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579
Manuscript/CI-F12.bbl
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Manuscript/CI-F12.bbl
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Chem. Phys.}\ }\textbf {\bibinfo {volume} {94}},\ \bibinfo {pages} {1985}
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Chem. Phys.}\ }\textbf {\bibinfo {volume} {94}},\ \bibinfo {pages} {2020}
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Chem. Phys.}\ }\textbf {\bibinfo {volume} {101}},\ \bibinfo {pages} {7738}
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||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~P.}\ \bibnamefont
|
||||
{Tew}}\ and\ \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Klopper}},\
|
||||
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J. Chem. Phys.}\
|
||||
}\textbf {\bibinfo {volume} {123}},\ \bibinfo {pages} {074101} (\bibinfo
|
||||
{year} {2005})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {May}\ \emph {et~al.}(2005)\citenamefont {May},
|
||||
\citenamefont {Valeev}, \citenamefont {Polly},\ and\ \citenamefont
|
||||
{Manby}}]{May05}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.~J.}\ \bibnamefont
|
||||
{May}}, \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {Valeev}},
|
||||
\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Polly}}, \ and\ \bibinfo
|
||||
{author} {\bibfnamefont {F.~R.}\ \bibnamefont {Manby}},\ }\href@noop {}
|
||||
{\bibfield {journal} {\bibinfo {journal} {Phys. Chem. Chem. Phys.}\
|
||||
}\textbf {\bibinfo {volume} {7}},\ \bibinfo {pages} {2710} (\bibinfo {year}
|
||||
{2005})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Ten-no}\ and\ \citenamefont {Noga}(2012)}]{Tenno12a}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
|
||||
{Ten-no}}\ and\ \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Noga}},\
|
||||
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {WIREs Comput. Mol.
|
||||
Sci.}\ }\textbf {\bibinfo {volume} {2}},\ \bibinfo {pages} {114} (\bibinfo
|
||||
{year} {2012})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Ten-no}(2012)}]{Tenno12b}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
|
||||
{Ten-no}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Theor.
|
||||
Chem. Acc.}\ }\textbf {\bibinfo {volume} {131}},\ \bibinfo {pages} {1070}
|
||||
(\bibinfo {year} {2012})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Hattig}\ \emph {et~al.}(2012)\citenamefont {Hattig},
|
||||
\citenamefont {Klopper}, \citenamefont {Kohn},\ and\ \citenamefont
|
||||
{Tew}}]{Hattig12}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont
|
||||
{Hattig}}, \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Klopper}},
|
||||
\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Kohn}}, \ and\ \bibinfo
|
||||
{author} {\bibfnamefont {D.~P.}\ \bibnamefont {Tew}},\ }\href@noop {}
|
||||
{\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)}]{Kong12}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {L.}~\bibnamefont
|
||||
{Kong}}, \bibinfo {author} {\bibfnamefont {F.~A.}\ \bibnamefont {Bischo}}, \
|
||||
and\ \bibinfo {author} {\bibfnamefont {E.~F.}\ \bibnamefont {Valeev}},\
|
||||
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Chem. Rev.}\
|
||||
}\textbf {\bibinfo {volume} {112}},\ \bibinfo {pages} {75} (\bibinfo {year}
|
||||
{2012})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Tew}\ \emph {et~al.}(2007)\citenamefont {Tew},
|
||||
\citenamefont {Klopper}, \citenamefont {Neiss},\ and\ \citenamefont
|
||||
{Hattig}}]{Tew07b}%
|
||||
\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 {Huron}, \citenamefont {Malrieu},\ and\ \citenamefont
|
||||
{Rancurel}(1973)}]{Huron73}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {B.}~\bibnamefont
|
||||
{Huron}}, \bibinfo {author} {\bibfnamefont {J.~P.}\ \bibnamefont {Malrieu}},
|
||||
\ and\ \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont {Rancurel}},\
|
||||
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J. Chem. Phys.}\
|
||||
}\textbf {\bibinfo {volume} {58}},\ \bibinfo {pages} {5745} (\bibinfo {year}
|
||||
{1973})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Evangelisti}, \citenamefont {Daudey},\ and\
|
||||
\citenamefont {Malrieu}(1983)}]{Evangelisti83}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
|
||||
{Evangelisti}}, \bibinfo {author} {\bibfnamefont {J.-P.}\ \bibnamefont
|
||||
{Daudey}}, \ and\ \bibinfo {author} {\bibfnamefont {J.-P.}\ \bibnamefont
|
||||
{Malrieu}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Chem.
|
||||
Phys.}\ }\textbf {\bibinfo {volume} {75}},\ \bibinfo {pages} {91} (\bibinfo
|
||||
{year} {1983})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Heuilly}\ and\ \citenamefont
|
||||
{Malrieu}(1992)}]{Heully92}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.-L.}\ \bibnamefont
|
||||
{Heuilly}}\ and\ \bibinfo {author} {\bibfnamefont {J.-P.}\ \bibnamefont
|
||||
{Malrieu}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Chem.
|
||||
Phys. Lett.}\ }\textbf {\bibinfo {volume} {199}},\ \bibinfo {pages} {545}
|
||||
(\bibinfo {year} {1992})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Giner}, \citenamefont {Scemama},\ and\ \citenamefont
|
||||
{Caffarel}(2013)}]{Giner13}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
|
||||
{Giner}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Scemama}}, \
|
||||
and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Caffarel}},\ }\href
|
||||
{\doibase 10.1139/cjc-2013-0017} {\bibfield {journal} {\bibinfo {journal}
|
||||
{Can. J. Chem.}\ }\textbf {\bibinfo {volume} {91}},\ \bibinfo {pages} {879}
|
||||
(\bibinfo {year} {2013})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Scemama}\ \emph
|
||||
{et~al.}(2013{\natexlab{a}})\citenamefont {Scemama}, \citenamefont
|
||||
{Caffarel}, \citenamefont {Oseret},\ and\ \citenamefont
|
||||
{Jalby}}]{Scemama13a}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
|
||||
{Scemama}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Caffarel}},
|
||||
\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {Oseret}}, \ and\ \bibinfo
|
||||
{author} {\bibfnamefont {W.}~\bibnamefont {Jalby}},\ }in\ \href {\doibase
|
||||
10.1007/978-3-642-38718-0_14} {\emph {\bibinfo {booktitle} {High Performance
|
||||
Computing for Computational Science - {VECPAR} 2012}}}\ (\bibinfo
|
||||
{publisher} {Springer Science Business Media},\ \bibinfo {year} {2013})\ pp.\
|
||||
\bibinfo {pages} {118--127}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Scemama}\ \emph
|
||||
{et~al.}(2013{\natexlab{b}})\citenamefont {Scemama}, \citenamefont
|
||||
{Caffarel}, \citenamefont {Oseret},\ and\ \citenamefont
|
||||
{Jalby}}]{Scemama13b}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
|
||||
{Scemama}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Caffarel}},
|
||||
\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {Oseret}}, \ and\ \bibinfo
|
||||
{author} {\bibfnamefont {W.}~\bibnamefont {Jalby}},\ }\href {\doibase
|
||||
10.1002/jcc.23216} {\bibfield {journal} {\bibinfo {journal} {J. Comput.
|
||||
Chem.}\ }\textbf {\bibinfo {volume} {34}},\ \bibinfo {pages} {938} (\bibinfo
|
||||
{year} {2013}{\natexlab{b}})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Scemama}\ \emph {et~al.}(2014)\citenamefont
|
||||
{Scemama}, \citenamefont {Applencourt}, \citenamefont {Giner},\ and\
|
||||
\citenamefont {Caffarel}}]{Scemama14}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
|
||||
{Scemama}}, \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
|
||||
{Applencourt}}, \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {Giner}},
|
||||
\ and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Caffarel}},\
|
||||
}\href {\doibase 10.1063/1.4903985} {\bibfield {journal} {\bibinfo
|
||||
{journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {141}},\ \bibinfo
|
||||
{pages} {244110} (\bibinfo {year} {2014})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Giner}, \citenamefont {Scemama},\ and\ \citenamefont
|
||||
{Caffarel}(2015)}]{Giner15}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
|
||||
{Giner}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Scemama}}, \
|
||||
and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Caffarel}},\ }\href
|
||||
{\doibase 10.1063/1.4905528} {\bibfield {journal} {\bibinfo {journal} {J.
