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%merlin.mbs aipnum4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked
%Control: key (0)
%Control: author (8) initials jnrlst
%Control: editor formatted (1) identically to author
%Control: production of article title (-1) disabled
%Control: page (0) single
%Control: year (1) truncated
%Control: production of eprint (0) enabled
\begin{thebibliography}{82}%
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%</preamble>
\bibitem [{\citenamefont {Pople}(1999)}]{Pop-RMP-99}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~A.}\ \bibnamefont
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(\bibinfo {year} {1999})}\BibitemShut {NoStop}%
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\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont
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(\bibinfo {year} {1999})}\BibitemShut {NoStop}%
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\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont
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(\bibinfo {year} {1965})}\BibitemShut {NoStop}%
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\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.~A.}\ \bibnamefont
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\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont
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\bibitem [{\citenamefont {Kutzelnigg}\ and\ \citenamefont
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont
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(\bibinfo {year} {1991})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Noga}\ and\ \citenamefont
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
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(\bibinfo {year} {1994})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Helgaker}\ \emph {et~al.}(1997)\citenamefont
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\BibitemOpen
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{NoStop}%
\bibitem [{\citenamefont {Halkier}\ \emph {et~al.}(1998)\citenamefont
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\BibitemOpen
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\bibnamefont {Wilson}},\ }\href@noop {} {\bibfield {journal} {\bibinfo
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{pages} {243} (\bibinfo {year} {1998})}\BibitemShut {NoStop}%
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\BibitemOpen
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}\href@noop {} {\enquote {\bibinfo {title} {On the duality of ring and ladder
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{arXiv:1903.05458 [cond-mat.mtrl-sci]} \BibitemShut {NoStop}%
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
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(\bibinfo {year} {2012})}\BibitemShut {NoStop}%
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{year} {2012})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Hattig}\ \emph {et~al.}(2012)\citenamefont {Hattig},
\citenamefont {Klopper}, \citenamefont {Kohn},\ and\ \citenamefont
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\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont
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{\bibfield {journal} {\bibinfo {journal} {Chem. Rev.}\ }\textbf {\bibinfo
{volume} {112}},\ \bibinfo {pages} {4} (\bibinfo {year} {2012})}\BibitemShut
{NoStop}%
\bibitem [{\citenamefont {Kong}, \citenamefont {Bischo},\ and\ \citenamefont
{Valeev}(2012)}]{KonBisVal-CR-12}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {L.}~\bibnamefont
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}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Chem. Rev.}\
}\textbf {\bibinfo {volume} {112}},\ \bibinfo {pages} {75} (\bibinfo {year}
{2012})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Gr\"uneis}\ \emph {et~al.}(2017)\citenamefont
{Gr\"uneis}, \citenamefont {Hirata}, \citenamefont {Ohnishi},\ and\
\citenamefont {Ten-no}}]{GruHirOhnTen-JCP-17}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
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\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Ten-no}},\ }\href
{\doibase 10.1063/1.4976974} {\bibfield {journal} {\bibinfo {journal} {J.
Chem. Phys.}\ }\textbf {\bibinfo {volume} {146}},\ \bibinfo {pages} {080901}
(\bibinfo {year} {2017})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Ma}\ and\ \citenamefont
{Werner}(2018)}]{MaWer-WIREs-18}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Q.}~\bibnamefont
{Ma}}\ and\ \bibinfo {author} {\bibfnamefont {H.-J.}\ \bibnamefont
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Comput. Mol. Sci.}\ }\textbf {\bibinfo {volume} {8}},\ \bibinfo {pages}
{e1371} (\bibinfo {year} {2018})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Tew}\ \emph {et~al.}(2007)\citenamefont {Tew},
\citenamefont {Klopper}, \citenamefont {Neiss},\ and\ \citenamefont
{Hattig}}]{TewKloNeiHat-PCCP-07}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~P.}\ \bibnamefont
{Tew}}, \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Klopper}},
\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {Neiss}}, \ and\ \bibinfo
{author} {\bibfnamefont {C.}~\bibnamefont {Hattig}},\ }\href@noop {}
{\bibfield {journal} {\bibinfo {journal} {Phys. Chem. Chem. Phys.}\
}\textbf {\bibinfo {volume} {9}},\ \bibinfo {pages} {1921} (\bibinfo {year}
{2007})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Ten-no}(2007)}]{Ten-CPL-07}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
{Ten-no}},\ }\href {\doibase https://doi.org/10.1016/j.cplett.2007.09.006}
{\bibfield {journal} {\bibinfo {journal} {Chemical Physics Letters}\
}\textbf {\bibinfo {volume} {447}},\ \bibinfo {pages} {175 } (\bibinfo {year}
{2007})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Shiozaki}\ and\ \citenamefont
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\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
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{volume} {133}},\ \bibinfo {pages} {141103} (\bibinfo {year} {2010})},\
\Eprint {http://arxiv.org/abs/https://doi.org/10.1063/1.3489000}
{https://doi.org/10.1063/1.3489000} \BibitemShut {NoStop}%
\bibitem [{\citenamefont {Shiozaki}, \citenamefont {Knizia},\ and\
\citenamefont {Werner}(2011)}]{TorKniWer-JCP-11}%
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
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}\href {\doibase 10.1063/1.3528720} {\bibfield {journal} {\bibinfo
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{134}},\ \bibinfo {pages} {034113} (\bibinfo {year} {2011})},\ \Eprint
{http://arxiv.org/abs/https://doi.org/10.1063/1.3528720}
{https://doi.org/10.1063/1.3528720} \BibitemShut {NoStop}%
\bibitem [{\citenamefont {Demel}\ \emph {et~al.}(2012)\citenamefont {Demel},
\citenamefont {Kedžuch}, \citenamefont {Švaňa}, \citenamefont {Ten-no},
\citenamefont {Pittner},\ and\ \citenamefont
{Noga}}]{DemStanMatTenPitNog-PCCP-12}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {O.}~\bibnamefont
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\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Švaňa}}, \bibinfo
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{\bibfnamefont {J.}~\bibnamefont {Pittner}}, \ and\ \bibinfo {author}
{\bibfnamefont {J.}~\bibnamefont {Noga}},\ }\href {\doibase
10.1039/C2CP23198K} {\bibfield {journal} {\bibinfo {journal} {Phys. Chem.
