and the basis-set convergence is much faster\cite{FraMusLupTou-JCP-15}. A number of approximate RSDFT schemes have been developed involving single-reference \cite{AngGerSavTou-PRA-05, GolWerSto-PCCP-05, TouGerJanSavAng-PRL-09,JanHenScu-JCP-09, TouZhuSavJanAng-JCP-11, MusReiAngTou-JCP-15,KalTou-JCP-18,KalMusTou-JCP-19} or multi-reference \cite{LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, HedTouJen-JCP-18, FerGinTou-JCP-18} WFT methods. Nevertheless, there are still some open issues in RSDFT, such as remaining fractional-charge and fractional-spin errors in the short-range density functionals~\cite{MusTou-MP-17} or the dependence of the quality of the results on the value of the range-separation parameter $\mu$.
% which can be seen as an empirical parameter.
Following this path, a very recent solution to the basis set convergence problem has been proposed by some of the preset authors\cite{GinPraFerAssSavTou-JCP-18} where they proposed to use RSDFT to take into account only the correlation effects outside a given basis set. The key idea in such a work is to realize that a wave function developed in an incomplete basis set is cusp-less could also come from a Hamiltonian with a non divergent electron-electron interaction. Therefore, the authors proposed a mapping with RSDFT through the introduction of an effective non-divergent interaction representing the usual coulomb interaction projected in an incomplete basis set. First applications to weakly correlated molecular systems have been successfully carried recently\cite{LooPraSceTouGin-JCPL-19} together with the first attempt to generalize this approach to excited states\cite{GinSceTouLoo-JCP-19}.
The goal of the present work is to push the development of this new theory toward the description of strongly correlated systems.
The paper is organized as follows: in section \ref{sec:theory} we recall the mathematical framework of the basis set correction and we expose the extension for strongly correlated systems. Within the present development, two important formal properties are imposed: the extensivity of the correlation energies together with the $S_z$ independence of the results.
Then in section \ref{sec:results} we discuss the potential energy surfaces (PES) of the C$_2$, N$_2$, O$_2$, F$_2$ and H$_{10}$ molecules up to full dissociation as a prototype of strongly correlated problems. Finally, we conclude in section \ref{sec:conclusion}.
Building on the development of RSDFT, a possible solution to the basis-set convergence problem has been recently proposed by some of the present authors~\cite{GinPraFerAssSavTou-JCP-18} where RSDFT functionals are used to recover only the correlation effects outside a given basis set. The key point here is to realize that a wave function developed in an incomplete basis set is cuspless and could also come from a Hamiltonian with a non divergent electron-electron interaction. Therefore, a mapping with RSDFT can be introduced through the introduction of an effective non-divergent interaction representing the usual Coulomb electron-electron interaction projected in an incomplete basis set. First applications to weakly correlated molecular systems have been successfully carried out~\cite{LooPraSceTouGin-JCPL-19}, together with extensions of this approach to the calculations of excitation energies~\cite{GinSceTouLoo-JCP-19} and ionization potentials~\cite{LooPraSceGinTou-ARX-19}. The goal of the present work is to further develop this approach for the description of strongly correlated systems. The paper is organized as follows. In Section \ref{sec:theory} we recall the mathematical framework of the basis-set correction and we present the extension for strongly correlated systems. In particular, we focus on imposition of two important formal properties: size-consistency and spin-multiplet degeneracy.
Then, in Section \ref{sec:results} we apply the method to the calculation of the potential energy curves of the C$_2$, N$_2$, O$_2$, F$_2$, and H$_{10}$ molecules up to the dissociation limit, representing prototypes of strongly correlated systems. Finally, we conclude in Section \ref{sec:conclusion}.
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theory}
\label{sec:theory}
%%%%%%%%%%%%%%%%%%%%%%%%
As the theoretical framework of the basis set correction has been exposed in details in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we briefly recall the main equations and concepts needed for this study in sections \ref{sec:basic}, \ref{sec:wee} and \ref{sec:mur}.
More specifically, in section \ref{sec:basic} we recall the basic mathematical framework of the present theory by introducing the density functional complementary to a basis set $\Bas$.
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}.
Then, in section \ref{sec:functional} we expose the new approximated functionals complementary to a basis set $\Bas$ based on RSDFT functionals. The generic form of such functionals is exposed in section \ref{sec:functional_form}, their properties in the context of the basis set correction is discussed in \ref{sec:functional_prop}, and the requirements for strong correlation is discussed in section \ref{sec:requirements}. Then, the actual form of the functionals used in this work are introduced in section \ref{sec:final_def_func}.
As the theory of the basis-set correction has been exposed in details in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we only briefly recall the main equations and concepts needed for this study in Section \ref{sec:basic}, \ref{sec:wee}, and \ref{sec:mur}. More specifically, in Section \ref{sec:basic} we recall the basic mathematical framework of the present theory by introducing the density functional complementary to a basis set $\Bas$. In Section \ref{sec:wee} we introduce the effective non-divergent interaction in the basis set $\Bas$, which leads us to the definition of the effective local range-separation parameter in Section \ref{sec:mur}. Then, in Section \ref{sec:functional} we expose the new approximate complementary functionals based on RSDFT. The generic form of such functionals is exposed in Section \ref{sec:functional_form}, their properties in the context of the basis-set correction are discussed in Section \ref{sec:functional_prop}, and the specific requirements for strong correlation are discussed in Section \ref{sec:requirements}. Finally, the actual form of the functionals used in this work are introduced in Section \ref{sec:final_def_func}.