|
||||
Chem. Phys.}\ }\textbf {\bibinfo {volume} {142}},\ \bibinfo {pages} {044115}
|
||||
(\bibinfo {year} {2015})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Caffarel}\ \emph {et~al.}(2016)\citenamefont
|
||||
{Caffarel}, \citenamefont {Applencourt}, \citenamefont {Giner},\ and\
|
||||
\citenamefont {Scemama}}]{Caffarel16}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
|
||||
{Caffarel}}, \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
|
||||
{Applencourt}}, \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {Giner}},
|
||||
\ and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Scemama}},\
|
||||
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J. Chem. Phys.}\
|
||||
}\textbf {\bibinfo {volume} {144}},\ \bibinfo {pages} {151103} (\bibinfo
|
||||
{year} {2016})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Shiozaki}, \citenamefont {Knizia},\ and\
|
||||
\citenamefont {Werner}(2011)}]{Shiozaki11}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
|
||||
{Shiozaki}}, \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont {Knizia}}, \
|
||||
and\ \bibinfo {author} {\bibfnamefont {H.-J.}\ \bibnamefont {Werner}},\
|
||||
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J. Chem. Phys.}\
|
||||
}\textbf {\bibinfo {volume} {134}},\ \bibinfo {pages} {034113} (\bibinfo
|
||||
{year} {2011})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Kato}(1951)}]{Kato51}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
|
||||
{Kato}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Trans.
|
||||
Am. Math. Soc.}\ }\textbf {\bibinfo {volume} {70}},\ \bibinfo {pages} {195}
|
||||
(\bibinfo {year} {1951})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Kato}(1957)}]{Kato57}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
|
||||
{Kato}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Commun.
|
||||
Pure Appl. Math.}\ }\textbf {\bibinfo {volume} {10}},\ \bibinfo {pages} {151}
|
||||
(\bibinfo {year} {1957})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Pack}\ and\ \citenamefont {{Byers
|
||||
Brown}}(1966)}]{Pack66}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.~T.}\ \bibnamefont
|
||||
{Pack}}\ and\ \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {{Byers
|
||||
Brown}}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J.
|
||||
Chem. Phys.}\ }\textbf {\bibinfo {volume} {45}},\ \bibinfo {pages} {556}
|
||||
(\bibinfo {year} {1966})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {{Morgan III}}\ and\ \citenamefont
|
||||
{Kutzelnigg}(1993)}]{Morgan93}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~D.}\ \bibnamefont
|
||||
{{Morgan III}}}\ and\ \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont
|
||||
{Kutzelnigg}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J.
|
||||
Phys. Chem.}\ }\textbf {\bibinfo {volume} {97}},\ \bibinfo {pages} {2425}
|
||||
(\bibinfo {year} {1993})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Ten-no}(2004{\natexlab{b}})}]{Tenno04a}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
|
||||
{Ten-no}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Chem.
|
||||
Phys. Lett.}\ }\textbf {\bibinfo {volume} {398}},\ \bibinfo {pages} {56}
|
||||
(\bibinfo {year} {2004}{\natexlab{b}})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Pulay}(1982)}]{Pulay82}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
|
||||
{Pulay}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J.
|
||||
Comput. Chem.}\ }\textbf {\bibinfo {volume} {3}},\ \bibinfo {pages} {556}
|
||||
(\bibinfo {year} {1982})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Szabo}\ and\ \citenamefont
|
||||
{Ostlund}(1989)}]{SzaboBook}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
|
||||
{Szabo}}\ and\ \bibinfo {author} {\bibfnamefont {N.~S.}\ \bibnamefont
|
||||
{Ostlund}},\ }\href@noop {} {\emph {\bibinfo {title} {Modern quantum
|
||||
chemistry}}}\ (\bibinfo {publisher} {McGraw-Hill},\ \bibinfo {address} {New
|
||||
York},\ \bibinfo {year} {1989})\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Klopper}\ and\ \citenamefont
|
||||
{Rohse}(1992)}]{Klopper92}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont
|
||||
{Klopper}}\ and\ \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont
|
||||
{Rohse}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Theor.
|
||||
Chem. Acc.}\ }\textbf {\bibinfo {volume} {83}},\ \bibinfo {pages} {441}
|
||||
(\bibinfo {year} {1992})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Klopper}\ and\ \citenamefont
|
||||
{Samson}(2002)}]{Klopper02}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont
|
||||
{Klopper}}\ and\ \bibinfo {author} {\bibfnamefont {C.~C.~M.}\ \bibnamefont
|
||||
{Samson}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J.
|
||||
Chem. Phys.}\ }\textbf {\bibinfo {volume} {116}},\ \bibinfo {pages} {6397}
|
||||
(\bibinfo {year} {2002})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Manby}(2003)}]{Manby03}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {F.~R.}\ \bibnamefont
|
||||
{Manby}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J.
|
||||
Chem. Phys.}\ }\textbf {\bibinfo {volume} {119}},\ \bibinfo {pages} {4607}
|
||||
(\bibinfo {year} {2003})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Werner}, \citenamefont {Manby},\ and\ \citenamefont
|
||||
{Knowles}(2003)}]{Werner03}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.~J.}\ \bibnamefont
|
||||
{Werner}}, \bibinfo {author} {\bibfnamefont {F.~R.}\ \bibnamefont {Manby}}, \
|
||||
and\ \bibinfo {author} {\bibfnamefont {P.~J.}\ \bibnamefont {Knowles}},\
|
||||
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J. Chem. Phys.}\
|
||||
}\textbf {\bibinfo {volume} {118}},\ \bibinfo {pages} {8149} (\bibinfo {year}
|
||||
{2003})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Klopper}(2004)}]{Klopper04}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont
|
||||
{Klopper}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J.
|
||||
Chem. Phys.}\ }\textbf {\bibinfo {volume} {120}},\ \bibinfo {pages} {10890}
|
||||
(\bibinfo {year} {2004})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Manby}\ \emph {et~al.}(2006)\citenamefont {Manby},
|
||||
\citenamefont {Werner}, \citenamefont {Adler},\ and\ \citenamefont
|
||||
{May}}]{Manby06}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {F.~R.}\ \bibnamefont
|
||||
{Manby}}, \bibinfo {author} {\bibfnamefont {H.~J.}\ \bibnamefont {Werner}},
|
||||
\bibinfo {author} {\bibfnamefont {T.~B.}\ \bibnamefont {Adler}}, \ and\
|
||||
\bibinfo {author} {\bibfnamefont {A.~J.}\ \bibnamefont {May}},\ }\href@noop
|
||||
{} {\bibfield {journal} {\bibinfo {journal} {J. Chem. Phys.}\ }\textbf
|
||||
{\bibinfo {volume} {124}},\ \bibinfo {pages} {094103} (\bibinfo {year}
|
||||
{2006})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Ten-no}(2007)}]{Tenno07}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
|
||||
{Ten-no}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J.
|
||||
Chem. Phys.}\ }\textbf {\bibinfo {volume} {126}},\ \bibinfo {pages} {014108}
|
||||
(\bibinfo {year} {2007})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Komornicki}\ and\ \citenamefont
|
||||
{King}(2011)}]{Komornicki11}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
|
||||
{Komornicki}}\ and\ \bibinfo {author} {\bibfnamefont {H.~F.}\ \bibnamefont
|
||||
{King}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J. Chem.
