changes in theory
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@ -12734,6 +12734,16 @@ year = {2014}
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volume = {}
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}
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%
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@article{StaDav-CPL-01,
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author = {Viktor N. Staroverov and Ernest R. Davidson},
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title = {A density functional method for degenerate spin-multiplet components},
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journal = {Chem. Phys. Lett.},
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volume = {340},
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pages = {142},
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year = {2001}
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}
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@article{StaScuPerDavKat-PRA-06,
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author = {V. N. Staroverov and G. E. Scuseria and J. P. Perdew and E. R. Davidson and J. Katriel},
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journal = {Phys. Rev. A},
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@ -6,7 +6,7 @@
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%Control: page (0) single
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%Control: year (1) truncated
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%Control: production of eprint (0) enabled
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\begin{thebibliography}{70}%
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\begin{thebibliography}{82}%
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\makeatletter
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\providecommand \@ifxundefined [1]{%
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\@ifx{#1\undefined}
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@ -613,7 +613,7 @@
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{\bibfield {journal} {\bibinfo {journal} {Theoret. Chim. Acta}\ }\textbf
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{\bibinfo {volume} {{91}}},\ \bibinfo {pages} {147} (\bibinfo {year}
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{1995})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Savin}(1996)}]{Sav-INC-96a}%
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\bibitem [{\citenamefont {Savin}(1996{\natexlab{a}})}]{Sav-INC-96a}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
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{Savin}},\ }in\ \href@noop {} {\emph {\bibinfo {booktitle} {Recent Advances
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@ -621,6 +621,108 @@
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{editor} {\bibfnamefont {D.~P.}\ \bibnamefont {Chong}}}\ (\bibinfo
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{publisher} {World Scientific},\ \bibinfo {year} {1996})\ pp.\ \bibinfo
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{pages} {129--148}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Savin}(1996{\natexlab{b}})}]{Sav-INC-96}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
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{Savin}},\ }in\ \href@noop {} {\emph {\bibinfo {booktitle} {Recent
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Developments of Modern Density Functional Theory}}},\ \bibinfo {editor}
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{edited by\ \bibinfo {editor} {\bibfnamefont {J.~M.}\ \bibnamefont
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{Seminario}}}\ (\bibinfo {publisher} {Elsevier},\ \bibinfo {address}
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{Amsterdam},\ \bibinfo {year} {1996})\ pp.\ \bibinfo {pages}
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{327--357}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Miehlich}, \citenamefont {Stoll},\ and\ \citenamefont
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{Savin}(1997)}]{MieStoSav-MP-97}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {B.}~\bibnamefont
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{Miehlich}}, \bibinfo {author} {\bibfnamefont {H.}~\bibnamefont {Stoll}}, \
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and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Savin}},\
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}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Mol. Phys.}\
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}\textbf {\bibinfo {volume} {{91}}},\ \bibinfo {pages} {527} (\bibinfo {year}
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{1997})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Takeda}, \citenamefont {Yamanaka},\ and\
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\citenamefont {Yamaguchi}(2002)}]{TakYamYam-CPL-02}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont
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{Takeda}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Yamanaka}}, \
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and\ \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Yamaguchi}},\
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}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Chem. Phys.
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Lett.}\ }\textbf {\bibinfo {volume} {366}},\ \bibinfo {pages} {321} (\bibinfo
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{year} {2002})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Takeda}, \citenamefont {Yamanaka},\ and\
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\citenamefont {Yamaguchi}(2004)}]{TakYamYam-IJQC-04}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont
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{Takeda}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Yamanaka}}, \
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and\ \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Yamaguchi}},\
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}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Int. J. Quantum
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Chem.}\ }\textbf {\bibinfo {volume} {{96}}},\ \bibinfo {pages} {463}
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(\bibinfo {year} {2004})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Gr\"afenstein}\ and\ \citenamefont
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{Cremer}(2005)}]{GraCre-MP-05}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
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{Gr\"afenstein}}\ and\ \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
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{Cremer}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Mol.
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Phys.}\ }\textbf {\bibinfo {volume} {103}},\ \bibinfo {pages} {279} (\bibinfo
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{year} {2005})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Tsuchimochi}, \citenamefont {Scuseria},\ and\
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\citenamefont {Savin}(2010)}]{TsuScuSav-JCP-10}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
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{Tsuchimochi}}, \bibinfo {author} {\bibfnamefont {G.~E.}\ \bibnamefont
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{Scuseria}}, \ and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
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{Savin}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J.