Chem. Phys.}\ }\textbf {\bibinfo {volume} {14}},\ \bibinfo {pages} {4753}
(\bibinfo {year} {2012})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Guo}\ \emph {et~al.}(2017)\citenamefont {Guo},
\citenamefont {Sivalingam}, \citenamefont {Valeev},\ and\ \citenamefont
{Neese}}]{GuoSivValNee-JCP-17}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont
{Guo}}, \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Sivalingam}},
\bibinfo {author} {\bibfnamefont {E.~F.}\ \bibnamefont {Valeev}}, \ and\
\bibinfo {author} {\bibfnamefont {F.}~\bibnamefont {Neese}},\ }\href
{\doibase 10.1063/1.4996560} {\bibfield {journal} {\bibinfo {journal} {The
Journal of Chemical Physics}\ }\textbf {\bibinfo {volume} {147}},\ \bibinfo
{pages} {064110} (\bibinfo {year} {2017})},\ \Eprint
{http://arxiv.org/abs/https://doi.org/10.1063/1.4996560}
{https://doi.org/10.1063/1.4996560} \BibitemShut {NoStop}%
\bibitem [{\citenamefont {Torheyden}\ and\ \citenamefont
{Valeev}(2009)}]{TorVal-JCP-09}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
{Torheyden}}\ and\ \bibinfo {author} {\bibfnamefont {E.~F.}\ \bibnamefont
{Valeev}},\ }\href {\doibase 10.1063/1.3254836} {\bibfield {journal}
{\bibinfo {journal} {The Journal of Chemical Physics}\ }\textbf {\bibinfo
{volume} {131}},\ \bibinfo {pages} {171103} (\bibinfo {year} {2009})},\
\Eprint {http://arxiv.org/abs/https://doi.org/10.1063/1.3254836}
{https://doi.org/10.1063/1.3254836} \BibitemShut {NoStop}%
\bibitem [{\citenamefont {Kong}\ and\ \citenamefont
{Valeev}(2011)}]{KonVal-JCP-11}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {L.}~\bibnamefont
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{Valeev}},\ }\href {\doibase 10.1063/1.3664729} {\bibfield {journal}
{\bibinfo {journal} {The Journal of Chemical Physics}\ }\textbf {\bibinfo
{volume} {135}},\ \bibinfo {pages} {214105} (\bibinfo {year} {2011})},\
\Eprint {http://arxiv.org/abs/https://doi.org/10.1063/1.3664729}
{https://doi.org/10.1063/1.3664729} \BibitemShut {NoStop}%
\bibitem [{\citenamefont {Haunschild}\ \emph {et~al.}(2012)\citenamefont
{Haunschild}, \citenamefont {Mao}, \citenamefont {Mukherjee},\ and\
\citenamefont {Klopper}}]{HauMaoMukKlo-CPL-12}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont
{Haunschild}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Mao}},
\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Mukherjee}}, \ and\
\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Klopper}},\ }\href
{\doibase https://doi.org/10.1016/j.cplett.2012.02.020} {\bibfield {journal}
{\bibinfo {journal} {Chemical Physics Letters}\ }\textbf {\bibinfo {volume}
{531}},\ \bibinfo {pages} {247 } (\bibinfo {year} {2012})}\BibitemShut
{NoStop}%
\bibitem [{\citenamefont {Booth}\ \emph {et~al.}(2012)\citenamefont {Booth},
\citenamefont {Cleland}, \citenamefont {Alavi},\ and\ \citenamefont
{Tew}}]{BooCleAlaTew-JCP-12}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.~H.}\ \bibnamefont
{Booth}}, \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Cleland}},
\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Alavi}}, \ and\ \bibinfo
{author} {\bibfnamefont {D.~P.}\ \bibnamefont {Tew}},\ }\href {\doibase
10.1063/1.4762445} {\bibfield {journal} {\bibinfo {journal} {The Journal of
Chemical Physics}\ }\textbf {\bibinfo {volume} {137}},\ \bibinfo {pages}
{164112} (\bibinfo {year} {2012})},\ \Eprint
{http://arxiv.org/abs/https://doi.org/10.1063/1.4762445}
{https://doi.org/10.1063/1.4762445} \BibitemShut {NoStop}%
\bibitem [{\citenamefont {Toulouse}, \citenamefont {Colonna},\ and\
\citenamefont {Savin}(2004)}]{TouColSav-PRA-04}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
{Toulouse}}, \bibinfo {author} {\bibfnamefont {F.}~\bibnamefont {Colonna}}, \
and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Savin}},\
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. A}\
}\textbf {\bibinfo {volume} {70}},\ \bibinfo {pages} {062505} (\bibinfo
{year} {2004})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Gori-Giorgi}\ and\ \citenamefont
{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})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Franck}\ \emph {et~al.}(2015)\citenamefont {Franck},
\citenamefont {Mussard}, \citenamefont {Luppi},\ and\ \citenamefont
{Toulouse}}]{FraMusLupTou-JCP-15}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {O.}~\bibnamefont
{Franck}}, \bibinfo {author} {\bibfnamefont {B.}~\bibnamefont {Mussard}},
\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {Luppi}}, \ and\ \bibinfo
{author} {\bibfnamefont {J.}~\bibnamefont {Toulouse}},\ }\href@noop {}
{\bibfield {journal} {\bibinfo {journal} {J. Chem. Phys.}\ }\textbf
{\bibinfo {volume} {142}},\ \bibinfo {pages} {074107} (\bibinfo {year}
{2015})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {\'Angy\'an}\ \emph {et~al.}(2005)\citenamefont
{\'Angy\'an}, \citenamefont {Gerber}, \citenamefont {Savin},\ and\
\citenamefont {Toulouse}}]{AngGerSavTou-PRA-05}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~G.}\ \bibnamefont
{\'Angy\'an}}, \bibinfo {author} {\bibfnamefont {I.~C.}\ \bibnamefont
{Gerber}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Savin}}, \
and\ \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Toulouse}},\
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. A}\
}\textbf {\bibinfo {volume} {72}},\ \bibinfo {pages} {012510} (\bibinfo
{year} {2005})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Goll}, \citenamefont {Werner},\ and\ \citenamefont
{Stoll}(2005)}]{GolWerSto-PCCP-05}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
{Goll}}, \bibinfo {author} {\bibfnamefont {H.-J.}\ \bibnamefont {Werner}}, \
and\ \bibinfo {author} {\bibfnamefont {H.}~\bibnamefont {Stoll}},\
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Phys. Chem. Chem.
Phys.}\ }\textbf {\bibinfo {volume} {7}},\ \bibinfo {pages} {3917} (\bibinfo
{year} {2005})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Toulouse}\ \emph {et~al.}(2009)\citenamefont
{Toulouse}, \citenamefont {Gerber}, \citenamefont {Jansen}, \citenamefont
{Savin},\ and\ \citenamefont {\'Angy\'an}}]{TouGerJanSavAng-PRL-09}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
{Toulouse}}, \bibinfo {author} {\bibfnamefont {I.~C.}\ \bibnamefont
{Gerber}}, \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont {Jansen}},
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\bibitem [{\citenamefont {Paziani}\ \emph {et~al.}(2006)\citenamefont
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\bibitem [{\citenamefont {Moscard\'o}\ and\ \citenamefont
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\bibitem [{\citenamefont {Gr\"afenstein}\ and\ \citenamefont
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\bibitem [{\citenamefont {Garza}\ \emph
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{\bibfnamefont {G.}~\bibnamefont {Frenking}}, \bibinfo {editor}
{\bibfnamefont {K.~S.}\ \bibnamefont {Kim}}, \ and\ \bibinfo {editor}
{\bibfnamefont {G.~E.}\ \bibnamefont {Scuseria}}}\ (\bibinfo {publisher}
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\bibinfo {pages} {1167--1189}\BibitemShut {NoStop}%
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718
Manuscript/srDFT_SC.bib
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@ -1,13 +1,55 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2019-04-03 22:18:33 +0200
%% Created for Pierre-Francois Loos at 2019-12-12 03:48:35 +0100
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@article{StaDav-CPL-01,
Author = {Viktor N. Staroverov and Ernest R. Davidson},
Date-Added = {2019-12-12 03:48:34 +0100},
Date-Modified = {2019-12-12 03:48:34 +0100},
Journal = {Chem. Phys. Lett.},
Pages = {142},
Title = {A density functional method for degenerate spin-multiplet components},
Volume = {340},
Year = {2001}}
@article{LooPraSceGinTou-ARX-19,
Author = {P.-F. Loos and B. Pradines and A. Scemama and E. Giner and J. Toulouse},
Date-Added = {2019-12-12 03:48:12 +0100},
Date-Modified = {2019-12-12 03:48:12 +0100},
Journal = {arXiv:1910.12238},
Title = {A density-based basis-set incompleteness correction for GW methods}}
@article{KalMusTou-JCP-19,
Author = {C. Kalai and B. Mussard and J. Toulouse},
Date-Added = {2019-12-12 03:47:01 +0100},
Date-Modified = {2019-12-12 03:47:01 +0100},
Doi = {10.1063/1.5108536},
Journal = {J. Chem. Phys.},
Pages = {074102},
Title = {Range-separated double-hybrid density-functional theory with coupled-cluster and random-phase approximations},
Volume = {151},