|
||||
Phys.}\ }\textbf {\bibinfo {volume} {134}},\ \bibinfo {pages} {244115}
|
||||
(\bibinfo {year} {2011})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Reine}, \citenamefont {Helgaker},\ and\ \citenamefont
|
||||
{Lind}(2012)}]{Reine12}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
|
||||
{Reine}}, \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont {Helgaker}}, \
|
||||
and\ \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Lind}},\ }\href@noop
|
||||
{} {\bibfield {journal} {\bibinfo {journal} {WIREs Comput. Mol. Sci.}\
|
||||
}\textbf {\bibinfo {volume} {2}},\ \bibinfo {pages} {290} (\bibinfo {year}
|
||||
{2012})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Barca}\ and\ \citenamefont {Gill}(2016)}]{GG16}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.~M.~J.}\
|
||||
\bibnamefont {Barca}}\ and\ \bibinfo {author} {\bibfnamefont {P.~M.~W.}\
|
||||
\bibnamefont {Gill}},\ }\href@noop {} {\bibfield {journal} {\bibinfo
|
||||
{journal} {J. Chem. Theory Comput.}\ }\textbf {\bibinfo {volume} {12}},\
|
||||
\bibinfo {pages} {4915} (\bibinfo {year} {2016})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Barca}, \citenamefont {Loos},\ and\ \citenamefont
|
||||
{Gill}(2016)}]{3ERI1}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.~M.~J.}\
|
||||
\bibnamefont {Barca}}, \bibinfo {author} {\bibfnamefont {P.~F.}\ \bibnamefont
|
||||
{Loos}}, \ and\ \bibinfo {author} {\bibfnamefont {P.~M.~W.}\ \bibnamefont
|
||||
{Gill}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J. Chem.
|
||||
Theory Comput.}\ }\textbf {\bibinfo {volume} {12}},\ \bibinfo {pages} {1725}
|
||||
(\bibinfo {year} {2016})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Barca}, \citenamefont {Loos},\ and\ \citenamefont
|
||||
{Gill}(tion)}]{3ERI2}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.~M.~J.}\
|
||||
\bibnamefont {Barca}}, \bibinfo {author} {\bibfnamefont {P.~F.}\ \bibnamefont
|
||||
{Loos}}, \ and\ \bibinfo {author} {\bibfnamefont {P.~M.~W.}\ \bibnamefont
|
||||
{Gill}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J. Chem.
|
||||
Theory Comput.}\ } (\bibinfo {year} {in preparation})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Barca}\ and\ \citenamefont {Loos}(ress)}]{4eRR}%
|
||||
\BibitemOpen
|
||||
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|
||||
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|
||||
\bibnamefont {Loos}},\ }\href@noop {} {\bibfield {journal} {\bibinfo
|
||||
{journal} {Adv. Quantum Chem.}\ } (\bibinfo {year} {in press})}\BibitemShut
|
||||
{NoStop}%
|
||||
\bibitem [{\citenamefont {Barca}\ and\ \citenamefont {Loos}(2017)}]{IntF12}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.~M.~J.}\
|
||||
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|
||||
\bibnamefont {Loos}},\ }\href@noop {} {\bibfield {journal} {\bibinfo
|
||||
{journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {147}},\ \bibinfo
|
||||
{pages} {024103} (\bibinfo {year} {2017})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Valeev}(2004)}]{Valeev04}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.~F.}\ \bibnamefont
|
||||
{Valeev}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Chem.
|
||||
Phys. Lett.}\ }\textbf {\bibinfo {volume} {395}},\ \bibinfo {pages} {190}
|
||||
(\bibinfo {year} {2004})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Werner}, \citenamefont {Adler},\ and\ \citenamefont
|
||||
{Manby}(2007)}]{Werner07}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.~J.}\ \bibnamefont
|
||||
{Werner}}, \bibinfo {author} {\bibfnamefont {T.~B.}\ \bibnamefont {Adler}}, \
|
||||
and\ \bibinfo {author} {\bibfnamefont {F.~R.}\ \bibnamefont {Manby}},\
|
||||
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J. Chem. Phys.}\
|
||||
}\textbf {\bibinfo {volume} {126}},\ \bibinfo {pages} {164102} (\bibinfo
|
||||
{year} {2007})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Garniron}\ \emph {et~al.}(2017)\citenamefont
|
||||
{Garniron}, \citenamefont {Scemama}, \citenamefont {Loos},\ and\
|
||||
\citenamefont {Caffarel.}}]{PT2}%
|
||||
\BibitemOpen
|
||||
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|
||||
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|
||||
\bibinfo {author} {\bibfnamefont {P.~F.}\ \bibnamefont {Loos}}, \ and\
|
||||
\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Caffarel.}},\ }\href@noop
|
||||
{} {\bibfield {journal} {\bibinfo {journal} {J. Chem. Phys.}\ }\textbf
|
||||
{\bibinfo {volume} {147}},\ \bibinfo {pages} {034101} (\bibinfo {year}
|
||||
{2017})}\BibitemShut {NoStop}%
|
||||
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|
||||
{Scemama}, \citenamefont {Giner}, \citenamefont {Applencourt}, \citenamefont
|
||||
{David},\ and\ \citenamefont {Caffarel}}]{QP}%
|
||||
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|
||||
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|
||||
{Scemama}}, \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {Giner}},
|
||||
\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont {Applencourt}}, \bibinfo
|
||||
{author} {\bibfnamefont {G.}~\bibnamefont {David}}, \ and\ \bibinfo {author}
|
||||
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|
||||
10.5281/zenodo.30624} {\enquote {\bibinfo {title} {Quantum package v0.6},}\ }
|
||||
(\bibinfo {year} {2015}),\ \bibinfo {note}
|
||||
{doi:10.5281/zenodo.30624}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Nakashima}\ and\ \citenamefont
|
||||
{Nakatsuji}(2007)}]{Nakashima07}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
|
||||
{Nakashima}}\ and\ \bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
|
||||
{Nakatsuji}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J.
|
||||
Chem. Phys.}\ }\textbf {\bibinfo {volume} {127}},\ \bibinfo {pages} {224104}
|
||||
(\bibinfo {year} {2007})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Puchalski}, \citenamefont {Kedziera},\ and\
|
||||
\citenamefont {Pachucki}(2009)}]{Puchalski09}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
|
||||
{Puchalski}}, \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Kedziera}},
|
||||
\ and\ \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Pachucki}},\
|
||||
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. A}\
|
||||
}\textbf {\bibinfo {volume} {80}},\ \bibinfo {pages} {032521} (\bibinfo
|
||||
{year} {2009})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Sharkey}\ and\ \citenamefont
|
||||
{Adamowicz}(2011)}]{Sharkey11}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {K.~L.}\ \bibnamefont
|
||||
{Sharkey}}\ and\ \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont
|
||||
{Adamowicz}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J.