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Chem. Phys.}\ }\textbf {\bibinfo {volume} {132}},\ \bibinfo {pages} {024111}
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(\bibinfo {year} {2010})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {{G. Li Manni, R. K. Carlson, S. Luo, D. Ma, J. Olsen,
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D. G. Truhlar, and L. Gagliardi}}(2014)}]{LimCarLuoMaOlsTruGag-JCTC-14}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibnamefont {{G. Li Manni, R. K.
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Carlson, S. Luo, D. Ma, J. Olsen, D. G. Truhlar, and L. Gagliardi}}},\
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}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J. Chem. Theory
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Comput.}\ }\textbf {\bibinfo {volume} {10}},\ \bibinfo {pages} {3669}
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(\bibinfo {year} {2014})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Garza}\ \emph
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{et~al.}(2015{\natexlab{a}})\citenamefont {Garza}, \citenamefont {Bulik},
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\citenamefont {Henderson},\ and\ \citenamefont
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{Scuseria}}]{GarBulHenScu-JCP-15}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.~J.}\ \bibnamefont
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{Garza}}, \bibinfo {author} {\bibfnamefont {I.~W.}\ \bibnamefont {Bulik}},
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\bibinfo {author} {\bibfnamefont {T.~M.}\ \bibnamefont {Henderson}}, \ and\
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\bibinfo {author} {\bibfnamefont {G.~E.}\ \bibnamefont {Scuseria}},\
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}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J. Chem. Phys.}\
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}\textbf {\bibinfo {volume} {142}},\ \bibinfo {pages} {044109} (\bibinfo
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{year} {2015}{\natexlab{a}})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Garza}\ \emph
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{et~al.}(2015{\natexlab{b}})\citenamefont {Garza}, \citenamefont {Bulik},
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\citenamefont {Henderson},\ and\ \citenamefont
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{Scuseria}}]{GarBulHenScu-PCCP-15}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.~J.}\ \bibnamefont
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{Garza}}, \bibinfo {author} {\bibfnamefont {I.~W.}\ \bibnamefont {Bulik}},
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\bibinfo {author} {\bibfnamefont {T.~M.}\ \bibnamefont {Henderson}}, \ and\
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\bibinfo {author} {\bibfnamefont {G.~E.}\ \bibnamefont {Scuseria}},\
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}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Phys. Chem. Chem.
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Phys.}\ }\textbf {\bibinfo {volume} {17}},\ \bibinfo {pages} {22412}
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(\bibinfo {year} {2015}{\natexlab{b}})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Carlson}, \citenamefont {Truhlar},\ and\
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\citenamefont {Gagliardi}(2015)}]{CarTruGag-JCTC-15}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.~K.}\ \bibnamefont
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{Carlson}}, \bibinfo {author} {\bibfnamefont {D.~G.}\ \bibnamefont
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{Truhlar}}, \ and\ \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont
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{Gagliardi}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J.
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Chem. Theory Comput.}\ }\textbf {\bibinfo {volume} {11}},\ \bibinfo {pages}
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{4077} (\bibinfo {year} {2015})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {{L. Gagliardi, D. G. Truhlar, G. Li Manni, R. K.
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Carlson, C. E. Hoyer, and J. Lucas Bao}}(2017)}]{GagTruLiCarHoyBa-ACR-17}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibnamefont {{L. Gagliardi, D. G.