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5108536}}
@article{BarLoo-JCP-17,
Author = {Barca, Giuseppe MJ and Loos, Pierre-Fran{\c c}ois},
Date-Added = {2019-12-12 03:40:57 +0100},
Date-Modified = {2019-12-12 03:40:57 +0100},
File = {/Users/loos/Zotero/storage/DCFUMHWZ/56.pdf},
Journal = {J. Chem. Phys.},
Number = {2},
Pages = {024103},
Shorttitle = {Three-and Four-Electron Integrals Involving {{Gaussian}} Geminals},
Title = {Three-and Four-Electron Integrals Involving {{Gaussian}} Geminals: {{Fundamental}} Integrals, Upper Bounds, and Recurrence Relations},
Volume = {147},
Year = {2017}}
@article{PerRuzTaoStaScuCso-JCP-05,
Author = {J. P. Perdew and A. Ruzsinszky and J. Tao and V. N. Staroverov and G. E. Scuseria and G. I. Csonka},
Date-Added = {2019-04-03 22:17:53 +0200},
@ -207,7 +249,6 @@
Volume = {94},
Year = {1991}}
@article{Kut-TCA-85,
Author = {W. Kutzelnigg},
Date-Added = {2019-04-03 21:34:30 +0200},
@ -624,16 +665,6 @@
Volume = {62},
Year = {1997}}
@misc{AngGerSavTou-JJJ-XXa,
Author = {J. G. \'Angy\'an and I. Gerber and A. Savin and J. Toulouse},
Note = {in preparation},
Volume = {{}}}
@misc{AngGerSavTou-JJJ-XX,
Author = {J. G. \'Angy\'an and I. Gerber and A. Savin and J. Toulouse},
Note = {in preparation},
Volume = {{}}}
@article{AngGerSavTou-PRA-05,
Author = {J. G. \'Angy\'an and I. C. Gerber and A. Savin and J. Toulouse},
Journal = {Phys. Rev. A},
@ -11883,40 +11914,36 @@
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b00591}}
@article{GorSav-PRA-06,
title = {Properties of short-range and long-range correlation energy density functionals from electron-electron coalescence},
author = {Gori-Giorgi, Paola and Savin, Andreas},
journal = {Phys. Rev. A},
volume = {73},
issue = {3},
pages = {032506},
numpages = {9},
year = {2006},
month = {Mar},
publisher = {American Physical Society},
doi = {10.1103/PhysRevA.73.032506},
url = {https://link.aps.org/doi/10.1103/PhysRevA.73.032506}
}
Author = {Gori-Giorgi, Paola and Savin, Andreas},
Doi = {10.1103/PhysRevA.73.032506},
Issue = {3},
Journal = {Phys. Rev. A},
Month = {Mar},
Numpages = {9},
Pages = {032506},
Publisher = {American Physical Society},
Title = {Properties of short-range and long-range correlation energy density functionals from electron-electron coalescence},
Url = {https://link.aps.org/doi/10.1103/PhysRevA.73.032506},
Volume = {73},
Year = {2006},
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevA.73.032506},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevA.73.032506}}
@article{HalHelJorKloKocOls-CPL-98,
title = "Basis-set convergence in correlated calculations on Ne, N2, and H2O",
journal = "Chemical Physics Letters",
volume = "286",
number = "3",
pages = "243 - 252",
year = "1998",
issn = "0009-2614",
doi = "https://doi.org/10.1016/S0009-2614(98)00111-0",
url = "http://www.sciencedirect.com/science/article/pii/S0009261498001110",
author = "Asger Halkier and Trygve Helgaker and Poul Jørgensen and Wim Klopper and Henrik Koch and Jeppe Olsen and Angela K. Wilson",
abstract = "Valence and all-electron correlation energies of Ne, N2, and H2O at fixed experimental geometries are computed at the levels of second-order perturbation theory (MP2) and coupled cluster theory with singles and doubles excitations (CCSD), and singles and doubles excitations with a perturbative triples correction (CCSD(T)). Correlation-consistent polarized valence and core-valence basis sets up to sextuple zeta quality are employed. Guided by basis-set limits established by rij-dependent methods, a number of extrapolation schemes for use with the correlation-consistent basis sets are investigated. Among the schemes considered here, a linear least-squares procedure applied to the quintuple and sextuple zeta results yields the most accurate extrapolations."
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Selected CI %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Abstract = {Valence and all-electron correlation energies of Ne, N2, and H2O at fixed experimental geometries are computed at the levels of second-order perturbation theory (MP2) and coupled cluster theory with singles and doubles excitations (CCSD), and singles and doubles excitations with a perturbative triples correction (CCSD(T)). Correlation-consistent polarized valence and core-valence basis sets up to sextuple zeta quality are employed. Guided by basis-set limits established by rij-dependent methods, a number of extrapolation schemes for use with the correlation-consistent basis sets are investigated. Among the schemes considered here, a linear least-squares procedure applied to the quintuple and sextuple zeta results yields the most accurate extrapolations.},
Author = {Asger Halkier and Trygve Helgaker and Poul J{\o}rgensen and Wim Klopper and Henrik Koch and Jeppe Olsen and Angela K. Wilson},
Doi = {https://doi.org/10.1016/S0009-2614(98)00111-0},
Issn = {0009-2614},
Journal = {Chemical Physics Letters},
Number = {3},
Pages = {243 - 252},
Title = {Basis-set convergence in correlated calculations on Ne, N2, and H2O},
Url = {http://www.sciencedirect.com/science/article/pii/S0009261498001110},
Volume = {286},
Year = {1998},
Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0009261498001110},
Bdsk-Url-2 = {https://doi.org/10.1016/S0009-2614(98)00111-0}}
@article{BenErn-PhysRev-1969,
Author = {Charles F. Bender and Ernest R. Davidson},
@ -11978,7 +12005,6 @@
Bdsk-Url-1 = {https://doi.org/10.1016%2F0301-0104%2883%2985011-3},
Bdsk-Url-2 = {https://doi.org/10.1016/0301-0104(83)85011-3}}
@article{Cim-JCP-1985,
Author = {Cimiraglia, Renzo},
Date-Added = {2018-12-01 13:44:03 +0100},
@ -12047,8 +12073,6 @@
Year = {1992},
Bdsk-Url-1 = {https://doi.org/10.1007/BF01113255}}
@article{BunCarRam-JCP-06,
Author = {Bunge, Carlos F. and Carb{\'o}-Dorca, Ramon},
Date-Added = {2018-12-01 13:44:28 +0100},
@ -12085,14 +12109,11 @@
Year = {2005},
Bdsk-Url-1 = {http://dx.doi.org/10.1016/j.cplett.2005.06.107}}
@article{MusEngels-JCC-06,
Author = {Musch, Patrick and Engels, Bernd},
Journal = {{Journal of Computational Chemistry}},
Number = {{10}},
Year = {{2006}}
}
Year = {{2006}}}
@article{BytRue-CP-09,
Author = {Laimutis Bytautas and Klaus Ruedenberg},
@ -12143,7 +12164,6 @@
Year = {2014},
Bdsk-Url-1 = {http://dx.doi.org/10.1021/ct5004252}}
@article{GinSceCaf-JCP-15,
Author = {Emmanuel Giner and Anthony Scemama and Michel Caffarel},
Date-Added = {2018-11-29 14:22:53 +0100},
@ -12187,7 +12207,6 @@
Bdsk-Url-1 = {https://doi.org/10.1063%2F1.4947093},
Bdsk-Url-2 = {https://doi.org/10.1063/1.4947093}}
@article{SchEva-JCP-16,
Author = {Schriber, Jeffrey B. and Evangelista, Francesco A.},
Date-Added = {2018-12-01 13:44:38 +0100},
@ -12223,7 +12242,6 @@
Bdsk-Url-1 = {https://doi.org/10.1021%2Facs.jctc.5b01099},
Bdsk-Url-2 = {https://doi.org/10.1021/acs.jctc.5b01099}}
@article{ShaHolJeaAlaUmr-JCTC-17,
Author = {Sharma, Sandeep and Holmes, Adam A. and Jeanmairet, Guillaume and Alavi, Ali and Umrigar, C. J.},
Date-Added = {2018-12-01 13:35:29 +0100},
@ -12286,7 +12304,6 @@
Year = {2017},
Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4981527}}
@article{OhtJun-JCP-17,
Author = {Yuhki Ohtsuka and Jun-ya Hasegawa},
Doi = {10.1063/1.4993214},
@ -12302,7 +12319,6 @@
Bdsk-Url-1 = {https://doi.org/10.1063%2F1.4993214},
Bdsk-Url-2 = {https://doi.org/10.1063/1.4993214}}
@article{Zim-JCP-17,
Author = {Zimmerman, Paul M.},
Date-Added = {2018-12-01 13:45:04 +0100},
@ -12332,7 +12348,6 @@
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5055390}}
@article{ChiHolOttUmrShaZim-JPCA-18,
Author = {Chien, Alan D. and Holmes, Adam A. and Otten, Matthew and Umrigar, C. J. and Sharma, Sandeep and Zimmerman, Paul M.},
Date-Added = {2018-12-01 13:35:29 +0100},
@ -12366,7 +12381,6 @@
Bdsk-Url-1 = {https://doi.org/10.1063%2F1.5041327},
Bdsk-Url-2 = {https://doi.org/10.1063/1.5041327}}
@article{LooSceBloGarCafJac-JCTC-18,
Author = {Pierre-Fran{\c{c}}ois Loos and Anthony Scemama and Aymeric Blondel and Yann Garniron and Michel Caffarel and Denis Jacquemin},
Date-Modified = {2019-02-05 11:23:26 +0100},
@ -12407,7 +12421,6 @@
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.7b01250}}
@article{GarGinMalSce-JCP-16,
Author = {Yann Garniron and Emmanuel Giner and Jean-Paul Malrieu and Anthony Scemama},
Doi = {10.1063/1.4980034},
@ -12423,7 +12436,6 @@
Bdsk-Url-1 = {https://doi.org/10.1063%2F1.4980034},
Bdsk-Url-2 = {https://doi.org/10.1063/1.4980034}}
@article{LooBogSceCafJac-JCTC-19,