|
||||
Chem. Phys.}\ }\textbf {\bibinfo {volume} {134}},\ \bibinfo {pages} {094104}
|
||||
(\bibinfo {year} {2011})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Bubin}\ and\ \citenamefont
|
||||
{Adamowicz}(2011)}]{Bubin11}%
|
||||
\BibitemOpen
|
||||
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|
||||
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|
||||
{Adamowicz}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal}
|
||||
{Phys. Rev. A}\ }\textbf {\bibinfo {volume} {83}},\ \bibinfo {pages} {022505}
|
||||
(\bibinfo {year} {2011})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Sharkey}, \citenamefont {Bubin},\ and\ \citenamefont
|
||||
{Adamowicz}(2010)}]{Sharkey10}%
|
||||
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|
||||
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|
||||
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|
||||
and\ \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont {Adamowicz}},\
|
||||
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|
||||
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|
||||
{year} {2010})}\BibitemShut {NoStop}%
|
||||
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|
||||
{Klopper}, \citenamefont {Bachorz}, \citenamefont {Tew},\ and\ \citenamefont
|
||||
{Hattig}}]{Klopper10}%
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
\BibitemOpen
|
||||
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|
||||
{Pachucki}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal}
|
||||
{Phys. Rev. A}\ }\textbf {\bibinfo {volume} {82}},\ \bibinfo {pages} {032509}
|
||||
(\bibinfo {year} {2010})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Cleland}\ \emph {et~al.}(2012)\citenamefont
|
||||
{Cleland}, \citenamefont {Booth}, \citenamefont {Overy},\ and\ \citenamefont
|
||||
{Alavi}}]{Cleland12}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
|
||||
{Cleland}}, \bibinfo {author} {\bibfnamefont {G.~H.}\ \bibnamefont {Booth}},
|
||||
\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {Overy}}, \ and\ \bibinfo
|
||||
{author} {\bibfnamefont {A.}~\bibnamefont {Alavi}},\ }\href@noop {}
|
||||
{\bibfield {journal} {\bibinfo {journal} {J. Chem. Theory Comput.}\
|
||||
}\textbf {\bibinfo {volume} {8}},\ \bibinfo {pages} {4138} (\bibinfo {year}
|
||||
{2012})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Almora-Diaz}(2014)}]{AlmoraDiaz14}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {C.~X.}\ \bibnamefont
|
||||
{Almora-Diaz}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal}
|
||||
{J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {140}},\ \bibinfo {pages}
|
||||
{184302} (\bibinfo {year} {2014})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Yousaf}\ and\ \citenamefont
|
||||
{Peterson}(2008)}]{Yousaf08}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {K.~E.}\ \bibnamefont
|
||||
{Yousaf}}\ and\ \bibinfo {author} {\bibfnamefont {K.~A.}\ \bibnamefont
|
||||
{Peterson}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J.
|
||||
Chem. Phys.}\ }\textbf {\bibinfo {volume} {129}},\ \bibinfo {pages} {184108}
|
||||
(\bibinfo {year} {2008})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Yousaf}\ and\ \citenamefont
|
||||
{Peterson}(2009)}]{Yousaf09}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {K.~E.}\ \bibnamefont
|
||||
{Yousaf}}\ and\ \bibinfo {author} {\bibfnamefont {K.~A.}\ \bibnamefont
|
||||
{Peterson}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal}
|
||||
{Chem. Phys. Lett.}\ }\textbf {\bibinfo {volume} {476}},\ \bibinfo {pages}
|
||||
{303} (\bibinfo {year} {2009})}\BibitemShut {NoStop}%
|
||||
\end{thebibliography}%
|
||||
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|
||||
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||||
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|
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|
||||
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||||
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|
||||
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|
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|
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|
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|
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|
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|
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|
||||
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|
||||
5918 wiz_defined-function locations,
|
||||
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|
||||
and the built_in function-call counts, 60829 in all, are:
|
||||
= -- 4101
|
||||
> -- 1786
|
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< -- 361
|
||||
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|
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|
||||
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|
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|
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|
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|
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|
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|
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\BOOKMARK [0][-]{section*.2}{Dressing the configuration interaction matrix with explicit correlation}{}% 2
|
||||
\BOOKMARK [1][-]{section*.1}{Abstract}{section*.2}% 1
|
||||
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||||
\documentclass[aip,jcp,reprint,showkeys]{revtex4-1}
|
||||
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,hyperref,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem}
|
||||
|
||||
\usepackage{mathpazo,libertine}
|
||||
\newcommand{\alert}[1]{\textcolor{red}{#1}}
|
||||
\newcommand{\cdash}{\multicolumn{1}{c}{---}}
|
||||
\newcommand{\mc}{\multicolumn}
|
||||
\newcommand{\mcc}[1]{\multicolumn{1}{c}{#1}}
|
||||
\newcommand{\mr}{\multirow}
|
||||
|
||||
\newcommand{\eps}{\varepsilon}
|
||||
\newcommand{\la}{\lambda}
|
||||
|
||||
\newcommand{\br}{\bm{r}}
|
||||
|
||||
\newcommand{\oH}{\mathring{H}}
|
||||
\newcommand{\EPT}{E_\text{PT2}}
|
||||
\newcommand{\EHF}{E_\text{HF}}
|
||||
\newcommand{\NGG}{N_\text{GG}}
|
||||
|
||||
\newcommand{\hH}{\Hat{H}}
|
||||
\newcommand{\hO}{\Hat{O}}
|
||||
\newcommand{\hT}{\Hat{T}}
|
||||
\newcommand{\hV}{\Hat{V}}
|
||||
\newcommand{\hU}{\Hat{U}}
|
||||
\newcommand{\hQ}{\Hat{Q}}
|
||||
\newcommand{\hS}{\Hat{S}}
|
||||
\newcommand{\hP}{\Hat{P}}
|
||||
\newcommand{\hI}{\Hat{I}}
|
||||
|
||||
\newcommand{\mA}{\mathcal{A}}
|
||||
\newcommand{\mC}{\mathcal{C}}
|
||||
\newcommand{\mD}{\mathcal{D}}
|
||||
\newcommand{\mE}{\mathcal{E}}
|
||||
\newcommand{\mK}{\mathcal{K}}
|
||||
\newcommand{\mF}{\mathcal{F}}
|
||||
\newcommand{\mL}{\mathcal{L}}
|
||||
|
||||
\newcommand{\kA}{\ket{A}}
|
||||
\newcommand{\kD}{\ket{D}}
|
||||
\newcommand{\kI}{\ket{I}}
|
||||
\newcommand{\kJ}{\ket{J}}
|
||||
\newcommand{\kE}{\ket{E}}
|
||||
\newcommand{\kK}{\ket{K}}
|
||||
\newcommand{\kL}{\ket{L}}
|
||||
\newcommand{\kF}{\ket{F}}
|
||||
\newcommand{\kO}{\ket{0}}
|
||||
|
||||
% addresses
|
||||
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\title{Dressing the configuration interaction matrix with explicit correlation}
|
||||
|
||||
\author{Anthony Scemama}
|
||||
\affiliation{\LCPQ}
|
||||
\author{Michel Caffarel}
|
||||
\affiliation{\LCPQ}
|
||||
\author{Pierre-Fran{\c c}ois Loos}
|
||||
\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr}
|
||||
\affiliation{\LCPQ}
|
||||
|
||||
\begin{abstract}
|
||||
We present an explicitly-correlated version of the configuration interaction (CI) method.
|
||||
An explicitly-correlated term is introduced via dressing of the CI matrix.
|
||||
The dressing is guided by electron-electron cusp conditions.
|
||||
This greatly enhances the convergence with respect to the one-electron basis set compared to conventional CI methods.
|
||||
The performance of the newly-designed explicitly-correlated dressing CI method is illustrated on atoms and molecules.
|
||||
\end{abstract}
|
||||
|
||||
\keywords{configuration interaction; explicitly-correlated methods; dressed Hamiltonian}
|
||||
|
||||
\maketitle
|
||||
|
||||
%----------------------------------------------------------------
|
||||
\textit{Introduction.---}
|
||||
%----------------------------------------------------------------
|
||||
One of the most fundamental problem of conventional electronic structure methods is their slow energy convergence with respect to the size of the one-electron basis set.
|
||||
This problem was already spotted thirty years ago by Kutzelnigg \cite{Kutzelnigg85} who proposed to introduce explicitly the correlation between electrons via the introduction of the interelectronic distance $r_{12} = \abs{\br_1 - \br_2}$ as a basis function \cite{Kutzelnigg91, Termath91, Klopper91a, Klopper91b, Noga94}.
|
||||
This yields a prominent improvement of the energy convergence from $O(L^{-3})$ to $O(L^{-7})$ (where $L$ is the maximum angular momentum of the one-electron basis).
|
||||
This idea was later generalised to more accurate correlation factors $f_{12} \equiv f(r_{12})$ \cite{Persson96, Persson97, May04, Tenno04b, Tew05, May05}.
|
||||
The resulting F12 methods achieve chemical accuracy for small organic molecules with relatively small Gaussian basis sets \cite{Tenno12a, Tenno12b, Hattig12, Kong12}.