\subsection{Basic formal equations}
\label{sec:basic}
The exact ground state energy $E_0$ of a $N-$electron system can be obtained by an elegant mathematical framework connecting WFT and DFT, that is the Levy-Lieb constrained search formalism which reads
The exact ground-state energy $E_0$ of a $N$-electron system can in principle be obtained in DFT by a minimization over $N$-electron density $\denr$
\begin{equation}
\label{eq:levy}
E_0 = \min_{\denr} \bigg\{ F[\denr] + (v_{\text{ne}} (\br{}) |\denr) \bigg\},
E_0 = \min_{\den} \bigg\{ F[\den] + \int \d \br{} v_{\text{ne}} (\br{}) \denr \bigg\},
\end{equation}
where $(v_{ne}(\br{})|\denr)$ is the nuclei-electron interaction for a given density $\denr$ and $F[\denr]$ is the so-called Levy-Liev universal density functional
where $v_{ne}(\br{})$ is the nuclei-electron potential, and $F[\den]$ is the universal Levy-Lieb density functional written with the constrained search formalism as~\cite{Lev-PNAS-79,Lie-IJQC-83}
\begin{equation}
\label{eq:levy_func}
F[\denr] = \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop +\weeop }{\Psi}.
F[\den] = \min_{\Psi \rightarrow \den} \elemm{\Psi}{\kinop +\weeop }{\Psi},
\end{equation}
The minimizing density $n_0$ of eq. \eqref{eq:levy} is the exact ground state density.
Nevertheless, in practical calculations the minimization is performed over the set $\setdenbasis$ which are the densities representable in a basis set $\Bas$ and we assume from thereon that the densities used in the equations belong to $\setdenbasis$.
where the notation $\Psi \rightarrow \den$ means that the wave function $\Psi$ yields the density $n$. The minimizing density $n_0$ in Eq.~\eqref{eq:levy} is the exact ground-state density. Nevertheless, in practical calculations, the accessible densities are necessarily restricted to the set of densities ``representable in a basis set $\Bas$'', i.e. densities coming from wave functions expandable in the Hilbert space generated by the basis set $\Bas$. In the following, we always implicitly consider only such densities representable in $\Bas$. With this restriction, Eq.~\eqref{eq:levy} gives then an upper bound $E_0^\Bas$ of the exact ground-state energy. Since the density has a fast convergence with the size of the basis set, this restriction is a rather weak one and we can consider that $E_0^\Bas$ is a very good approximation to the exact ground-state energy: $E_0^\Bas \approx E_0$.
In the present context it is important to notice that in order to recover the \textit{exact} ground state energy, the wave functions $\Psi$ involved in the definition of eq. \eqref{eq:levy_func} must be developed in a complete basis set.
An important step proposed originally by some of the present authors in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}
was to propose to split the minimization in the definition of $F[\denr]$ using $\wf{}{\Bas}$ which are wave functions developed in $\basis$
In the present context, it is important to notice that in the definition of Eq.~\eqref{eq:levy_func} the wave functions $\Psi$ involved have no restriction to a finite basis set, i.e. they should be expanded in a complete basis set. In Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, it was then propose to decompose $F[\den]$ as
\begin{equation}
\label{eq:def_levy_bas}
F[\denr] = \min_{\wf{}{\Bas} \rightarrow \denr} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\denr},
F[\den] = \min_{\wf{}{\Bas} \rightarrow \den} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\den},
\end{equation}
which leads to the following definition of $\efuncden{\denr}$ which is the density functional complementary to the basis set $\Bas$
where $\wf{}{\Bas}$ are wave functions expandable in the Hilbert space generated by $\basis$, and $\efuncden{\den}$ is the density functional complementary to the basis set $\Bas$ defined as
\begin{equation}
\begin{aligned}
\efuncden{\denr} =& \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop +\weeop }{\Psi} \\ 
&- \min_{\Psi^{\Bas} \rightarrow \denr} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}}.
\efuncden{\den} = \min_{\Psi \rightarrow \den} \elemm{\Psi}{\kinop +\weeop }{\Psi}  
- \min_{\Psi^{\Bas} \rightarrow \den} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}}.
\end{aligned}
\end{equation}
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.
Introducing the decomposition in Eq. \eqref{eq:def_levy_bas} back into Eq.~\eqref{eq:levy}, we arrive at the following expression for $E_0^\Bas$
\begin{eqnarray}
\label{eq:E0basminPsiB}
E_0^\Bas &=& \min_{\Psi^{\Bas}} \bigg\{ \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\den_{{\Psi^{\Bas}}}}
\nonumber\\
&&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + \int \d \br{} v_{\text{ne}} (\br{}) \den_{\Psi^{\Bas}}(\br{}) \bigg\},
\end{eqnarray}
where the minimization is only over wave functions $\wf{}{\Bas}$ restricted to the basis set $\basis$ and $\den_{{\Psi^{\Bas}}}(\br{})$ refers to the density extracted from $\wf{}{\Bas}$. Therefore, with Eq.~\eqref{eq:E0basminPsiB}, one can properly combine a WFT calculation in a finite basis set with a density functional accounting for the correlation effects not included in the basis set.