Author = {P. F. Loos and M. Boggio-Pasqua and A. Scemama and M. Caffarel and D. Jacquemin},
Date-Added = {2019-02-05 09:37:37 +0100},
@ -12436,131 +12448,128 @@
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b01205}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% stochastic CC
@article{Thom-PRL-10,
title = {Stochastic Coupled Cluster Theory},
author = {Thom, Alex J. W.},
journal = {Phys. Rev. Lett.},
volume = {105},
issue = {26},
pages = {263004},
numpages = {4},
year = {2010},
month = {Dec},
publisher = {American Physical Society},
doi = {10.1103/PhysRevLett.105.263004},
url = {https://link.aps.org/doi/10.1103/PhysRevLett.105.263004}
}
Author = {Thom, Alex J. W.},
Doi = {10.1103/PhysRevLett.105.263004},
Issue = {26},
Journal = {Phys. Rev. Lett.},
Month = {Dec},
Numpages = {4},
Pages = {263004},
Publisher = {American Physical Society},
Title = {Stochastic Coupled Cluster Theory},
Url = {https://link.aps.org/doi/10.1103/PhysRevLett.105.263004},
Volume = {105},
Year = {2010},
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.105.263004},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.105.263004}}
@article{ScoTho-JCP-17,
author = {Scott,Charles J. C. and Thom,Alex J. W. },
title = {Stochastic coupled cluster theory: Efficient sampling of the coupled cluster expansion},
journal = {The Journal of Chemical Physics},
volume = {147},
number = {12},
pages = {124105},
year = {2017},
doi = {10.1063/1.4991795},
URL = {https://doi.org/10.1063/1.4991795},
eprint = {https://doi.org/10.1063/1.4991795}
}
@article{SpeNeuVigFraTho-JCP-18,
author = {Spencer,J. S. and Neufeld,V. A. and Vigor,W. A. and Franklin,R. S. T. and Thom,A. J. W. },
title = {Large scale parallelization in stochastic coupled cluster},
journal = {The Journal of Chemical Physics},
volume = {149},
number = {20},
pages = {204103},
year = {2018},
doi = {10.1063/1.5047420},
URL = {https://doi.org/10.1063/1.5047420},
eprint = {https://doi.org/10.1063/1.5047420}
}
Author = {Scott,Charles J. C. and Thom,Alex J. W.},
Doi = {10.1063/1.4991795},
Eprint = {https://doi.org/10.1063/1.4991795},
Journal = {The Journal of Chemical Physics},
Number = {12},
Pages = {124105},
Title = {Stochastic coupled cluster theory: Efficient sampling of the coupled cluster expansion},
Url = {https://doi.org/10.1063/1.4991795},
Volume = {147},
Year = {2017},
Bdsk-Url-1 = {https://doi.org/10.1063/1.4991795}}
@article{SpeNeuVigFraTho-JCP-18,
Author = {Spencer,J. S. and Neufeld,V. A. and Vigor,W. A. and Franklin,R. S. T. and Thom,A. J. W.},
Doi = {10.1063/1.5047420},
Eprint = {https://doi.org/10.1063/1.5047420},
Journal = {The Journal of Chemical Physics},
Number = {20},
Pages = {204103},
Title = {Large scale parallelization in stochastic coupled cluster},
Url = {https://doi.org/10.1063/1.5047420},
Volume = {149},
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5047420}}
%%%%%%%%%%%%%%%%%%%%%%% Piotr Piecuch
@article{DeuEmiShePie-PRL-17,
title = {Converging High-Level Coupled-Cluster Energetics by Monte Carlo Sampling and Moment Expansions},
author = {Deustua, J. Emiliano and Shen, Jun and Piecuch, Piotr},
journal = {Phys. Rev. Lett.},
volume = {119},
issue = {22},
pages = {223003},
numpages = {5},
year = {2017},
month = {Nov},
publisher = {American Physical Society},
doi = {10.1103/PhysRevLett.119.223003},
url = {https://link.aps.org/doi/10.1103/PhysRevLett.119.223003}
}
Author = {Deustua, J. Emiliano and Shen, Jun and Piecuch, Piotr},
Doi = {10.1103/PhysRevLett.119.223003},
Issue = {22},
Journal = {Phys. Rev. Lett.},
Month = {Nov},
Numpages = {5},
Pages = {223003},
Publisher = {American Physical Society},
Title = {Converging High-Level Coupled-Cluster Energetics by Monte Carlo Sampling and Moment Expansions},
Url = {https://link.aps.org/doi/10.1103/PhysRevLett.119.223003},
Volume = {119},
Year = {2017},
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.119.223003},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.119.223003}}
@article{DeuEmiMagShePie-JCP-18,
author = {Deustua,J. Emiliano and Magoulas,Ilias and Shen,Jun and Piecuch,Piotr },
title = {Communication: Approaching exact quantum chemistry by cluster analysis of full configuration interaction quantum Monte Carlo wave functions},
journal = {The Journal of Chemical Physics},
volume = {149},
number = {15},
pages = {151101},
year = {2018},
doi = {10.1063/1.5055769},
URL = {https://doi.org/10.1063/1.5055769},
eprint = {https://doi.org/10.1063/1.5055769}
}
Author = {Deustua,J. Emiliano and Magoulas,Ilias and Shen,Jun and Piecuch,Piotr},
Doi = {10.1063/1.5055769},
Eprint = {https://doi.org/10.1063/1.5055769},
Journal = {The Journal of Chemical Physics},
Number = {15},
Pages = {151101},
Title = {Communication: Approaching exact quantum chemistry by cluster analysis of full configuration interaction quantum Monte Carlo wave functions},
Url = {https://doi.org/10.1063/1.5055769},
Volume = {149},
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5055769}}
@article{DeuEmiYumShePie-JCP-19,
author = {Deustua,J. Emiliano and Yuwono,Stephen H. and Shen,Jun and Piecuch,Piotr },
title = {Accurate excited-state energetics by a combination of Monte Carlo sampling and equation-of-motion coupled-cluster computations},
journal = {The Journal of Chemical Physics},
volume = {150},
number = {11},
pages = {111101},
year = {2019},
doi = {10.1063/1.5090346},
URL = {https://doi.org/10.1063/1.5090346},
eprint = {https://doi.org/10.1063/1.5090346}
}
Author = {Deustua,J. Emiliano and Yuwono,Stephen H. and Shen,Jun and Piecuch,Piotr},
Doi = {10.1063/1.5090346},
Eprint = {https://doi.org/10.1063/1.5090346},
Journal = {The Journal of Chemical Physics},
Number = {11},
Pages = {111101},
Title = {Accurate excited-state energetics by a combination of Monte Carlo sampling and equation-of-motion coupled-cluster computations},
Url = {https://doi.org/10.1063/1.5090346},
Volume = {150},
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5090346}}
%%%%%%%%%%%%%%% Scuseria
@article{QiuHenZhaScu-JCP-17,
author = {Qiu,Yiheng and Henderson,Thomas M. and Zhao,Jinmo and Scuseria,Gustavo E. },
title = {Projected coupled cluster theory},
journal = {The Journal of Chemical Physics},
volume = {147},
number = {6},
pages = {064111},
year = {2017},
doi = {10.1063/1.4991020},
URL = {https://doi.org/10.1063/1.4991020},
eprint = {https://doi.org/10.1063/1.4991020}
}
Author = {Qiu,Yiheng and Henderson,Thomas M. and Zhao,Jinmo and Scuseria,Gustavo E.},
Doi = {10.1063/1.4991020},
Eprint = {https://doi.org/10.1063/1.4991020},
Journal = {The Journal of Chemical Physics},
Number = {6},
Pages = {064111},
Title = {Projected coupled cluster theory},
Url = {https://doi.org/10.1063/1.4991020},
Volume = {147},
Year = {2017},
Bdsk-Url-1 = {https://doi.org/10.1063/1.4991020}}
@article{QiuHenZhaScu-JCP-18,
author = {Qiu,Yiheng and Henderson,Thomas M. and Zhao,Jinmo and Scuseria,Gustavo E. },
title = {Projected coupled cluster theory: Optimization of cluster amplitudes in the presence of symmetry projection},
journal = {The Journal of Chemical Physics},
volume = {149},
number = {16},
pages = {164108},
year = {2018},
doi = {10.1063/1.5053605},
URL = {https://doi.org/10.1063/1.5053605},
eprint = {https://doi.org/10.1063/1.5053605}
}
Author = {Qiu,Yiheng and Henderson,Thomas M. and Zhao,Jinmo and Scuseria,Gustavo E.},
Doi = {10.1063/1.5053605},
Eprint = {https://doi.org/10.1063/1.5053605},
Journal = {The Journal of Chemical Physics},
Number = {16},
Pages = {164108},
Title = {Projected coupled cluster theory: Optimization of cluster amplitudes in the presence of symmetry projection},
Url = {https://doi.org/10.1063/1.5053605},
Volume = {149},
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5053605}}
@article{GomHenScu-JCP-19,
author = {Gomez,John A. and Henderson,Thomas M. and Scuseria,Gustavo E. },
title = {Polynomial-product states: A symmetry-projection-based factorization of the full coupled cluster wavefunction in terms of polynomials of double excitations},
journal = {The Journal of Chemical Physics},
volume = {150},
number = {14},
pages = {144108},
year = {2019},
doi = {10.1063/1.5085314},
URL = {https://doi.org/10.1063/1.5085314},
eprint = {https://doi.org/10.1063/1.5085314}
}
Author = {Gomez,John A. and Henderson,Thomas M. and Scuseria,Gustavo E.},
Doi = {10.1063/1.5085314},
Eprint = {https://doi.org/10.1063/1.5085314},
Journal = {The Journal of Chemical Physics},
Number = {14},
Pages = {144108},
Title = {Polynomial-product states: A symmetry-projection-based factorization of the full coupled cluster wavefunction in terms of polynomials of double excitations},