|
||||
For example, as illustrated by Tew and coworkers, one can obtain, at the CCSD(T) level, quintuple-zeta quality correlation energies with a triple-zeta basis \cite{Tew07b}.
|
||||
|
||||
In the present study, following Kutzelnigg's idea, we propose to introduce the explicit correlation between electrons within the configuration interaction (CI) method via a dressing of the CI matrix \cite{Huron73, Evangelisti83}.
|
||||
This method, involving effective Hamiltonian theory, has been shown to be successful in other scenarios \cite{Heully92}.
|
||||
Compared to other explicitly-correlated methods, this dressing strategy has the advantage of introducing the explicit correlation at a low computational cost.
|
||||
The present explicitly-correlated dressing CI method is completely general and can be applied to any type of truncated, full, or even selected CI methods \cite{Giner13, Scemama13a, Scemama13b, Scemama14, Giner15, Caffarel16}.
|
||||
However, for the sake of generality, we will discuss here the dressing of the full CI (FCI) matrix.
|
||||
%Here, we focus on systems well described by a single (reference) determinant $\kO$ assumed to be a Hartree-Fock (HF) determinant.
|
||||
%The multireference version of the present method will be reported in a separate study.
|
||||
Atomic units are used throughout.
|
||||
|
||||
%----------------------------------------------------------------
|
||||
\textit{Ansatz.---}
|
||||
%----------------------------------------------------------------
|
||||
Inspired by a number of previous research \cite{Shiozaki11}, our electronic wave function ansatz $\ket{\Psi} = \kD + \kF$ is simply written as the sum of a ``conventional'' part
|
||||
\begin{equation}
|
||||
\label{eq:D}
|
||||
\kD = \sum_{I} c_I \kI
|
||||
\end{equation}
|
||||
composed by a linear combination of determinants $\kI$ of coefficients $c_I$ and an ``explicitly-correlated'' part
|
||||
\begin{equation}
|
||||
\label{eq:WF-F12-CIPSI}
|
||||
\kF = \sum_{I} t_I \hQ f \kI
|
||||
\end{equation}
|
||||
with coefficients $t_I$.
|
||||
The projector
|
||||
\begin{equation}
|
||||
\hQ = \hI - \sum_{I} \dyad{I}
|
||||
\end{equation}
|
||||
ensures the orthogonality between $\kD$ and $\kF$, and
|
||||
\begin{equation}
|
||||
\label{eq:Ja}
|
||||
f = \sum_{i < j} \gamma_{ij} f_{ij}
|
||||
\end{equation}
|
||||
is a correlation factor, and
|
||||
\begin{equation}
|
||||
\gamma_{ij} =
|
||||
\begin{cases}
|
||||
1/2, & \text{for opposite-spin electrons},
|
||||
\\
|
||||
1/4, & \text{for same-spin electrons}.
|
||||
\end{cases}
|
||||
\end{equation}
|
||||
|
||||
As first shown by Kato \cite{Kato51, Kato57} (and further elaborated by various authors \cite{Pack66, Morgan93}), for small $r_{12}$, the two-electron correlation factor $f_{12}$ in Eq.~\eqref{eq:Ja} must behave as
|
||||
\begin{equation}
|
||||
f_{12} = \gamma_{12}\,r_{12} + \order{r_{12}^2}.
|
||||
\end{equation}
|
||||
|
||||
%----------------------------------------------------------------
|
||||
\textit{Dressing.---}
|
||||
%----------------------------------------------------------------
|
||||
Our primary goal is to introduce the explicit correlation between electrons at low computational cost.
|
||||
Therefore, assuming that $\hH \ket{\Psi} = E \Psi$, one can write, by projection over $\bra{I}$,
|
||||
\begin{equation}
|
||||
c_I \qty[ H_{II} + c_I^{-1} \mel*{I}{\hH}{F} - E] + \sum_{J \ne I} c_J H_{IJ} = 0.
|
||||
\end{equation}
|
||||
where $H_{IJ} = \mel{I}{\hH}{J}$.
|
||||
Hence, we obtain the desired energy by diagonalizing the dressed Hamiltonian:
|
||||
\begin{equation}
|
||||
\label{eq:DrH}
|
||||
\oH_{IJ} =
|
||||
\begin{cases}
|
||||
H_{II} + c_I^{-1}\mel*{I}{\hH}{F}, & \text{if $I = J$},
|
||||
\\
|
||||
H_{IJ}, & \text{otherwise},
|
||||
\end{cases}
|
||||
\end{equation}
|
||||
with
|
||||
\begin{equation}
|
||||
\label{eq:IHF}
|
||||
\mel{I}{\hH}{F} = \sum_J t_J \qty[ \mel{I}{\hH f}{J} - \sum_{K} H_{IK} f_{KJ} ],
|
||||
\end{equation}
|
||||
and $f_{IJ} = \mel{I}{f}{J}$.
|
||||
It is interesting to note that, in an infinite basis, we have $\mel{I}{\hH}{F} = 0$, which demonstrates that our dressed CI method becomes exact in the limit of a complete one-electron basis.
|
||||
|
||||
At this stage, two key comments are in order.
|
||||
First, as one may have realized, the coefficients $t_I$ are unknown.
|
||||
\alert{However, they can be set to ensure the $s$- and $p$-wave electron-electron cusp conditions (SP ansatz) \cite{Tenno04a}.}
|
||||
|
||||
\alert{This yields the following linear system of equations
|
||||
\begin{equation}
|
||||
\label{eq:tI}
|
||||
\sum_J (\delta_{IJ} + f_{IJ}) t_J = c_I,
|
||||
\end{equation}
|
||||
which can be easily solved using standard linear algebra packages.}
|
||||
|
||||
Second, because Eq.~\eqref{eq:DrH} depends on the CI coefficient $c_I$, one must iterate the diagonalization process self-consistently until convergence of the desired eigenvalues of the dressed Hamiltonian $\oH$.
|
||||
At each iteration, we solve Eq.~\eqref{eq:tI} to obtain the coefficients $t_I$ and dress the Hamiltonian [see Eq.~\eqref{eq:DrH}].
|
||||
In practice, we initially start with a CI vector obtained by the diagonalization of the undressed Hamiltonian, and convergence is usually reached within few cycles.
|
||||
For pathological cases, a DIIS-like procedure may be employed \cite{Pulay82}.
|
||||
|
||||
%%% FIG 1 %%%
|
||||
%\begin{figure}
|
||||
% \includegraphics[width=\linewidth]{fig1}
|
||||
% \caption{
|
||||
% \label{fig:CBS}
|
||||
% Schematic representation of the various orbital spaces and their notation.
|
||||
% The arrows represent the three types of excited determinants contributing to the dressing: the pure doubles $\ket*{_{ij}^{\alpha \beta}}$ (green), the mixed doubles $\ket*{_{ij}^{a \beta}}$ (magenta) and the pure singles $\ket*{_{i}^{\alpha}}$ (orange).}
|
||||
%\end{figure}
|
||||
%%% %%%
|
||||
|
||||
%----------------------------------------------------------------
|
||||
\textit{Matrix elements.---}
|
||||
%----------------------------------------------------------------
|
||||
Compared to a conventional CI calculation, new matrix elements are required.
|
||||
The simplest of them $f_{IJ}$ --- required in Eqs.~\eqref{eq:IHF} and \eqref{eq:tI} --- can be easily computed by applying Condon-Slater rules \cite{SzaboBook}.
|
||||
They involve two-electron integrals over the geminal factor $f_{12}$.
|
||||
Their computation has been thoroughly studied in the literature in the last thirty years \cite{Kutzelnigg91, Klopper92, Persson97, Klopper02, Manby03, Werner03, Klopper04, Tenno04a, Tenno04b, May05, Manby06, Tenno07, Komornicki11, Reine12, GG16}.
|
||||
These can be more or less expensive to compute depending on the choice of the correlation factor.
|
||||
|
||||
As shown in Eq.~\eqref{eq:IHF}, the present explicitly-correlated CI method also requires matrix elements of the form $\mel{I}{\hH f}{ J}$.