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 energy (see equations 12-15 of Ref. \onlinecite{GinPraFerAssSavTou-JCP-18})
As a simple non-self-consistent version of this approach, we can approximate the minimizing wave function $\Psi^{\Bas}$ in Eq.~\eqref{eq:E0basminPsiB} by the ground-state FCI wave function $\psifci$ within $\Bas$, and we then obtain the following approximation for the exact ground-state energy (see Eqs. (12)-(15) of Ref. \onlinecite{GinPraFerAssSavTou-JCP-18})
\begin{equation}
\label{eq:e0approx}
E_0 = \efci + \efuncbasisFCI
E_0 \approx E_0^\Bas \approx \efci + \efuncbasisFCI,
\end{equation}
where $\efci$ is the ground state FCI energy within $\Bas$. As it was originally shown in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18} and further emphasized in Ref. \onlinecite{LooPraSceTouGin-JCPL-19,GinSceTouLoo-JCP-19}, the main role of $\efuncbasisFCI$ is to correct for the basis set incompleteness errors, a large part of which originates from the lack of cusp in any wave function developed in an incomplete basis set.
The whole purpose of this paper is to determine approximations for $\efuncbasisFCI$ which are suited for treating strong correlation regimes. The two requirement for such conditions are that i) it must provide size extensive energies, ii) it is invariant of the $S_z$ component of a given spin multiplicity.
where $\efci$ and $n_\text{FCI}^\Bas$ are the ground-state FCI energy and density, respectively. As it was originally shown in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18} and further emphasized in Ref. \onlinecite{LooPraSceTouGin-JCPL-19,GinSceTouLoo-JCP-19}, the main role of $\efuncbasisFCI$ is to correct for the basis-set incompleteness error, a large part of which originating from the lack of electron-electron cusp in the wave function expanded in an incomplete basis set. The whole purpose of this work is to determine approximations for $\efuncbasisFCI$ which are suitable for treating strong correlation regimes. Two requirements on the approximations for this purpose are i) size consistency and ii) spin-multiplet degeneracy.
\subsection{Definition of an effective interaction within $\Bas$}
\label{sec:wee}
As 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 impact of 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 originally shown by Kato\cite{Kat-CPAM-57}, the cusp in the exact wave function originates from the divergence of the Coulomb interaction at the coalescence point. Therefore, a cuspless wave function $\wf{}{\Bas}$ could also be obtained from a Hamiltonian with a non-divergent electron-electron interaction. In other words, the impact of the incompleteness of a finite basis set can be understood as the removal of the divergence of the usual Coulomb electron-electron interaction at the coalescence point.
As 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, the effective interaction associated to a given wave function $\wf{}{\Bas}$ is defined as
As originally derived in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18} (Section D and Appendices), one can obtain an effective non-divergent electron-electron interaction, here referred to as $\wbasis$, which reproduces the expectation value of the Coulomb electron-electron interaction operator over a given wave function $\wf{}{\Bas}$. As we are interested in the behaviour at the coalescence point, we focus on the opposite-spin part of the electron-electron interaction. More specifically, the effective electron-electron interaction associated to a given wave function $\wf{}{\Bas}$ is defined as
\begin{equation}
\label{eq:wbasis}
\wbasis =
@ -416,56 +386,56 @@ More specifically, the effective interaction associated to a given wave function
\infty, & \text{otherwise,}
\end{cases}
\end{equation}
where $\twodmrdiagpsi$ is the opposite spin two-body density associated to $\wf{}{\Bas}$
where $\twodmrdiagpsi$ is the opposite-spin pair density associated with $\wf{}{\Bas}$
\begin{equation}
\twodmrdiagpsi = \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
\end{equation}
$\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$ its associated two-body tensor, $\SO{p}{}$ are the spatial orthonormal orbitals,
and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$ its associated tensor in a basis of spatial orthonormal orbitals $\{\SO{p}{}\}$, and $\fbasis$ is
\begin{equation}
\label{eq:fbasis}
\fbasis
= \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
\end{equation}
and $\V{pq}{rs}=\langle pq | rs \rangle$ are the usual two-electron Coulomb integrals.
with the usual two-electron Coulomb integrals $\V{pq}{rs}=\langle pq | rs \rangle$.
With such a definition, one can show that $\wbasis$ satisfies
\begin{equation}
\int \int \dr{1} \dr{2} \wbasis \twodmrdiagpsi = \int \int \dr{1} \dr{2} \frac{\twodmrdiagpsi}{|\br{1}-\br{2}|}.
\end{equation}
As it was shown in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, the effective interaction $\wbasis$ is necessary finite at coalescence for an incomplete basis set, and tends to the regular coulomb interaction in the limit of a complete basis set for any choice of wave function $\psibasis$, that is
\begin{eqnarray}
\frac{1}{2}\iint \dr{1} \dr{2} \wbasis \twodmrdiagpsi =
\nonumber\\
\frac{1}{2} \iint \dr{1} \dr{2} \frac{\twodmrdiagpsi}{|\br{1}-\br{2}|}.