Url = {https://doi.org/10.1063/1.5085314},
Volume = {150},
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5085314}}
@article{Hyl-ZP-29,
Author = {E. A. Hylleraas},
@ -12583,13 +12592,13 @@ eprint = {https://doi.org/10.1063/1.5085314}
Year = {1991}}
@misc{IrmHulGru-arxiv-19,
title={On the duality of ring and ladder diagrams and its importance for many-electron perturbation theories},
author={Andreas Irmler and Felix Hummel and Andreas Grüneis},
year={2019},
eprint={1903.05458},
archivePrefix={arXiv},
primaryClass={cond-mat.mtrl-sci}
}
Archiveprefix = {arXiv},
Author = {Andreas Irmler and Felix Hummel and Andreas Gr{\"u}neis},
Eprint = {1903.05458},
Primaryclass = {cond-mat.mtrl-sci},
Title = {On the duality of ring and ladder diagrams and its importance for many-electron perturbation theories},
Year = {2019}}
@article{GruHirOhnTen-JCP-17,
Author = {A. Gr\"uneis and S. Hirata and Y.-Y. Ohnishi and S. Ten-no},
Date-Added = {2019-05-08 10:24:45 +0200},
@ -12599,7 +12608,8 @@ eprint = {https://doi.org/10.1063/1.5085314}
Pages = {080901},
Title = {Perspective: Explicitly correlated electronic structure theory for complex systems},
Volume = {146},
Year = {2017}}
Year = {2017},
Bdsk-Url-1 = {https://doi.org/10.1063/1.4976974}}
@article{MaWer-WIREs-18,
Author = {Q. Ma and H.-J. Werner},
@ -12613,18 +12623,18 @@ eprint = {https://doi.org/10.1063/1.5085314}
Year = {2018}}
@article{LooPraSceTouGin-JCPL-19,
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volume = {10},
number = {11},
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year = {2019},
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note ={PMID: 31090432},
URL = {https://doi.org/10.1021/acs.jpclett.9b01176},
eprint = {https://doi.org/10.1021/acs.jpclett.9b01176}
}
Author = {Loos, Pierre-Fran{\c c}ois and Pradines, Barth{\'e}l{\'e}my and Scemama, Anthony and Toulouse, Julien and Giner, Emmanuel},
Doi = {10.1021/acs.jpclett.9b01176},
Eprint = {https://doi.org/10.1021/acs.jpclett.9b01176},
Journal = {The Journal of Physical Chemistry Letters},
Note = {PMID: 31090432},
Number = {11},
Pages = {2931-2937},
Title = {A Density-Based Basis-Set Correction for Wave Function Theory},
Url = {https://doi.org/10.1021/acs.jpclett.9b01176},
Volume = {10},
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpclett.9b01176}}
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@ -12642,33 +12652,34 @@ eprint = {https://doi.org/10.1021/acs.jpclett.9b01176}
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevA.73.032506},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevA.73.032506}}
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Journal = {Phys. Rev. A},
Month = {Dec},
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Pages = {062510},
Publisher = {American Physical Society},
Title = {Efficient evaluation of electron correlation along the bond-dissociation coordinate in the ground and excited ionic states with dynamic correlation suppression and enhancement functions of the on-top pair density},
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Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevA.98.062510},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevA.98.062510}}
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journal = {The Journal of Chemical Physics},
volume = {151},
number = {14},
pages = {144118},
year = {2019},
doi = {10.1063/1.5122976},
URL = {https://doi.org/10.1063/1.5122976},
eprint = {https://doi.org/10.1063/1.5122976}
}
Author = {Giner,Emmanuel and Scemama,Anthony and Toulouse,Julien and Loos,Pierre-Fran{\c c}ois},
Doi = {10.1063/1.5122976},
Eprint = {https://doi.org/10.1063/1.5122976},
Journal = {The Journal of Chemical Physics},
Number = {14},
Pages = {144118},
Title = {Chemically accurate excitation energies with small basis sets},
Url = {https://doi.org/10.1063/1.5122976},
Volume = {151},
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5122976}}
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Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.9b00176}}
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Booktitle = {Theory and Applications of Computational Chemistry: the first forty years},
Editor = {Dykstra, C. E. and Frenking, G. and Kim, K. S. and Scuseria, G. E.},
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Publisher = {Elsevier},
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Journal = {Phys. Rev. X},
Month = {Sep},
Numpages = {28},
Pages = {031059},
Publisher = {American Physical Society},
Title = {Towards the Solution of the Many-Electron Problem in Real Materials: Equation of State of the Hydrogen Chain with State-of-the-Art Many-Body Methods},
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Year = {2017},
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Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevX.7.031059}}
@article{ShiWer-JCP-10,
author = {Shiozaki,Toru and Werner,Hans-Joachim },
title = {Communication: Second-order multireference perturbation theory with explicit correlation: CASPT2-F12},
journal = {The Journal of Chemical Physics},
volume = {133},
number = {14},
pages = {141103},
year = {2010},
doi = {10.1063/1.3489000},
URL = {https://doi.org/10.1063/1.348900},
eprint = {https://doi.org/10.1063/1.3489000}
}
Author = {Shiozaki,Toru and Werner,Hans-Joachim},
Doi = {10.1063/1.3489000},
Eprint = {https://doi.org/10.1063/1.3489000},
Journal = {The Journal of Chemical Physics},
Number = {14},
Pages = {141103},
Title = {Communication: Second-order multireference perturbation theory with explicit correlation: CASPT2-F12},
Url = {https://doi.org/10.1063/1.348900},
Volume = {133},
Year = {2010},
Bdsk-Url-1 = {https://doi.org/10.1063/1.348900},
Bdsk-Url-2 = {https://doi.org/10.1063/1.3489000}}
@article{GuoSivValNee-JCP-17,
author = {Guo,Yang and Sivalingam,Kantharuban and Valeev,Edward F. and Neese,Frank },
title = {Explicitly correlated N-electron valence state perturbation theory (NEVPT2-F12)},
journal = {The Journal of Chemical Physics},
volume = {147},
number = {6},
pages = {064110},
year = {2017},
doi = {10.1063/1.4996560},
URL = {https://doi.org/10.1063/1.4996560},
eprint = {https://doi.org/10.1063/1.4996560}
}
Author = {Guo,Yang and Sivalingam,Kantharuban and Valeev,Edward F. and Neese,Frank},
Doi = {10.1063/1.4996560},
Eprint = {https://doi.org/10.1063/1.4996560},
Journal = {The Journal of Chemical Physics},
Number = {6},
Pages = {064110},
Title = {Explicitly correlated N-electron valence state perturbation theory (NEVPT2-F12)},
Url = {https://doi.org/10.1063/1.4996560},
Volume = {147},
Year = {2017},
Bdsk-Url-1 = {https://doi.org/10.1063/1.4996560}}
@article{TorVal-JCP-09,
author = {Torheyden,Martin and Valeev,Edward F. },
title = {Universal perturbative explicitly correlated basis set incompleteness correction},
journal = {The Journal of Chemical Physics},
volume = {131},
number = {17},
pages = {171103},
year = {2009},
doi = {10.1063/1.3254836},
URL = {https://doi.org/10.1063/1.3254836},
eprint = {https://doi.org/10.1063/1.3254836}
}
Author = {Torheyden,Martin and Valeev,Edward F.},
Doi = {10.1063/1.3254836},
Eprint = {https://doi.org/10.1063/1.3254836},
Journal = {The Journal of Chemical Physics},
Number = {17},
Pages = {171103},
Title = {Universal perturbative explicitly correlated basis set incompleteness correction},
Url = {https://doi.org/10.1063/1.3254836},
Volume = {131},
Year = {2009},
Bdsk-Url-1 = {https://doi.org/10.1063/1.3254836}}
@article{Ten-CPL-07,
title = "A simple F12 geminal correction in multi-reference perturbation theory",
journal = "Chemical Physics Letters",
volume = "447",
number = "1",
pages = "175 - 179",
year = "2007",
issn = "0009-2614",
doi = "https://doi.org/10.1016/j.cplett.2007.09.006",
url = "http://www.sciencedirect.com/science/article/pii/S0009261407012286",
author = "Seiichiro Ten-no",
abstract = "We propose a simple F12 geminal correction in multi-reference perturbation theory. An explicitly correlated term is introduced in the external excitations of the first order wave function in an internally contracted manner. By the use of the s- and p-wave cusp conditions, the F12 correction is expressed as the expectation value of a two-body effective operator, which reduces to the MP2-F12/A∗(SP) energy in the single reference limit. The performance of the F12 multi-reference perturbation method is demonstrated for C, CH2, O2, and SiC3."