|
||||
These are more problematic, as they involve the computation of numerous three-electron integrals over the operator $r_{12}^{-1}f_{13}$, as well as new two-electron integrals \cite{Kutzelnigg91, Klopper92}.
|
||||
We have recently developed recurrence relations and efficient upper bounds in order to compute these types of integrals \cite{3ERI1, 3ERI2, 4eRR, IntF12}.
|
||||
|
||||
However, we will explore here a different route.
|
||||
We propose to compute them using the resolution of the identity (RI) approximation \cite{Kutzelnigg91, Klopper02, Valeev04, Werner07, Hattig12}, which requires a complete basis set (CBS).
|
||||
This CBS is built as the union of the orbital basis set (OBS) $\qty{p}$ (divided as occupied $\qty{i}$ and virtual $\qty{a}$ subspaces) augmented by a complementary auxiliary basis set (CABS) $\qty{\alpha}$, such as $ \qty{p} \cap \qty{\alpha} = \varnothing$ and $\braket{p}{\alpha} = 0$ \cite{Klopper02, Valeev04}.% (see Fig.~\ref{fig:CBS}).
|
||||
|
||||
In the CBS, one can write
|
||||
\begin{equation}
|
||||
\label{eq:RI}
|
||||
\hI = \sum_{A \in \mA} \dyad{A}{A}
|
||||
\end{equation}
|
||||
where $\mA$ is the set of all the determinants $\kA$ corresponding to electronic excitations from occupied orbitals $\qty{i}$ to the extended virtual orbital space $\qty{a} \cup \qty{\alpha}$.
|
||||
Substituting \eqref{eq:RI} into the first term of the right-hand side of Eq.~\eqref{eq:IHF}, one gets
|
||||
\begin{equation}
|
||||
\label{eq:IHF-RI}
|
||||
\begin{split}
|
||||
\mel{I}{\hH}{F}
|
||||
& = \sum_J t_J \qty[ \sum_{A \in \mA} H_{IA} f_{AJ} - \sum_{K \in \mD} H_{IK} f_{KJ} ]
|
||||
\\
|
||||
& = \sum_J t_J \sum_{A \in \mC} H_{IA} f_{AJ},
|
||||
\end{split}
|
||||
\end{equation}
|
||||
where $\mD$ is the set of ``conventional'' determinants obtained by excitations from the occupied space $\qty{i}$ to the virtual one $\qty{a}$, and $\mC = \mA \setminus \mD$.
|
||||
Because $f$ is a two-electron operator, the way to compute efficiently Eq.~\eqref{eq:IHF-RI} is actually very similar to what is done within second-order multireference perturbation theory \cite{PT2}.
|
||||
|
||||
%The set $\mC$ is defined by two simple rules.
|
||||
%First, because $f$ is a two-electron operator (and thanks to the matrix element $f_{AJ}$ in \eqref{eq:IHF-RI}), we know that the sum over $A$ is restricted to the singly- or doubly-excited determinants with respect to the determinant $\kJ$.
|
||||
%Second, to ensure that $H_{IA} \neq 0$, $A$ must be connected to $\kI$, i.e.~differs from $\kI$ by no more than two spin orbitals.
|
||||
%Three types of determinants have these two properties:
|
||||
%i) the pure doubles $\ket*{_{ij}^{\alpha \beta}}$;
|
||||
%ii) the mixed doubles $\ket*{_{ij}^{a \beta}}$;
|
||||
%iii) the pure singles $\ket*{_{i}^{\alpha}}$.
|
||||
|
||||
Although $\mel{0}{\hH}{_{i}^{a}} = 0$, note that the Brillouin theorem does not hold in the CABS, i.e.~$\mel{0}{\hH}{_{i}^{\alpha}} \neq 0$.
|
||||
Here, we will eschew the generalized Brillouin condition (GBC) which set these to zero \cite{Kutzelnigg91}.
|
||||
|
||||
%\begin{gather}
|
||||
% \mel*{0}{\hH}{_i^\alpha} = \mel{i}{h}{\alpha} + \sum_{j} \mel{ij}{}{\alpha j}
|
||||
% \\
|
||||
% \mel*{0}{\hH}{_{ij}^{\alpha\beta}} = \mel{ij}{}{\alpha \beta}
|
||||
% \\
|
||||
% \mel*{0}{\hH}{_{ij}^{a\beta}} = \mel{ij}{}{a \beta}
|
||||
%\end{gather}
|
||||
%
|
||||
%\begin{gather}
|
||||
% \mel*{_k^c}{\hH}{_i^\alpha} = \mel{c}{h}{\alpha} + \sum_{j} \mel{cj}{}{\alpha j}
|
||||
% \\
|
||||
% \mel*{_k^c}{\hH}{_{ij}^{\alpha\beta}} = 0
|
||||
% \\
|
||||
% \mel*{_k^c}{\hH}{_{ij}^{a\beta}} =
|
||||
%\end{gather}
|
||||
%
|
||||
%\begin{gather}
|
||||
% \mel*{_{kl}^{cd}}{\hH}{_i^\alpha} =
|
||||
% \\
|
||||
% \mel*{_{kl}^{cd}}{\hH}{_{ij}^{\alpha\beta}} =
|
||||
% \\
|
||||
% \mel*{_{kl}^{cd}}{\hH}{_{ij}^{a\beta}} =
|
||||
%\end{gather}
|
||||
%
|
||||
%\begin{gather}
|
||||
% \mel*{_{klm}^{cde}}{\hH}{_i^\alpha} =
|
||||
% \\
|
||||
% \mel*{_{klm}^{cde}}{\hH}{_{ij}^{\alpha\beta}} =
|
||||
% \\
|
||||
% \mel*{_{klm}^{cde}}{\hH}{_{ij}^{a\beta}} =
|
||||
%\end{gather}
|
||||
|
||||
%----------------------------------------------------------------
|
||||
\textit{Results.---}
|
||||
%----------------------------------------------------------------
|
||||
In all the calculations presented below, we consider the following Slater-type correlation factor \cite{Tenno04a}
|
||||
\begin{equation}
|
||||
f_{12} = \frac{1 - \exp( - \la r_{12} )}{\la},
|
||||
\end{equation}
|
||||
which is fitted using $N_\text{GG}$ Gaussian geminals fo computational convenience \cite{Persson96, Persson97, May04, Tenno04b, Tew05, May05}, i.e.
|
||||
\begin{equation}
|
||||
\exp( - \la r_{12} ) \approx \sum_{\nu=1}^{\NGG} a_\nu \exp( - \la_\nu r_{12}^2 ).
|
||||
\end{equation}
|
||||
The coefficients $a_\nu$ can be found in Ref.~\onlinecite{Tew05} for various $\NGG$, but we consider $\NGG = 6$ in this study.
|
||||
All the calculations have been performed with Quantum Package \cite{QP}.
|
||||
|
||||
%%% TABLE 1 %%%
|
||||
\begin{table}
|
||||
\caption{
|
||||
\label{tab:atoms}
|
||||
FCI-F12, CIPSI and FCI total ground-state energy of the neutral atoms for $Z = 2$ to $10$ calculated with Dunning's cc-pVXZ basis set.