\end{eqnarray}
As shown in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, the effective interaction $\wbasis$ is necessarily finite at coalescence for an incomplete basis set, and tends to the usual Coulomb interaction in the CBS limit for any choice of wave function $\psibasis$, i.e.
\begin{equation}
\label{eq:cbs_wbasis}
\lim_{\Bas \rightarrow \text{CBS}} \wbasis = \frac{1}{|\br{1}-\br{2}|}\quad \forall\,\psibasis.
\lim_{\Bas \rightarrow \text{CBS}} \wbasis = \frac{1}{|\br{1}-\br{2}|},\quad \forall\,\psibasis.
\end{equation}
The condition of eq. \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 in Eq.~\eqref{eq:cbs_wbasis} is fundamental as it guarantees the correct behavior of the theory in the CBS limit.
\subsection{Definition of a range-separation parameter varying in real space}
\subsection{Definition of a local range-separation parameter}
\label{sec:mur}
\subsubsection{General definition}
As the effective interaction within a basis set $\wbasis$ is non divergent, 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, it ressembles the long-range interaction used in RSDFT
\begin{equation}
\label{eq:weelr}
w_{ee}^{\lr}(\mu;r_{12}) = \frac{\text{erf}\big(\mu \,r_{12} \big)}{r_{12}}.
w_\text{ee}^{\lr}(\mu;r_{12}) = \frac{\text{erf}\big(\mu \,r_{12} \big)}{r_{12}},
\end{equation}
As originally proposed in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we use a range-separation parameter $\murpsi$ varying in real space
where $\mu$ is the range-separation parameter. As originally proposed in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we make the correspondance between these two interactions by using the local range-separation parameter $\murpsi$
\begin{equation}
\label{eq:def_mur}
\murpsi = \frac{\sqrt{\pi}}{2} \wbasiscoal
\murpsi = \frac{\sqrt{\pi}}{2} \wbasiscoal,
\end{equation}
such that
such that the interactions coincide at the electron-electron colescence point for each $\br{}$
\begin{equation}
w_{ee}^{\lr}(\murpsi;0) = \wbasiscoal \quad \forall \, \br{}.
w_\text{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})
Because of the very definition of $\wbasis$, one has the following property in the CBS limit (see Eq.~\eqref{eq:cbs_wbasis})
\begin{equation}
\label{eq:cbs_mu}
\lim_{\Bas \rightarrow \text{CBS}} \murpsi = \infty\quad \forall \,\psibasis,
\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.
which is again fundamental to guarantee the correct behavior of the theory in the CBS limit.
\subsubsection{Frozen core density approximation}
As all WFT calculations for the purpose of that work are performed within the frozen core approximation, we use the valence-only versions of the various quantities needed for the complementary basis set functional introduced in Ref. \cite{LooPraSceTouGin-JCPL-19}.
We split the basis set as $\Bas = \Cor \bigcup \BasFC$ (where $\Cor$ and $\BasFC$ are the sets of core and active MOs, respectively)
and define the valence only range separation parameter
\subsubsection{Frozen-core approximation}
As all WFT calculations in this work are performed within the frozen-core approximation, we use the valence-only version of the various quantities needed for the complementary basis functional introduced in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}. We partition the basis set as $\Bas = \Cor \bigcup \BasFC$, where $\Cor$ and $\BasFC$ are the sets of core and active orbitals, respectively, and define the valence-only local range-separation parameter as
\begin{equation}
\label{eq:def_mur_val}
\murpsival = \frac{\sqrt{\pi}}{2} \wbasiscoalval{},
@ -475,114 +445,121 @@ where $\wbasisval$ is the valence-only effective interaction defined as
\label{eq:wbasis_val}
\wbasisval =
\begin{cases}
\fbasisval /\twodmrdiagpsi, & \text{if $\twodmrdiagpsival \ne 0$,}
\fbasisval /\twodmrdiagpsival, & \text{if $\twodmrdiagpsival \ne 0$,}
\\
\infty, & \text{otherwise,}
\end{cases}
\end{equation}
where $\fbasisval$ is defined as
where $\fbasisval$ and $\twodmrdiagpsival$ are defined as
\begin{equation}
\label{eq:fbasis_val}
\fbasisval
= \sum_{pq\in \Bas} \sum_{rstu \in \BasFC} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
\end{equation}
and $\twodmrdiagpsival$
and
\begin{equation}
\label{eq:twordm_val}
\twodmrdiagpsival = \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
\twodmrdiagpsival = \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}.
\end{equation}
Notice the summations on the active set of orbitals in eqs. \eqref{eq:fbasis_val} and \eqref{eq:twordm_val}.
It is noteworthy that, within the present definition, $\wbasisval$ still tends to the regular Coulomb interaction as $\Bas \to \CBS$.
Notice the restrictions of the sums to the set of active orbitals in Eqs.~\eqref{eq:fbasis_val} and \eqref{eq:twordm_val}.
It is noteworthy that, with the present definition, $\wbasisval$ still tends to the usual Coulomb interaction as $\Bas \to \CBS$.
\subsection{Generic form and properties of the approximations for $\efuncden{\denr}$ }
\subsection{Generic form and properties of the approximations for $\efuncden{\den}$ }
\label{sec:functional}
\subsubsection{Generic form of the approximated functionals}
\subsubsection{Generic form of the approximate functionals}
\label{sec:functional_form}
As originally proposed and motivated in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the complementary basis 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 $\denr$, spin polarisation $\zeta(\br{})$, reduced density gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$ and the on-top pair density $\ntwo(\br{})$. In the present work, all the density-related quantities are computed with the same wave function $\psibasis$ used to define $\murpsi$.