}
Abstract = {We propose a simple F12 geminal correction in multi-reference perturbation theory. An explicitly correlated term is introduced in the external excitations of the first order wave function in an internally contracted manner. By the use of the s- and p-wave cusp conditions, the F12 correction is expressed as the expectation value of a two-body effective operator, which reduces to the MP2-F12/A∗(SP) energy in the single reference limit. The performance of the F12 multi-reference perturbation method is demonstrated for C, CH2, O2, and SiC3.},
Author = {Seiichiro Ten-no},
Doi = {https://doi.org/10.1016/j.cplett.2007.09.006},
Issn = {0009-2614},
Journal = {Chemical Physics Letters},
Number = {1},
Pages = {175 - 179},
Title = {A simple F12 geminal correction in multi-reference perturbation theory},
Url = {http://www.sciencedirect.com/science/article/pii/S0009261407012286},
Volume = {447},
Year = {2007},
Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0009261407012286},
Bdsk-Url-2 = {https://doi.org/10.1016/j.cplett.2007.09.006}}
@article{KedDemPitTenNog-CPL-11,
title = "Multireference F12 coupled cluster theory: The Brillouin-Wigner approach with single and double excitations",
journal = "Chemical Physics Letters",
volume = "511",
number = "4",
pages = "418 - 423",
year = "2011",
issn = "0009-2614",
doi = "https://doi.org/10.1016/j.cplett.2011.06.023",
url = "http://www.sciencedirect.com/science/article/pii/S0009261411007160",
author = "Stanislav Kedžuch and Ondřej Demel and Jiří Pittner and Seiichiro Ten-no and Jozef Noga",
abstract = "This Letter reports development and implementation of the explicitly correlated multireference Brillouin Wigner (MR BW-CC) coupled cluster method with Slater type geminals. The performance of the new approach is tested on the H4 model system and the dissociation curve of the fluorine molecule. Like in single reference methods, results show a dramatically improved convergence of total energies towards complete basis set limit as compared to a conventional MR BW-CC approach. In comparison with previously reported calculations with a linear correlation factor, there is a better performance for calculations in smaller basis sets."
}
Abstract = {This Letter reports development and implementation of the explicitly correlated multireference Brillouin Wigner (MR BW-CC) coupled cluster method with Slater type geminals. The performance of the new approach is tested on the H4 model system and the dissociation curve of the fluorine molecule. Like in single reference methods, results show a dramatically improved convergence of total energies towards complete basis set limit as compared to a conventional MR BW-CC approach. In comparison with previously reported calculations with a linear correlation factor, there is a better performance for calculations in smaller basis sets.},
Author = {Stanislav Ked{\v z}uch and Ond{\v r}ej Demel and Ji{\v r}{\'\i} Pittner and Seiichiro Ten-no and Jozef Noga},
Doi = {https://doi.org/10.1016/j.cplett.2011.06.023},
Issn = {0009-2614},
Journal = {Chemical Physics Letters},
Number = {4},
Pages = {418 - 423},
Title = {Multireference F12 coupled cluster theory: The Brillouin-Wigner approach with single and double excitations},
Url = {http://www.sciencedirect.com/science/article/pii/S0009261411007160},
Volume = {511},
Year = {2011},
Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0009261411007160},
Bdsk-Url-2 = {https://doi.org/10.1016/j.cplett.2011.06.023}}
@article{TorKniWer-JCP-11,
author = {Shiozaki,Toru and Knizia,Gerald and Werner,Hans-Joachim },
title = {Explicitly correlated multireference configuration interaction: MRCI-F12},
journal = {The Journal of Chemical Physics},
volume = {134},
number = {3},
pages = {034113},
year = {2011},
doi = {10.1063/1.3528720},
URL = {https://doi.org/10.1063/1.3528720},
eprint = {https://doi.org/10.1063/1.3528720}
}
Author = {Shiozaki,Toru and Knizia,Gerald and Werner,Hans-Joachim},
Doi = {10.1063/1.3528720},
Eprint = {https://doi.org/10.1063/1.3528720},
Journal = {The Journal of Chemical Physics},
Number = {3},
Pages = {034113},
Title = {Explicitly correlated multireference configuration interaction: MRCI-F12},
Url = {https://doi.org/10.1063/1.3528720},
Volume = {134},
Year = {2011},
Bdsk-Url-1 = {https://doi.org/10.1063/1.3528720}}
@Article{DemStanMatTenPitNog-PCCP-12,
author ="Demel, Ondřej and Kedžuch, Stanislav and Švaňa, Matej and Ten-no, Seiichiro and Pittner, Jiří and Noga, Jozef",
title ="An explicitly correlated Mukherjee{'}s state specific coupled cluster method: development and pilot applications",
journal ="Phys. Chem. Chem. Phys.",
year ="2012",
volume ="14",
issue ="14",
pages ="4753-4762",
publisher ="The Royal Society of Chemistry",
doi ="10.1039/C2CP23198K",
url ="http://dx.doi.org/10.1039/C2CP23198K",
abstract ="This paper reports development of the explicitly correlated variant of Mukherjee{'}s state specific multireference coupled cluster method (MkCC-F12). The current implementation is restricted to conventional single and double excitations and to pseudo-double excitations related to the Slater Type Geminal (STG) correlation factor using the SP ansatz. The performance of the MkCCSD-F12 was tested on calculations of singlet methylene{,} dissociation curve of the fluorine molecule{,} and the BeH2 insertion pathway. As expected{,} the results of the newly developed method reconfirm the significantly faster convergence with respect to the basis set limit compared to the traditional expansion in Slater determinants. Results prove that treating the correlation factor separately for each reference is appropriate."}
@article{DemStanMatTenPitNog-PCCP-12,
Abstract = {This paper reports development of the explicitly correlated variant of Mukherjee{'}s state specific multireference coupled cluster method (MkCC-F12). The current implementation is restricted to conventional single and double excitations and to pseudo-double excitations related to the Slater Type Geminal (STG) correlation factor using the SP ansatz. The performance of the MkCCSD-F12 was tested on calculations of singlet methylene{,} dissociation curve of the fluorine molecule{,} and the BeH2 insertion pathway. As expected{,} the results of the newly developed method reconfirm the significantly faster convergence with respect to the basis set limit compared to the traditional expansion in Slater determinants. Results prove that treating the correlation factor separately for each reference is appropriate.},
Author = {Demel, Ond{\v r}ej and Ked{\v z}uch, Stanislav and {\v S}va{\v n}a, Matej and Ten-no, Seiichiro and Pittner, Ji{\v r}{\'\i} and Noga, Jozef},
Doi = {10.1039/C2CP23198K},
Issue = {14},
Journal = {Phys. Chem. Chem. Phys.},
Pages = {4753-4762},
Publisher = {The Royal Society of Chemistry},
Title = {An explicitly correlated Mukherjee{'}s state specific coupled cluster method: development and pilot applications},
Url = {http://dx.doi.org/10.1039/C2CP23198K},
Volume = {14},
Year = {2012},
Bdsk-Url-1 = {http://dx.doi.org/10.1039/C2CP23198K}}
@article{HauMaoMukKlo-CPL-12,
title = "A universal explicit electron correlation correction applied to Mukherjee’s multi-reference perturbation theory",
journal = "Chemical Physics Letters",
volume = "531",
pages = "247 - 251",
year = "2012",
issn = "0009-2614",
doi = "https://doi.org/10.1016/j.cplett.2012.02.020",
url = "http://www.sciencedirect.com/science/article/pii/S0009261412002072",
author = "Robin Haunschild and Shuneng Mao and Debashis Mukherjee and Wim Klopper",
abstract = "We present a universally applicable explicit electron correlation (F12) correction and apply it to Mukherjee’s multi-reference perturbation theory (Mk-MRPT2). Two different F12 corrections are proposed: one is a universal F12 correction which is added to the conventional orbital correction, which is referred to as Mk-MRPT2+F12. In the second type of F12 correction the individual F12 contributions are added to each matrix element of the effective Hamiltonian. Subsequent diagonalization yields the Mk-MRPT2-F12 correction. Thereby, we achieve for both F12 corrections the accuracy of a quadruple-ζ basis set calculation when a triple-ζ basis set is employed and the F12 correction is added. Both F12 corrections reduce to MP2-F12/1A (fixed) in the single-reference limit."