|
||||
The corresponding cc-pVXZ\_OPTRI auxiliary basis is used as CABS.}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{lcccc}
|
||||
Atom & cc-pVXZ & FCI-F12 & CIPSI & FCI \\
|
||||
\hline
|
||||
\ce{He} & D & & & $-2.887\,595$\footnotemark[1] \\
|
||||
& T & & & $-2.900\,232$\footnotemark[1] \\
|
||||
& Q & & & $-2.902\,411$\footnotemark[1] \\
|
||||
& 5 & & & $-2.903\,152$\footnotemark[1] \\
|
||||
& 6 & & & $-2.903\,432$\footnotemark[1] \\
|
||||
& $\infty$ & & & $-2.903\,724$\footnotemark[2] \\
|
||||
\hline
|
||||
\ce{Li} & D & & & \\
|
||||
& T & & & \\
|
||||
& Q & & & \\
|
||||
& 5 & & & \\
|
||||
& $\infty$ & & & $-7.478\,060$\footnotemark[3] \\
|
||||
\hline
|
||||
\ce{Be} & D & & & \\
|
||||
& T & & & \\
|
||||
& Q & & & \\
|
||||
& 5 & & & \\
|
||||
& $\infty$ & & & $-14.667\,356$\footnotemark[4] \\
|
||||
\hline
|
||||
\ce{B} & D & & & \\
|
||||
& T & & & \\
|
||||
& Q & & & \\
|
||||
& 5 & & & \\
|
||||
& $\infty$ & & & $-24.653\,866$\footnotemark[5] \\
|
||||
\hline
|
||||
\ce{C} & D & & & \\
|
||||
& T & & & \\
|
||||
& Q & & & \\
|
||||
& 5 & & & \\
|
||||
& $\infty$ & & & $-37.840\,129$\footnotemark[6] \\
|
||||
\hline
|
||||
\ce{N} & D & & & \\
|
||||
& T & & & \\
|
||||
& Q & & & \\
|
||||
& 5 & & & \\
|
||||
& $\infty$ & & & $ 54.588917$\footnotemark[7] \\
|
||||
\hline
|
||||
\ce{O} & D & & & \\
|
||||
& T & & & \\
|
||||
& Q & & & \\
|
||||
& 5 & & & \\
|
||||
& $\infty$ & & & $-75.066892$\footnotemark[7] \\
|
||||
\hline
|
||||
\ce{F} & D & & & \\
|
||||
& T & & & \\
|
||||
& Q & & & \\
|
||||
& 5 & & & \\
|
||||
& $\infty$ & & & $-99.733\,424$\footnotemark[7] \\
|
||||
\hline
|
||||
\ce{Ne} & D & & & \\
|
||||
& T & & & \\
|
||||
& Q & & & \\
|
||||
& 5 & & & \\
|
||||
& $\infty$ & & & $-128.937\,274$\footnotemark[7] \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\footnotetext[1]{Reference \onlinecite{Kong12}}
|
||||
\footnotetext[2]{Reference \onlinecite{Nakashima07}}
|
||||
\footnotetext[3]{Reference \onlinecite{Puchalski09}}
|
||||
\footnotetext[4]{Reference \onlinecite{Sharkey11}}
|
||||
\footnotetext[5]{Reference \onlinecite{Bubin11}}
|
||||
\footnotetext[6]{Reference \onlinecite{Sharkey10}}
|
||||
\footnotetext[7]{Reference \onlinecite{Klopper10}}
|
||||
\end{table}
|
||||
%%%
|
||||
|
||||
%%% TABLE 2 %%%
|
||||
\begin{table*}
|
||||
\caption{
|
||||
\label{tab:molecules}
|
||||
CIPSI, FCI-F12 i-FCIQMC and FCI total ground-state energy of the \ce{H2}, \ce{F2} and \ce{H2)} molecules at experimental geometry with Dunning's cc-pVXZ basis set.
|
||||
The corresponding cc-pVXZ\_OPTRI auxiliary basis is used as CABS.}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{lccccc}
|
||||
Molecule & cc-pVXZ & \mcc{CIPSI} & \mcc{FCI-F12} & \mcc{i-FCIQMC} & \mcc{FCI} \\
|
||||
\hline
|
||||
\ce{H2} & D & & & \\
|
||||
& T & & & \\
|
||||
& Q & & & \\
|
||||
& 5 & & & \\
|
||||
& $\infty$ & & & & $-1.174\,476$\footnotemark[1] \\
|
||||
\hline
|
||||
\ce{F2} & D & $-199.099\,28$\footnotemark[2] & & $-199.099\,41(9)$\footnotemark[3] \\
|
||||
& T & $-199.296\,5$\footnotemark[2] & & $-199.297\,7(1)$\footnotemark[3] \\
|
||||
\hline
|
||||
\ce{H2O} & D & $-76.282\,136$\footnotemark[4] & & & $-76.282\,865$\footnotemark[5] \\
|
||||
& T & $-76.388\,287$\footnotemark[4] & & & $-76.390\,158$\footnotemark[5] \\
|
||||
& Q & $-76.419\,324$\footnotemark[4] & & & $-76.421\,148$\footnotemark[5] \\
|
||||
& 5 & $-76.428\,550$\footnotemark[4] & & & $-76.431\,105$\footnotemark[5] \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\footnotetext[1]{Reference \onlinecite{Pachucki10}}
|
||||
\footnotetext[2]{Reference \onlinecite{Giner15}}
|
||||
\footnotetext[3]{Reference \onlinecite{Cleland12}}
|
||||
\footnotetext[4]{Reference \onlinecite{Caffarel16}}
|
||||
\footnotetext[5]{Reference \onlinecite{AlmoraDiaz14}}
|
||||
\end{table*}
|
||||
%%%
|
||||
|
||||
In Table \ref{tab:atoms}, we report the total atomic energy of the neutral atoms from $Z = 2$ to $10$ for various Dunning's basis sets.
|
||||
In all calculations, the associated OPTRI basis is used as CABS \cite{Yousaf08, Yousaf09}.
|
||||
|
||||
In Table \ref{tab:molecules}, we report the total energy of the \ce{H2}, \ce{F2} and \ce{H2O} molecules at experimental geometry \cite{Giner13, Giner15, Caffarel16}.
|
||||
|
||||
%----------------------------------------------------------------
|
||||
\textit{Conclusion.---}
|
||||
%----------------------------------------------------------------
|
||||
We have introduced a dressed version of the well-established CI method to incorporate explicitly the correlation between electrons.
|
||||
We have shown that the new CI-F12 method allows to fix one of the main issue of conventional CI methods, i.e.~the slow convergence of the electronic energy with respect to the size of the one-electron basis set. Albeit not variational, our method is able to catch a large fraction of the basis set incompleteness error at a low computational cost compared to other variants.
|
||||
In particular, one eschew the computation of four-electron integrals as well as some types of three-electron integrals.
|
||||
We believe that the present approach is a significant step towards the development of an accurate and efficient explicitly-correlated full CI methods.
|
||||
|
||||
%----------------------------------------------------------------
|
||||
\textit{Acknowledgments.---}
|
||||
This work was performed using HPC resources from CALMIP (Toulouse) under allocation 2016-0510 and from GENCI-TGCC (Grant 2016-08s015).