Therefore, a given approximation X of $\efuncden{\denr}$ have the following generic form
As originally proposed and motivated in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the complementary basis functional $\efuncden{\den}$ by using the so-called correlation energy functional with multideterminant reference (ECMD) introduced by Toulouse \textit{et al.}~\cite{TouGorSav-TCA-05}. Following the recent work in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}, we propose to use a Perdew-Burke-Ernzerhof (PBE)-like functional which uses the total density $\denr$, the spin polarisation $\zeta(\br{})=[n_\uparrow(\br{})-n_\downarrow(\br{})]/\denr$, the reduced density gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$, and the on-top pair density $\ntwo(\br{})\equiv \ntwo(\br{},\br{})$. In the present work, all these quantities are computed with the same wave function $\psibasis$ used to define $\mur \equiv\murpsi$. Therefore, a given approximation X of $\efuncden{\den}$ will have the following generic local form
\begin{equation}
\begin{aligned}
\label{eq:def_ecmdpbebasis}
\efuncdenpbe{\argebasis} = &\int d\br{} \,\denr \\ & \ecmd(\argrebasis)
&\efuncdenpbe{\argebasis} = \;\;\;\;\;\;\;\; \\ &\int \d\br{} \,\denr \ecmd(\argrebasis),
\end{aligned}
\end{equation}
where $\ecmd(\argecmd)$ is the ECMD correlation energy density defined as
where $\ecmd(\argecmd)$ is the correlation energy per particle taken as
\begin{equation}
\label{eq:def_ecmdpbe}
\ecmd(\argecmd) = \frac{\varepsilon_{\text{c,PBE}}(\argepbe)}{1+ \mu^3 \beta(\argepbe)}
\ecmd(\argecmd) = \frac{\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)}{1+ \beta(\argepbe) \; \mu^3},
\end{equation}
with
\begin{equation}
\label{eq:def_beta}
\beta(\argebasis) = \frac{3}{2\sqrt{\pi}(1 - \sqrt{2})}\frac{\varepsilon_{\text{c,PBE}}(\argepbe)}{\ntwo/\den},
\beta(\argepbe) = \frac{3}{2\sqrt{\pi}(1 - \sqrt{2})}\frac{\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)}{\ntwo/\den},
\end{equation}
and where $\varepsilon_{\text{c,PBE}}(\argepbe)$ is the usual PBE correlation energy density\cite{PerBurErn-PRL-96}. Before introducing the different flavour of approximated functionals that we will use here (see \ref{sec:def_func}), we would like to give some motivations for the such a choice of functional form.
where $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ is the usual PBE correlation energy per particle~\cite{PerBurErn-PRL-96}. Before introducing the different flavors of approximate functionals that we will use here (see Section~\ref{sec:def_func}), we would like to give some motivations for this choice of functional form.
The 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
The functional form of $\ecmd(\argecmd)$ in Eq.~\ref{eq:def_ecmdpbe} has been originally proposed in Ref.~\onlinecite{FerGinTou-JCP-18} in the context of RSDFT. In the $\mu\to 0$ limit, it reduces to the usual PBE correlation functional
\begin{equation}
\lim_{\mu \rightarrow 0} \ecmd(\argecmd) = \varepsilon_{\text{c,PBE}}(\argepbe),
\lim_{\mu \rightarrow 0} \ecmd(\argecmd) = \varepsilon_{\text{c}}^{\text{PBE}}(\argepbe),
\end{equation}
which can be qualified as the weak correlation regime, and the large $\mu$ limit
which is relevant in the weak-correlation (or high-density) limit. In the large-$\mu$ limit, it behaves as
\begin{equation}
\label{eq:lim_mularge}
\ecmd(\argecmd) = \frac{1}{\mu^3} \ntwo + o(\frac{1}{\mu^5}),
\ecmd(\argecmd) \isEquivTo{\mu\to\infty} \frac{2\sqrt{\pi}(1 - \sqrt{2})}{3 \mu^3} \frac{\ntwo}{n},
\end{equation}
which, as it was previously shown\cite{TouColSav-PRA-04, GoriSav-PRA-06,PazMorGorBac-PRB-06} by various authors, is the exact expression for the ECMD in the limit of large $\mu$, provided that $\ntwo$ is the \textit{exact} on-top pair density of the system.
In the context of RSDFT, some of the present authors have illustrated in Ref.~\onlinecite{FerGinTou-JCP-18} that the on-top pair density involved in eq. \eqref{eq:def_ecmdpbe} plays a crucial role when reaching the strong correlation regime. The importance of the on-top pair density in the strong correlation regime have been also acknowledged by Pernal and co-workers\cite{GritMeePer-PRA-18} and Gagliardi and co-workers\cite{CarTruGag-JPCA-17}.