}
Abstract = {We present a universally applicable explicit electron correlation (F12) correction and apply it to Mukherjee's multi-reference perturbation theory (Mk-MRPT2). Two different F12 corrections are proposed: one is a universal F12 correction which is added to the conventional orbital correction, which is referred to as Mk-MRPT2+F12. In the second type of F12 correction the individual F12 contributions are added to each matrix element of the effective Hamiltonian. Subsequent diagonalization yields the Mk-MRPT2-F12 correction. Thereby, we achieve for both F12 corrections the accuracy of a quadruple-ζ basis set calculation when a triple-ζ basis set is employed and the F12 correction is added. Both F12 corrections reduce to MP2-F12/1A (fixed) in the single-reference limit.},
Author = {Robin Haunschild and Shuneng Mao and Debashis Mukherjee and Wim Klopper},
Doi = {https://doi.org/10.1016/j.cplett.2012.02.020},
Issn = {0009-2614},
Journal = {Chemical Physics Letters},
Pages = {247 - 251},
Title = {A universal explicit electron correlation correction applied to Mukherjee's multi-reference perturbation theory},
Url = {http://www.sciencedirect.com/science/article/pii/S0009261412002072},
Volume = {531},
Year = {2012},
Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0009261412002072},
Bdsk-Url-2 = {https://doi.org/10.1016/j.cplett.2012.02.020}}
@article{KonVal-JCP-11,
author = {Kong,Liguo and Valeev,Edward F. },
title = {SF-[2]R12: A spin-adapted explicitly correlated method applicable to arbitrary electronic states},
journal = {The Journal of Chemical Physics},
volume = {135},
number = {21},
pages = {214105},
year = {2011},
doi = {10.1063/1.3664729},
URL = {https://doi.org/10.1063/1.3664729},
eprint = {https://doi.org/10.1063/1.3664729}
}
Author = {Kong,Liguo and Valeev,Edward F.},
Doi = {10.1063/1.3664729},
Eprint = {https://doi.org/10.1063/1.3664729},
Journal = {The Journal of Chemical Physics},
Number = {21},
Pages = {214105},
Title = {SF-[2]R12: A spin-adapted explicitly correlated method applicable to arbitrary electronic states},
Url = {https://doi.org/10.1063/1.3664729},
Volume = {135},
Year = {2011},
Bdsk-Url-1 = {https://doi.org/10.1063/1.3664729}}
@article{BooCleAlaTew-JCP-12,
author = {Booth,George H. and Cleland,Deidre and Alavi,Ali and Tew,David P. },
title = {An explicitly correlated approach to basis set incompleteness in full configuration interaction quantum Monte Carlo},
journal = {The Journal of Chemical Physics},
volume = {137},
number = {16},
pages = {164112},
year = {2012},
doi = {10.1063/1.4762445},
URL = {https://doi.org/10.1063/1.4762445},
eprint = {https://doi.org/10.1063/1.4762445}
}
Author = {Booth,George H. and Cleland,Deidre and Alavi,Ali and Tew,David P.},
Doi = {10.1063/1.4762445},
Eprint = {https://doi.org/10.1063/1.4762445},
Journal = {The Journal of Chemical Physics},
Number = {16},
Pages = {164112},
Title = {An explicitly correlated approach to basis set incompleteness in full configuration interaction quantum Monte Carlo},
Url = {https://doi.org/10.1063/1.4762445},
Volume = {137},
Year = {2012},
Bdsk-Url-1 = {https://doi.org/10.1063/1.4762445}}

55
Manuscript/srDFT_SC.tex
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@ -12,6 +12,24 @@
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@ -281,7 +299,7 @@
\begin{abstract}
We extend to strongly correlated systems the recently introduced basis-set incompleteness correction based on density-functional theory (DFT) [E. Giner \textit{et al.}, \href{https://doi.org/10.1063/1.5052714}{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. This allows to use RSDFT-type complementary functionals to recover the dominant part of the short-range correlation effects missing in this finite basis. 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 explore various approximations of complementary functionals suited to describe strong correlation. These short-range correlation functionals fulfill two very desirable properties: invariance with respect to the spin operator $S_z$ and size consistency. Specifically, we systematically investigate the dependence of the functionals on different flavors of on-top pair densities and spin polarizations. The key result of this study 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.
We extend to strongly correlated systems the recently introduced basis-set incompleteness correction based on density-functional theory (DFT) [E. Giner \textit{et al.}, \href{https://doi.org/10.1063/1.5052714}{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. This allows to use RSDFT-type complementary functionals to recover the dominant part of the short-range correlation effects missing in this finite basis. 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 explore various approximations of complementary functionals suited to describe strong correlation. These short-range correlation functionals fulfill two very desirable properties: \titou{spin-multiplet degeneracy (\ie, invariance with respect to the spin operator $S_z$)} and size consistency. Specifically, we systematically investigate the dependence of the functionals on different flavors of on-top pair densities and spin polarizations. The key result of this study 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 here. Also, the present basis-set incompleteness correction provides smooth curves along the whole potential energy surfaces.
\end{abstract}
@ -290,11 +308,12 @@ In the general context of multiconfigurational DFT, this finding shows that one
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%
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 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 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.
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 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 (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.
\PFL{I think we should add some references in the paragraph above.}
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.
In practice, WFT uses a finite one-particle basis set (here denoted as $\basis$) to project the Schr\"odinger equation. The exact solution within this basis set is then provided by full configuration interaction (FCI) which consists in a linear-algebra eigenvalue 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 also to 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 \titou{spin-multiplet degeneracy (\ie, invariance with respect to the spin operator $S_z$)} 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.
@ -314,13 +333,15 @@ In practice WFT uses a finite one-particle basis set (here denoted as $\basis$)
%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.
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}.
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, \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 are strongly affected. To alleviate 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 explicitly correlated F12 (or R12) 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 depending explicitly on the interelectronic distances. \titou{This ensures a correct representation of the Coulomb correlation hole around the electron-electron coalescence points, and leads 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-$\zeta$ basis set is equivalent to using a quintuple-$\zeta$ 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. \cite{BarLoo-JCP-17} 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 explicit 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 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 $\mu$.