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Truhlar, G. Li Manni, R. K. Carlson, C. E. Hoyer, and J. Lucas Bao}}},\
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}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Acc. Chem. Res.}\
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}\textbf {\bibinfo {volume} {50}},\ \bibinfo {pages} {66} (\bibinfo {year}
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{2017})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Perdew}, \citenamefont {Savin},\ and\ \citenamefont
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{Burke}(1995)}]{PerSavBur-PRA-95}%
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\BibitemOpen
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@ -630,6 +732,14 @@
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}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. A}\
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}\textbf {\bibinfo {volume} {51}},\ \bibinfo {pages} {4531} (\bibinfo {year}
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{1995})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Staroverov}\ and\ \citenamefont
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{Davidson}(2001)}]{StaDav-CPL-01}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {V.~N.}\ \bibnamefont
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{Staroverov}}\ and\ \bibinfo {author} {\bibfnamefont {E.~R.}\ \bibnamefont
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{Davidson}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal}
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{Chem. Phys. Lett.}\ }\textbf {\bibinfo {volume} {340}},\ \bibinfo {pages}
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{142} (\bibinfo {year} {2001})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Holmes}, \citenamefont {Umrigar},\ and\ \citenamefont
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{Sharma}(2017)}]{HolUmrSha-JCP-17}%
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\BibitemOpen
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@ -62,7 +62,7 @@
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\newcommand{\efuncbasisfci}[0]{\bar{E}^\Bas[\denfci]}
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\newcommand{\efuncbasis}[0]{\bar{E}^\Bas[\den]}
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\newcommand{\efuncden}[1]{\bar{E}^\Bas[#1]}
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\newcommand{\efuncdenpbe}[1]{\bar{E}_{\text{X}}^\Bas[#1]}
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\newcommand{\efuncdenpbe}[1]{\bar{E}_{\text{}}^\Bas[#1]}
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\newcommand{\ecompmodel}[0]{\bar{E}^\Bas[\denmodel]}
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\newcommand{\ecmubis}[0]{\bar{E}_{\text{c,md}}^{\text{sr}}[\denr;\,\mu]}
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\newcommand{\ecmubisldapbe}[0]{\bar{E}_{\text{c,md}\,\text{PBE}}^{\text{sr}}[\denr;\,\mu]}
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@ -92,7 +92,7 @@
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%pbeuegXiCAS
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\newcommand{\pbeuegXi}{\text{PBE-UEG-}\tilde{\zeta}}
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\newcommand{\argpbeuegXi}[0]{\den,\tilde{\zeta},s,\ntwo^{\text{UEG}},\mu_{\text{CAS}}^{\basis}}
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\newcommand{\argrpbeuegXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),\ntwo^{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
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\newcommand{\argrpbeuegXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),\ntwo^{\text{UEG}}(\br{}),\mu_{\text{}}(\br{})}
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%pbeontxiCAS
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\newcommand{\pbeontxi}{\text{PBE-ot-}\zeta}
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\newcommand{\argpbeontxi}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
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@ -100,11 +100,11 @@
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%pbeontXiCAS
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\newcommand{\pbeontXi}{\text{PBE-ot-}\tilde{\zeta}}
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\newcommand{\argpbeontXi}[0]{\den,\tilde{\zeta},s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
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\newcommand{\argrpbeontXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
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\newcommand{\argrpbeontXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{}}^{}(\br{})}
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%pbeont0xiCAS
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\newcommand{\pbeontns}{\text{PBE-ot-}0\zeta}
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\newcommand{\argpbeontns}[0]{\den,0,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
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\newcommand{\argrpbeontns}[0]{\den(\br{}),0,s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
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\newcommand{\argrpbeontns}[0]{\den(\br{}),0,s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{}}^{}(\br{})}
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%%%%%% arguments
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@ -133,8 +133,8 @@
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\newcommand{\ntwo}[0]{n_{2}}
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\newcommand{\ntwohf}[0]{n_2^{\text{HF}}}
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\newcommand{\ntwophi}[0]{n_2^{{\phi}}}
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\newcommand{\ntwoextrap}[0]{\mathring{n}_{2}^{\psibasis}}
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\newcommand{\ntwoextrapcas}[0]{\mathring{n}_2^{\text{CAS},\basis}}
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\newcommand{\ntwoextrap}[0]{\mathring{n}_{2}^{\text{}}}
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\newcommand{\ntwoextrapcas}[0]{\mathring{n}_2^{\text{}}}
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\newcommand{\mur}[0]{\mu({\bf r})}
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\newcommand{\murr}[1]{\mu({\bf r}_{#1})}
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\newcommand{\murval}[0]{\mu_{\text{val}}({\bf r})}
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@ -535,79 +535,78 @@ Another important requirement is spin-multiplet degeneracy, i.e. the independenc
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\subsubsection{Condition for the functional $\efuncdenpbe{\argebasis}$ to obtain spin-multiplet degeneracy}
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A sufficient condition to achieve spin-multiplet degeneracy is to eliminate all dependence on $S_z$, which in the case of the functional $\ecmd(\argecmd)$ means removing the dependency on the spin polarization $\zeta(\br{})$ used in the PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ (see Eq. \eqref{eq:def_ecmdpbe}). It has been proposed to replace in functionals the dependence on the spin polarization by the dependence on the on-top pair density~\cite{MosSan-PRA-91,BecSavSto-TCA-95,Sav-INC-96a,Sav-INC-96a} (see, also, Refs.~\onlinecite{PerSavBur-PRA-95,StaDav-CPL-01}). Most often, the on-top pair density is used through an effective spin polarisation
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%Based on this reasoning, a similar approach has been used in the context of multi configurational DFT in order to remove the $S_z$ dependency.