Also, $\ecmd(\argecmd) $ vanishes when $\ntwo$ vanishes
which is the exact large-$\mu$ behavior of the exact ECMD correlation energy~\cite{PazMorGorBac-PRB-06,FerGinTou-JCP-18}. Of course, for a specific system, the large-$\mu$ behavior will be exact only if one uses for $n_2$ the \textit{exact} on-top pair density of this system. This large-$\mu$ limit in Eq.~\eqref{eq:lim_mularge} is relevant in the strong-correlation (or low-density) limit. In the context of RSDFT, some of the present authors have illustrated in Ref.~\onlinecite{FerGinTou-JCP-18} that the on-top pair density involved in Eq. \eqref{eq:def_ecmdpbe} plays indeed a crucial role when reaching the strong-correlation regime. The importance of the on-top pair density in the strong-correlation regime have been also recently acknowledged by Gagliardi and coworkers~\cite{CarTruGag-JPCA-17} and Pernal and coworkers\cite{GritMeePer-PRA-18}.
Note also that $\ecmd(\argecmd)$ vanishes when $\ntwo$ vanishes
\begin{equation}
\label{eq:lim_n2}
\lim_{\ntwo \rightarrow 0} \ecmd(\argecmd) = 0
\lim_{\ntwo \rightarrow 0} \ecmd(\argecmd) = 0,
\end{equation}
which is exact for systems with a vanishing on-top pair density, such as the totally dissociated H$_2$ which is the archetype of strongly correlated systems.
Also, the function $\ecmd(\argecmd)$ vanishes when $\mu \rightarrow \infty$ as all RSDFT functionals
\begin{equation}
which is expected for systems with a vanishing on-top pair density, such as the totally dissociated H$_2$ molecule which is the archetype of strongly correlated systems. Finally, the function $\ecmd(\argecmd)$ vanishes when $\mu \rightarrow \infty$ like all RSDFT short-range functionals \begin{equation}
\label{eq:lim_muinf}
\lim_{\mu \rightarrow \infty} \ecmd(\argecmd) = 0.
\end{equation}
\subsubsection{Properties of approximated functionals}
\subsubsection{Properties of approximate functionals}
\label{sec:functional_prop}
Within the definition of \eqref{eq:def_mur} and \eqref{eq:def_ecmdpbebasis}, any approximated complementary basis set functionals $\efuncdenpbe{\argecmd}$ satisfies two important properties.
Because of the properties \eqref{eq:cbs_mu} and \eqref{eq:lim_muinf}, $\efuncdenpbe{\argecmd}$ vanishes when reaching the complete basis set limit, whatever the wave function $\psibasis$ used to define the range separation parameter $\mu_{\Psi^{\basis}}$:
Within the definitions of Eqs.~\eqref{eq:def_mur} and \eqref{eq:def_ecmdpbebasis}, any approximate complementary basis functional $\efuncdenpbe{\argebasis}$ satisfies two important properties.
First, thanks to the properties in Eq.~\eqref{eq:cbs_mu} and~\eqref{eq:lim_muinf}, $\efuncdenpbe{\argebasis}$ vanishes in the CBS limit, whatever the wave function $\psibasis$ used to define the local range-separation parameter $\mu(\br{})$,
\begin{equation}
\label{eq:lim_ebasis}
\lim_{\basis \rightarrow \text{CBS}} \efuncdenpbe{\argecmd} = 0\quad \forall\, \psibasis,
\lim_{\basis \rightarrow \text{CBS}} \efuncdenpbe{\argebasis} = 0, \quad \forall\, \psibasis,
\end{equation}
which guarantees an unaltered limit when reaching the CBS limit.
Also, the $\efuncdenpbe{\argecmd}$ vanishes for systems with vanishing on-top pair density, which guarantees the good limit in the case of stretched H$_2$ and for one-electron system.
Such a property is guaranteed independently by i) the definition of the effective interaction $\wbasis$ (see eq. \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 eq. \eqref{eq:lim_n2}).
%<<<<<<< HEAD
%which guarantees an unaltered limit when reaching the CBS limit.
%Also, the $\efuncdenpbe{\argecmd}$ vanishes for systems with vanishing on-top pair density, which guarantees the good limit in the case of stretched H$_2$ and for one-electron system.
%Such a property is guaranteed independently by i) the definition of the effective interaction $\wbasis$ (see eq. \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 eq. \eqref{eq:lim_n2}).
%=======
%which guarantees an unaltered CBS limit.
%>>>>>>> e7e87f50643a28a43e403bc17effeaa48bb01e35
\subsection{Requirements for the approximated functionals in the strong correlation regime}
Second, the fact that $\efuncdenpbe{\argebasis}$ vanishes for systems with vanishing on-top pair density guarantees the correct limit for one-electron systems and for the stretched H$_2$ molecule. This property is guaranteed independently by i) the definition of the effective interaction $\wbasis$ (see Eq.~\eqref{eq:wbasis}) together with the condition in Eq.~\eqref{eq:lim_muinf}, ii) the fact that $\ecmd(\argecmd)$ vanishes when the on-top pair density vanishes (see Eq.~\eqref{eq:lim_n2}).
\subsection{Requirements for the approximate functionals in the strong-correlation regime}
\label{sec:requirements}
\subsubsection{Requirements: 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 mandatory to avoid any ambiguity in computing interaction energies.
When two subsystems $A$ and $B$ dissociate in closed shell systems, as in the case of weak interactions for instance, a simple RHF 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 RHF 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, which is also a property of any exact wave function.
Such a property is also important in the context of covalent bond breaking where the ground state of the super system $A+B$ is in general of low spin while the ground states of the fragments $A$ and $B$ are in high spin which can have multiple $S_z$ components.