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.
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}.
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 originate from a Hamiltonian with a non-divergent electron-electron interaction. Therefore, a mapping with RSDFT can be performed 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 Sec.~\ref{sec:theory} we recall the mathematical framework of the basis-set correction and we present its extension for strongly correlated systems. In particular, our focus is primarily set on imposing two key formal properties: spin-multiplet degeneracy and size-consistency.
Then, in Sec.~\ref{sec:results}, we apply the method to the calculation of the potential energy curves of the \ce{C2}, \ce{N2}, \ce{O2}, \ce{F2}, and \ce{H10} molecules up to the dissociation limit. These systems represent prototypes of strongly correlated systems. Finally, we conclude in Sec.~\ref{sec:conclusion}.
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theory}
@ -341,9 +362,9 @@ where $v_{ne}(\br{})$ is the nuclei-electron potential, and $F[\den]$ is the uni
\label{eq:levy_func}
F[\den] = \min_{\Psi \rightarrow \den} \elemm{\Psi}{\kinop +\weeop }{\Psi},
\end{equation}
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: $E_0^\Bas \approx E_0$.
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 $\Bas$'', \ie, 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 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
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, \ie, 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[\den] = \min_{\wf{}{\Bas} \rightarrow \den} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\den},
@ -402,7 +423,7 @@ With such a definition, one can show that $\wbasis$ satisfies
\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.
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 $\psibasis$, \ie,
\begin{equation}
\label{eq:cbs_wbasis}
\lim_{\Bas \rightarrow \text{CBS}} \wbasis = \frac{1}{|\br{1}-\br{2}|},\quad \forall\,\psibasis.
@ -528,9 +549,9 @@ Second, the fact that $\efuncdenpbe{\argebasis}$ vanishes for systems with vanis
\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.
An important requirement for any electronic-structure method is size-consistency, \ie, 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.
Another important requirement is spin-multiplet degeneracy, \ie, 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}
@ -551,12 +572,12 @@ expressed as a function of the density $n$ and the on-top pair density $n_2$, ca
%Nevertheless, the use of $\tilde{\zeta}$ presents several disadvantages as it can become complex when $1 - 2 \; n_{2}/n^2 < 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 way to eliminate the $S_z$ dependency is to simply set $\zeta=0$, i.e. to always use the spin-unpolarized functional. This lowers the accuracy for open-shell systems at $\mu=0$, i.e. for the usual PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$. Nevertheless, we argue that, for $\mu$ sufficiently large, it is a viable option. Indeed, the purpose of introducing the spin polarization in semilocal density-functional approximations is to mimic the exact on-top pair density~\cite{PerSavBur-PRA-95}, but our functional $\ecmd(\argecmd)$ already explicitly depends on the on-top pair density (see Eq.~\eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}). The dependencies on $\zeta$ and $n_2$ can thus be expected to be largely redundant. Therefore, we propose here to also test the $\ecmd$ functional with \textit{a zero spin polarization}. This ensures a $S_z$ independence and, as will be numerically shown, very weakly affects the accuracy of the functional.
An alternative way to eliminate the $S_z$ dependency is to simply set $\zeta=0$, \ie, to always use the spin-unpolarized functional. This lowers the accuracy for open-shell systems at $\mu=0$, \ie, for the usual PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$. Nevertheless, we argue that, for $\mu$ sufficiently large, it is a viable option. Indeed, the purpose of introducing the spin polarization in semilocal density-functional approximations is to mimic the exact on-top pair density~\cite{PerSavBur-PRA-95}, but our functional $\ecmd(\argecmd)$ already explicitly depends on the on-top pair density (see Eq.~\eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}). The dependencies on $\zeta$ and $n_2$ can thus be expected to be largely redundant. Therefore, we propose here to also test the $\ecmd$ functional with \textit{a zero spin polarization}. This ensures a $S_z$ independence and, as will be numerically shown, very weakly affects the accuracy of the functional.
\subsubsection{Conditions for size consistency}
Since $\efuncdenpbe{\argebasis}$ is a single integral over $\mathbb{R}^3$ of local quantities ($n(\br{})$, $\zeta(\br{})$, $s(\br{})$, $n_2(\br{})$,$\mu(\br{})$), 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 subsystem $A$ and the other one from the region of the subsystem $B$. Therefore, a sufficient condition for the size consistency is that these local quantities coincide in the isolated systems and in the subsystems of the supersystem $A\ldots B$. Since these local quantities are calculated from the wave function $\psibasis$, a sufficient condition is that the wave function is multiplicatively separable in the limit of non-interacting fragments, i.e. $\Psi_{A\ldots B}^{\basis} = \Psi_A^{\basis} \Psi_B^{\basis}$. In the case where the two subsystems $A$ and $B$ dissociate in closed-shell systems, a simple RHF wave function ensures this property, but when one or several covalent bonds are broken, the use of a properly chosen CAS wave function is sufficient to recover this property. The condition for the active space involved in the CAS wave function is that it has to lead to size-consistent energies in the limit of dissociated fragments.
Since $\efuncdenpbe{\argebasis}$ is a single integral over $\mathbb{R}^3$ of local quantities ($n(\br{})$, $\zeta(\br{})$, $s(\br{})$, $n_2(\br{})$,$\mu(\br{})$), 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 subsystem $A$ and the other one from the region of the subsystem $B$. Therefore, a sufficient condition for the size consistency is that these local quantities coincide in the isolated systems and in the subsystems of the supersystem $A\ldots B$. Since these local quantities are calculated from the wave function $\psibasis$, a sufficient condition is that the wave function is multiplicatively separable in the limit of non-interacting fragments, \ie, $\Psi_{A\ldots B}^{\basis} = \Psi_A^{\basis} \Psi_B^{\basis}$. In the case where the two subsystems $A$ and $B$ dissociate in closed-shell systems, a simple RHF wave function ensures this property, but when one or several covalent bonds are broken, the use of a properly chosen CAS wave function is sufficient to recover this property. The condition for the active space involved in the CAS wave function is that it has to lead to size-consistent energies in the limit of dissociated fragments.
\subsection{Different types of approximations for the functional}
@ -696,7 +717,7 @@ The study of the H$_{10}$ chain with equally distant atoms is a good prototype o
We report in Figure \ref{fig:H10} the potential energy curves computed using the cc-pVXZ (X=D,T,Q) basis sets for different levels of approximations. The computation of the atomization energies $D_0$ at each level of theory used here is reported in Table \ref{tab:d0}. A general trend that can be observed from these data is that, in a given basis set, the quality of the potential energy curves are globally improved by adding the basis-set correction, whatever the level of approximation used for the functional $\efuncbasisFCI$. Also, no bizarre behaviors are found when stretching the bonds, which shows that the functionals are robust when reaching the strong correlation regime.
More quantitatively, the values of $D_0$ are within chemical accuracy (i.e., an error below 1.4 mH) from the cc-pVTZ basis set when using the PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ functionals, whereas such an accuracy is not reached at the cc-pVQZ basis set using standard MRCI+Q.
More quantitatively, the values of $D_0$ are within chemical accuracy (\ie, an error below 1.4 mH) from the cc-pVTZ basis set when using the PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ functionals, whereas such an accuracy is not reached at the cc-pVQZ basis set using standard MRCI+Q.
Regarding in more details the performance of the different types of approximate functionals, the results show that PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ are very similar (the maximal difference on $D_0$ being 0.3 mH), and they give slightly more accurate results than PBE-UEG-$\tilde{\zeta}$. These findings bring two important clues on the role of the different physical ingredients used in the functionals: i) the explicit use of the on-top pair density coming from the CASSCF wave function (see Eq.~\eqref{eq:def_n2extrap}) is preferable to the use of the on-top pair density based on the UEG (see Eq.~\eqref{eq:def_n2ueg}); ii) removing the dependency on any kind of spin polarization does not lead to significant loss of accuracy provided that one uses a qualitatively correct on-top pair density. Point ii) is important as it shows that spin polarization in density-functional approximations essentially plays the same role as that of the on-top pair density.
@ -773,6 +794,6 @@ 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 information) that it is minor with respect to WFT methods for all systems and basis sets studied here. We thus believe that this approach is a significant step towards calculations near the CBS limit for strongly correlated systems.
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\end{document}