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%In practice, these approaches introduce the effective spin polarisation
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A sufficient condition to achieve spin-multiplet degeneracy is to eliminate all dependencies on $S_z$, which in the case of the functional $\ecmd(\argecmd)$ means removing the dependency on the spin polarization $\zeta(\br{})$ use the PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ (see Eq. \eqref{eq:def_ecmdpbe}).
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It has been proposed to replace in functionals the dependency on the spin polarization by the dependency on the on-top pair density. Most often, it is done by introducing an effective spin polarisation~\cite{MosSan-PRA-91,BecSavSto-TCA-95,Sav-INC-96a,Sav-INC-96,MieStoSav-MP-97,TakYamYam-CPL-02,TakYamYam-IJQC-04,GraCre-MP-05,TsuScuSav-JCP-10,LimCarLuoMaOlsTruGag-JCTC-14,GarBulHenScu-JCP-15,GarBulHenScu-PCCP-15,CarTruGag-JCTC-15,GagTruLiCarHoyBa-ACR-17} (see, also, Refs.~\onlinecite{PerSavBur-PRA-95,StaDav-CPL-01})
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\begin{equation}
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\label{eq:def_effspin}
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\tilde{\zeta}(n_{2}) =
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\tilde{\zeta}(n,n_{2}) =
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% \begin{cases}
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\sqrt{ 1 - 2 n_{2}/n^2 }
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\sqrt{ 1 - 2 \; n_{2}/n^2 },
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% 0 & \text{otherwise.}
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% \end{cases}
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\end{equation}
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which uses the on-top pair density $n_{2,\psibasis}$ of a given wave function $\psibasis$.
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expressed as a function of the density $n$ and the on-top pair density $n_2$, calculated from a given wave function. The advantage of this approach is that this effective spin polarisation $\tilde{\zeta}$ is independent from $S_z$, since the on-top pair density is independent from $S_z$. Nevertheless, the use of $\tilde{\zeta}$ in Eq.~\eqref{eq:def_effspin} presents the disadvantage that since this expression was derived for a single-determinant wave function, and it does not appear well justified to use it for a multideterminant wave function as well. In particular, for a multideterminant wave function, it may happen that $1 - 2 \; n_{2}/n^2 < 0 $ and thus in this cas Eq.~\eqref{eq:def_effspin} gives a complex value of $\tilde{\zeta}$~\cite{BecSavSto-TCA-95}.
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The advantages of this approach are at least two folds: i) the effective spin polarisation $\tilde{\zeta}$ is $S_z$ invariant, ii) it introduces an indirect dependency on the on-top pair density of the wave function $\psibasis$ which usually improves the treatment of strong correlation.
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Nevertheless, the use of $\tilde{\zeta}$ presents several disadvantages as it can become complex when $n^2 - 4 n_{2,\psibasis}<0$ and also
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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.
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%The advantage of this approach are at least two folds: i) the effective spin polarisation $\tilde{\zeta}$ is independent from $S_z$ since the on-top pair density is independent from $S_z$, ii) it introduces an indirect dependency on the on-top pair density of the wave function $\psibasis$ which usually improves the treatment of strong correlation.
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%Nevertheless, the use of $\tilde{\zeta}$ presents several disadvantages as it can become complex when $1 - 2 \; n_{2}/n^2 < 0 $ and also
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%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.
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An alternative to eliminate the $S_z$ dependency would be to simply set $\zeta(\br{})=0$, but this would lower the accuracy of the usual correlation functional, such as the PBE correlation functional used here $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$. Nevertheless, as the spin polarisation usually tries to mimic the on-top pair density and the function $\ecmd(\argecmd)$ explicitly depends on the on-top pair density (see equations \eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}), we propose here to use the $\ecmd$ functional with \textit{a zero spin polarisation}. This ensures a $S_z$ invariance and, as will be numerically shown, very weakly affect the accuracy of the functional.
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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 polarisation 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 polarisation}. This ensures a $S_z$ independence and, as will be numerically shown, very weakly affects the accuracy of the functional.
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\subsubsection{Conditions on $\psibasis$ for the extensivity}
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In the case of the present basis set correction, as $\efuncdenpbe{\argebasis}$ is an integral over $\mathbb{R}^3$ of local quantities, in the case of non overlapping fragments $A\ldots B$ it can be written as the sum of two local contributions: one coming from the integration over the region of the sub-system $A$ and the other one from the region of the sub-system $B$.