\subsubsection{Condition for the functional $\efuncdenpbe{\argebasis}$ to obtain $S_z$ invariance}
A sufficient condition to achieve $S_z$ invariance is to eliminate all dependency to $S_z$, which in the case of $\ecmd(\argecmd)$ is the spin polarisation $\zeta(\br{})$ involved in the correlation energy density $\varepsilon_{\text{c,PBE}}(\argepbe)$ (see eq. \eqref{eq:def_ecmdpbe}).
As originally shown by Perdew and co-workers\cite{PerSavBur-PRA-95}, the dependence on the spin polarisation in the KS-DFT framework can be removed by the rewriting the spin polarisation of a single Slater determinant with only the on-top pair density and the total density. In other terms, the spin density dependence usually introduced in the correlation functionals of KS-DFT tries to mimic the effect of the on-top pair density.
\subsubsection{Requirements: size-consistency and spin-multiplet degeneracy}
An important requirement for any electronic-structure method is size-consistency, i.e. the additivity of the energies of non-interacting fragments, which is mandatory to avoid any ambiguity in computing interaction energies. When two subsystems $A$ and $B$ dissociate in closed-shell systems, as in the case of weak intermolecular interactions for instance, spin-restricted Hartree-Fock (RHF) is size-consistent. When the two subsystems dissociate in open-shell systems, such as in covalent bond breaking, it is well known that the RHF approach fails and an alternative is to use a complete-active-space (CAS) wave function which, provided that the active space has been properly chosen, leads to additive energies.
Another important requirement is spin-multiplet degeneracy, i.e. the independence of the energy with respect to the $S_z$ component of a given spin state, which is also a property of any exact wave function. Such a property is also important in the context of covalent bond breaking where the ground state of the supersystem $A+B$ is generally low spin while the ground states of the fragments $A$ and $B$ are high spin and can have multiple $S_z$ components.
\subsubsection{Condition for the functional $\efuncdenpbe{\argebasis}$ to obtain spin-multiplet degeneracy}
A sufficient condition to achieve spin-multiplet degeneracy is to eliminate all dependencies on $S_z$, which in the case of the functional $\ecmd(\argecmd)$ means removing the dependence on the spin polarization $\zeta(\br{})$ used in the PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ (see Eq. \eqref{eq:def_ecmdpbe}).
As originally shown by Perdew and co-workers\cite{BecSavSto-TCA-95,PerSavBur-PRA-95}, the dependence on the spin polarisation in the KS-DFT framework can be removed by the rewriting the spin polarisation of a single Slater determinant with only the on-top pair density and the total density. In other terms, the spin density dependence usually introduced in the correlation functionals of KS-DFT tries to mimic the effect of the on-top pair density.
Based on this reasoning, a similar approach has been used in the context of multi configurational DFT in order to remove the $S_z$ dependency.
In practice, these approaches introduce the effective spin polarisation
\begin{equation}
\label{eq:def_effspin}
\tilde{\zeta}(n,\ntwo_{\psibasis}) =
\tilde{\zeta}(n,\ntwo^{\psibasis}) =
% \begin{cases}
\sqrt{ n^2 - 4 \ntwo_{\psibasis} }
\sqrt{ n^2 - 4 \ntwo^{\psibasis} }
% 0 & \text{otherwise.}
% \end{cases}
\end{equation}
which uses the on-top pair density $\ntwo_{\psibasis}$ of a given wave function $\psibasis$.
which uses the on-top pair density $\ntwo^{\psibasis}$ of a given wave function $\psibasis$.
The advantages of this approach are at least two folds: i) the effective spin polarisation $\tilde{\zeta}$ is $S_z$ invariant, ii) it introduces an indirect dependency on the on-top pair density of the wave function $\psibasis$ which usually improves the treatment of strong correlation.
Nevertheless, the use of $\tilde{\zeta}$ presents several disadvantages as it can become complex when $n^2 - 4 \ntwo_{\psibasis}<0$ and also
the formula of eq. \eqref{eq:def_effspin} is exact only when the density $n$ and on-top pair density $\ntwo_{\psibasis}$ are obtained from a single determinant\cite{PerSavBur-PRA-95}, but it is applied to multi configurational wave functions.
Nevertheless, the use of $\tilde{\zeta}$ presents several disadvantages as it can become complex when $n^2 - 4 \ntwo^{\psibasis}<0$ and also
the formula of equation \eqref{eq:def_effspin} is exact only when the density $n$ and on-top pair density $\ntwo^{\psibasis}$ are obtained from a single determinant\cite{PerSavBur-PRA-95}, but it is applied to multi configurational wave functions.
An alternative to eliminate the $S_z$ dependency would be to simply set $\zeta(\br{})=0$, but this would lower the accuracy of the usual correlation functional, such as the PBE correlation functional used here $\varepsilon_{\text{c,PBE}}(\argepbe)$. Nevertheless, as the spin polarisation usually tries to mimic the on-top pair density and the function $\ecmd(\argecmd)$ explicitly depends on the on-top pair density (see eqs. \eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}), we propose here to use the $\ecmd$ functional with \textit{a zero spin polarisation}. This ensures a $S_z$ invariance and, as will be numerically shown, very weakly affect the accuracy of the functional.