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Therefore, a sufficient condition for the extensivity is that these quantities coincide in the isolated systems and in the subsystem of the super system $A\ldots B$.
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As $\efuncdenpbe{\argebasis}$ depends only on quantities which are properties of the wave function $\psibasis$, a sufficient condition for the extensivity of these quantities is that the function factorise in the limit of non-interacting fragments, that is $\Psi_{A\ldots B}^{\basis} = \Psi_A^{\basis} \Psi_B^{\basis}$.
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In the case where the two subsystems $A$ and $B$ dissociate in closed shell systems, a simple HF wave function ensures this property, but when one or several covalent bonds are broken, the use of a properly chosen CASSCF wave function is sufficient to recover this property.
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The condition for the active space involved in the CASSCF wave function is that it has to lead to extensive energies in the limit of dissociated fragments.
|
||||
|
||||
\subsubsection{Conditions for size consistency}
|
||||
|
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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.
|
||||
|
||||
|
||||
\subsection{Different types of approximations for the functional}
|
||||
\label{sec:final_def_func}
|
||||
\subsubsection{Definition of the protocol to design functionals}
|
||||
As the present work focusses on the strong correlation regime, we propose here to investigate only approximated functionals which are $S_z$ invariant and size extensive in the case of covalent bond breaking. Therefore, the wave function $\psibasis$ used throughout this paper are of CASSCF type in order to ensure extensivity of all density related quantities.
|
||||
The difference between the different flavours of functionals are only on i) the type of on-top pair density used, and ii) the type of spin polarisation used.
|
||||
|
||||
Regarding the spin polarisation that enters into $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$, two different types of $S_z$ invariant formulations are used: i) the \textit{effective} spin polarization $\tilde{\zeta}$ defined in equation \eqref{eq:def_effspin}, and iii) a \textit{zero} spin polarization.
|
||||
As the present work focusses on the strong correlation regime, we propose here to investigate only approximate functionals which are $S_z$ independent and size-consistent in the case of covalent bond breaking. Therefore, the wave functions $\psibasis$ used throughout this paper are CAS wave functions in order to ensure size consistency of all local quantities. The difference between the different flavors of functionals are only on i) the type of spin polarisation used, and ii) the type of on-top pair density used.
|
||||
|
||||
Regarding the approximation to the \textit{exact} on-top pair density entering in equation \eqref{eq:def_beta}, we use two different approximations. The first one is based on the uniform electron gas (UEG) and reads
|
||||
Regarding the spin polarisation that enters into $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$, two different types of $S_z$-independent formulations are used: i) the \textit{effective} spin polarization $\tilde{\zeta}$ defined in Eq.~\eqref{eq:def_effspin} and calculated from the CAS wave function, and ii) a \textit{zero} spin polarization.
|
||||
|
||||
Regarding the on-top pair density entering in Eq.~\eqref{eq:def_beta}, we use two different approximations. The first one is based on the uniform-electron gas (UEG) and reads
|
||||
\begin{equation}
|
||||
\label{eq:def_n2ueg}
|
||||
\ntwo^{\text{UEG}}(n,\zeta,\br{}) = n(\br{})^2\big(1-\zeta(\br{})\big)g_0\big(n(\br{})\big)
|
||||
\ntwo^{\text{UEG}}(n,\zeta) \approx n^2\big(1-\zeta^2\big)g_0(n),
|
||||
\end{equation}
|
||||
where the pair-distribution function $g_0(n)$ is taken from equation (46) of Ref. \onlinecite{GorSav-PRA-06}. As some spin polarization appear in equation \eqref{eq:def_n2ueg}, we use the effective spin density $\tilde{\zeta}$ of equation \eqref{eq:def_effspin} in order to ensure $S_z$ invariance. Notice that, as we use a CASSCF wave function and $\tilde{\zeta}$ as spin polarization, the $\ntwo^{\text{UEG}}$ will depend indirectly on the on-top pair density of the CASSCF wave function as $\tilde{\zeta}$ depends on the on-top pair density.
|
||||
where the pair-distribution function $g_0(n)$ is taken from Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. As the spin polarization appears in Eq.~\eqref{eq:def_n2ueg}, we use the effective spin polarization $\tilde{\zeta}$ of Eq.~\eqref{eq:def_effspin} in order to ensure $S_z$ independence. Thus, $\ntwo^{\text{UEG}}$ will depend indirectly on the on-top pair density of the CAS wave function through $\tilde{\zeta}$.