An alternative to eliminate the $S_z$ dependency would be to simply set $\zeta(\br{})=0$, but this would lower the accuracy of the usual correlation functional, such as the PBE correlation functional used here $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$. Nevertheless, as the spin polarisation usually tries to mimic the on-top pair density and the function $\ecmd(\argecmd)$ explicitly depends on the on-top pair density (see equations \eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}), we propose here to use the $\ecmd$ functional with \textit{a zero spin polarisation}. This ensures a $S_z$ invariance and, as will be numerically shown, very weakly affect the accuracy of the functional.
\subsubsection{Conditions on $\psibasis$ for the extensivity}
In the case of the present basis set correction, as $\efuncdenpbe{\argebasis}$ is an integral over $\mathbb{R}^3$ of local quantities, in the case of non overlapping fragments $A\ldots B$ it can be written as the sum of two local contributions: one coming from the integration over the region of the sub-system $A$ and the other one from the region of the sub-system $B$.
@ -598,19 +575,19 @@ The condition for the active space involved in the CASSCF wave function is that
As the present work focusses on the strong correlation regime, we propose here to investigate only approximated functionals which are $S_z$ invariant and size extensive in the case of covalent bond breaking. Therefore, the wave function $\psibasis$ used throughout this paper are of CASSCF type in order to ensure extensivity of all density related quantities.
The difference between the different flavours of functionals are only on i) the type of on-top pair density used, and ii) the type of spin polarisation used.
Regarding the spin polarisation that enters into $\varepsilon_{\text{c,PBE}}(\argepbe)$, two different types of $S_z$ invariant formulations are used: i) the \textit{effective} spin polarization $\tilde{\zeta}$ defined in eq. \eqref{eq:def_effspin}, and iii) a \textit{zero} spin polarization.
Regarding the spin polarisation that enters into $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$, two different types of $S_z$ invariant formulations are used: i) the \textit{effective} spin polarization $\tilde{\zeta}$ defined in equation \eqref{eq:def_effspin}, and iii) a \textit{zero} spin polarization.
Regarding the approximation to the \textit{exact} on-top pair density entering in eq. \eqref{eq:def_beta}, we use two different approximations. The first one is based on the uniform electron gas (UEG) and reads
\begin{equation}
\label{eq:def_n2ueg}
\ntwo_{\text{UEG}}(n,\zeta,\br{}) = n(\br{})^2\big(1-\zeta(\br{})\big)g_0\big(n(\br{})\big)
\ntwo^{\text{UEG}}(n,\zeta,\br{}) = n(\br{})^2\big(1-\zeta(\br{})\big)g_0\big(n(\br{})\big)
\end{equation}
where the pair-distribution function $g_0(n)$ is taken from equation (46) of Ref. \onlinecite{GorSav-PRA-06}. As some spin polarization appear in eq. \eqref{eq:def_n2ueg}, we use the effective spin density $\tilde{\zeta}$ of eq. \eqref{eq:def_effspin} in order to ensure $S_z$ invariance. Notice that, as we use a CASSCF wave function and $\tilde{\zeta}$ as spin polarization, the $\ntwo_{\text{UEG}}$ will depend indirectly on the on-top pair density of the CASSCF wave function as $\tilde{\zeta}$ depends on the on-top pair density.
where the pair-distribution function $g_0(n)$ is taken from equation (46) of Ref. \onlinecite{GorSav-PRA-06}. As some spin polarization appear in equation \eqref{eq:def_n2ueg}, we use the effective spin density $\tilde{\zeta}$ of equation \eqref{eq:def_effspin} in order to ensure $S_z$ invariance. Notice that, as we use a CASSCF wave function and $\tilde{\zeta}$ as spin polarization, the $\ntwo^{\text{UEG}}$ will depend indirectly on the on-top pair density of the CASSCF wave function as $\tilde{\zeta}$ depends on the on-top pair density.
Another approach to approximate of the exact on top pair density consists in taking advantage of the on-top pair density of the wave function $\psibasis$. Following the work of some of the previous authors\cite{FerGinTou-JCP-18,GinSceTouLoo-JCP-19} we introduce the extrapolated on-top pair density $\ntwoextrap$ as
\begin{equation}
\label{eq:def_n2extrap}
\ntwoextrap(\ntwo_{\psibasis},\mu,\br{}) = \ntwo_{\wf{}{\Bas}}(\br{}) \bigg( 1 + \frac{2}{\sqrt{\pi}\murpsi} \bigg)^{-1}
\ntwoextrap(\ntwo^{\psibasis},\mu,\br{}) = \ntwo^{\wf{}{\Bas}}(\br{}) \bigg( 1 + \frac{2}{\sqrt{\pi}\murpsi} \bigg)^{-1}
\end{equation}
which directly follows from the large-$\mu$ extrapolation of the exact on-top pair density proposed by Gori-Giorgi and Savin\cite{GorSav-PRA-06} in the context of RSDFT.
When using $\ntwoextrap(\ntwo,\mu,\br{})$ in a functional, we will refer simply refer it as "ot".
@ -867,6 +844,7 @@ Also, it is shown that the basis set correction gives substantial differential c
Finally, regarding the computational cost of the present approach, it should be stressed (see supplementary materials) that it is minor with respect to WFT methods for all systems and basis set studied here. We believe that such approach is a significant step towards calculations near the CBS limit for strongly correlated systems.
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\end{document}