|
||||
|
||||
Another approach to approximate of the exact on top pair density consists in taking advantage of the on-top pair density of the wave function $\psibasis$. Following the work of some of the previous authors\cite{FerGinTou-JCP-18,GinSceTouLoo-JCP-19} we introduce the extrapolated on-top pair density $\ntwoextrap$ as
|
||||
Another approach to approximate the exact on top pair density consists in using directly the on-top pair density of the CAS wave function. Following the work of some of the previous authors~\cite{FerGinTou-JCP-18,GinSceTouLoo-JCP-19} we introduce the extrapolated on-top pair density $\ntwoextrap$ as
|
||||
\begin{equation}
|
||||
\label{eq:def_n2extrap}
|
||||
\ntwoextrap(\ntwo^{\psibasis},\mu,\br{}) = \ntwo^{\wf{}{\Bas}}(\br{}) \bigg( 1 + \frac{2}{\sqrt{\pi}\murpsi} \bigg)^{-1}
|
||||
\ntwoextrap(\ntwo,\mu) = \bigg( 1 + \frac{2}{\sqrt{\pi}\mu} \bigg)^{-1} \; \ntwo
|
||||
\end{equation}
|
||||
which directly follows from the large-$\mu$ extrapolation of the exact on-top pair density proposed by Gori-Giorgi and Savin\cite{GorSav-PRA-06} in the context of RSDFT.
|
||||
When using $\ntwoextrap(\ntwo,\mu,\br{})$ in a functional, we will refer simply refer it as "ot".
|
||||
which directly follows from the large-$\mu$ extrapolation of the exact on-top pair density derived by Gori-Giorgi and Savin\cite{GorSav-PRA-06} in the context of RSDFT. When using $\ntwoextrap(\ntwo,\mu)$ in a functional, we will simply refer it as ``ot''.
|
||||
|
||||
|
||||
\subsubsection{Definition of functionals with good formal properties}
|
||||
\label{sec:def_func}
|
||||
We define the following functionals:
|
||||
i) The PBE-UEG-$\tilde{\zeta}$ which uses the UEG-like on-top pair density defined in equation \eqref{eq:def_n2ueg}, the effective spin polarization of equation \eqref{eq:def_effspin} and which reads
|
||||
|
||||
We define the following functionals:\\
|
||||
|
||||
i) PBE-UEG-$\tilde{\zeta}$ which uses the UEG on-top pair density defined in Eq.~\eqref{eq:def_n2ueg} and the effective spin polarization of Eq.~\eqref{eq:def_effspin}
|
||||
\begin{equation}
|
||||
\label{eq:def_pbeueg}
|
||||
\begin{aligned}
|
||||
\pbeuegXi = &\int d\br{} \,\denr \\ & \ecmd(\argrpbeuegXi),
|
||||
\bar{E}^\Bas_{\pbeuegXi} = \int \d\br{} \,\denr \ecmd(\argrpbeuegXi),
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
ii) the PBE-ot-$\tilde{\zeta}$ where the on-top pair density of equation \eqref{eq:def_n2extrap} is used and which reads
|
||||
ii) PBE-ot-$\tilde{\zeta}$ which uses the on-top pair density of Eq.~\eqref{eq:def_n2extrap}
|
||||
\begin{equation}
|
||||
\label{eq:def_pbeueg}
|
||||
\begin{aligned}
|
||||
\pbeontXi = &\int d\br{} \,\denr \\ & \ecmd(\argrpbeontXi),
|
||||
\bar{E}^\Bas_{\pbeontXi} = \int \d\br{} \,\denr \ecmd(\argrpbeontXi),
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
iii) and the PBE-ot-$0{\zeta}$ where no spin polarization is used and which therefore uses only the total density and the on-top pair density of equation \eqref{eq:def_n2extrap} and which reads
|
||||
iii) PBE-ot-$0{\zeta}$ where uses zero spin polarization and the on-top pair density of Eq.~\eqref{eq:def_n2extrap}
|
||||
\begin{equation}
|
||||
\label{eq:def_pbeueg}
|
||||
\begin{aligned}
|
||||
\pbeontns = &\int d\br{} \,\denr \\ & \ecmd(\argrpbeontns).
|
||||
\bar{E}^\Bas_{\pbeontns} = \int \d\br{} \,\denr \ecmd(\argrpbeontns).
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
|
||||
|
||||
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