|
||||
%----------------------------------------------------------------
|
||||
|
||||
\bibliography{CI-F12}
|
||||
|
||||
\end{document}
|
||||
2
Manuscript/CI-F12Notes.bib
Normal file
2
Manuscript/CI-F12Notes.bib
Normal file
@ -0,0 +1,2 @@
|
||||
@CONTROL{REVTEX41Control}
|
||||
@CONTROL{aip41Control,pages="0",title=""}
|
||||
BIN
Manuscript/fig1.pdf
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Manuscript/fig1.pdf
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19
Manuscript/fig1/fig1.aux
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19
Manuscript/fig1/fig1.aux
Normal file
@ -0,0 +1,19 @@
|
||||
\relax
|
||||
\providecommand\hyper@newdestlabel[2]{}
|
||||
\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
|
||||
\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
|
||||
\global\let\oldcontentsline\contentsline
|
||||
\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
|
||||
\global\let\oldnewlabel\newlabel
|
||||
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
|
||||
\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
|
||||
\AtEndDocument{\ifx\hyper@anchor\@undefined
|
||||
\let\contentsline\oldcontentsline
|
||||
\let\newlabel\oldnewlabel
|
||||
\fi}
|
||||
\fi}
|
||||
\global\let\hyper@last\relax
|
||||
\gdef\HyperFirstAtBeginDocument#1{#1}
|
||||
\providecommand\HyField@AuxAddToFields[1]{}
|
||||
\providecommand\HyField@AuxAddToCoFields[2]{}
|
||||
\providecommand \oddpage@label [2]{}
|
||||
48
Manuscript/fig1/fig1.blg
Normal file
48
Manuscript/fig1/fig1.blg
Normal file
@ -0,0 +1,48 @@
|
||||
This is BibTeX, Version 0.99d (TeX Live 2015)
|
||||
Capacity: max_strings=35307, hash_size=35307, hash_prime=30011
|
||||
The top-level auxiliary file: fig1.aux
|
||||
I found no \citation commands---while reading file fig1.aux
|
||||
I found no \bibdata command---while reading file fig1.aux
|
||||
I found no \bibstyle command---while reading file fig1.aux
|
||||
You've used 0 entries,
|
||||
0 wiz_defined-function locations,
|
||||
83 strings with 482 characters,
|
||||
and the built_in function-call counts, 0 in all, are:
|
||||
= -- 0
|
||||
> -- 0
|
||||
< -- 0
|
||||
+ -- 0
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- -- 0
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||||
* -- 0
|
||||
:= -- 0
|
||||
add.period$ -- 0
|
||||
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|
||||
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|
||||
chr.to.int$ -- 0
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
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|
||||
int.to.str$ -- 0
|
||||
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|
||||
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|
||||
num.names$ -- 0
|
||||
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|
||||
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|
||||
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|
||||
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||||
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||||
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||||
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|
||||
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||||
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|
||||
text.prefix$ -- 0
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||||
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|
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|
||||
warning$ -- 0
|
||||
while$ -- 0
|
||||
width$ -- 0
|
||||
write$ -- 0
|
||||
(There were 3 error messages)
|
||||
1873
Manuscript/fig1/fig1.log
Normal file
1873
Manuscript/fig1/fig1.log
Normal file
File diff suppressed because it is too large
Load Diff
BIN
Manuscript/fig1/fig1.pdf
Normal file
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Manuscript/fig1/fig1.pdf
Normal file
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Manuscript/fig1/fig1.synctex.gz
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Manuscript/fig1/fig1.synctex.gz
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127
Manuscript/fig1/fig1.tex
Normal file
127
Manuscript/fig1/fig1.tex
Normal file
@ -0,0 +1,127 @@
|
||||
\documentclass{standalone}
|
||||
\usepackage{adjustbox,graphicx,bm,color,microtype,hyperref,physics,amsmath,amssymb,amsfonts}
|
||||
\usepackage{tikz}
|
||||
\usetikzlibrary{decorations.pathreplacing,decorations.pathmorphing,arrows,backgrounds,fit,shapes}
|
||||
|
||||
\tikzset{snake it/.style={
|
||||
decoration={snake,
|
||||
amplitude = .4mm,
|
||||
segment length = 2mm},decorate}
|
||||
}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\begin{tikzpicture}[scale=1,
|
||||
double`1/.style={ellipse,draw,fill opacity=0.5,fill=green!10},
|
||||
double2/.style={ellipse,draw,fill opacity=0.5,fill=magenta!10},
|
||||
single/.style={ellipse,draw,fill opacity=0.5,fill=orange!10},
|
||||
]
|
||||
\scriptsize
|
||||
% Draw windows
|
||||
% Occupied
|
||||
\draw[thick,blue,fill=blue!15!white!] (0,0) -- (0,1) -- (2,1) -- (2,0) -- (0,0);%
|
||||
% Virtual
|
||||
\draw[thick,yellow,fill=yellow!15!white!] (0,1.1) -- (0,3.1) -- (2,3.1) -- (2,1.1) -- (0,1.1);%
|
||||
% CABS
|
||||
\draw[thick,red,fill=red!15!white!] (0,3.2) -- (0,6.2) -- (2,6.2) -- (2,3.2) -- (0,3.2) node[xshift=1cm,yshift=1.5cm]{\textbf{CABS}};%
|
||||
|
||||
% Draw orbitals
|
||||
% Occupied
|
||||
\foreach \y in {0.15,0.4,0.8}
|
||||
\draw[black]
|
||||
(0.8,\y) -- (1.2,\y);
|
||||
\foreach \y in {0.6}
|
||||
\draw[black]
|
||||
(0.3,\y) -- (0.7,\y)
|
||||
(1.3,\y) -- (1.7,\y);
|
||||
% Virtual
|
||||
\foreach \y in {1.4,1.8,2.3,2.5,2.8,3.0}
|
||||
\draw[black]
|
||||
(0.8,\y) -- (1.2,\y);
|
||||
\foreach \y in {1.25,1.95,2.15,2.4}
|
||||
\draw[black]
|
||||
(0.3,\y) -- (0.7,\y)
|
||||
(1.3,\y) -- (1.7,\y);
|
||||
|
||||
% Draw electrons
|
||||
\foreach \y in {0.15,0.4,0.8}
|
||||
\draw[<-,black]
|
||||
(0.95,\y-0.1) -- (0.95,\y+0.1);
|
||||
\foreach \y in {0.15,0.4,0.8}
|
||||
\draw[->,black]
|
||||
(1.05,\y-0.1) -- (1.05,\y+0.1);
|
||||
\foreach \y in {0.6}
|
||||
\draw[<-,black]
|
||||
(0.45,\y-0.1) -- (0.45,\y+0.1);
|
||||
\foreach \y in {0.6}
|
||||
\draw[->,black]
|
||||
(0.55,\y-0.1) -- (0.55,\y+0.1);
|
||||
\foreach \y in {0.6}
|
||||
\draw[<-,black]
|
||||
(1.45,\y-0.1) -- (1.45,\y+0.1);
|
||||
\foreach \y in {0.6}
|
||||
\draw[->,black]
|
||||
(1.55,\y-0.1) -- (1.55,\y+0.1);
|
||||
|
||||
% Draw curly brackets
|
||||
% occupied space
|
||||
\draw[thick,decorate,decoration={brace,amplitude=4pt}]
|
||||
(2.1,1) --
|
||||
node[xshift=0.8cm,yshift=+0.15cm] {occupied}
|
||||
node[xshift=0.85cm,yshift=-0.15cm] {space $\qty{i}$} (2.1,0);%
|
||||
% virtual space
|
||||
\draw[thick,decorate,decoration={brace,amplitude=4pt}]
|
||||
(2.1,3.1) --
|
||||
node[xshift=0.65cm,yshift=+0.15cm] {virtual}
|
||||
node[xshift=0.85cm,yshift=-0.15cm] {space $\qty{a}$} (2.1,1.1);%
|
||||
% orbital space
|
||||
\draw[thick,decorate,decoration={brace,amplitude=4pt}]
|
||||
(3.6,3.1) --
|
||||
node[xshift=1.3cm,yshift=+0.2cm] {orbital space $\qty{p}$}
|
||||
node[xshift=1.3cm,yshift=-0.2cm] {$\qty{p} = \qty{i} \cup \qty{a}$}
|
||||
(3.6,0);%
|
||||
% complementary virtual space
|
||||
\draw[thick,decorate,decoration={brace,amplitude=4pt}]
|
||||
(2.1,6.2) --
|
||||
node[xshift=1.25cm,yshift=+0.4cm] {complementary}
|
||||
node[xshift=1.4cm] {virtual space $\qty{\alpha}$}
|
||||
node[xshift=1.4cm,yshift=-0.4cm] {$\qty{p} \cap \qty{\alpha} = \varnothing$}
|
||||
(2.1,3.2);%
|
||||
% complete space
|
||||
\draw[thick,decorate,decoration={brace,amplitude=4pt}]
|
||||
(6.1,6.2) --
|
||||
node[xshift=0.85cm,yshift=+0.3cm] {complete}
|
||||
node[xshift=0.7cm] {orbital}
|
||||
node[xshift=0.65cm,yshift=-0.35cm] {space}
|
||||
(6.1,0);%
|
||||
|
||||
% Draw dotted lines
|
||||
\draw[thick,dashed] (2.1,3.1) -- (3.6,3.1);
|
||||
|
||||
\draw[thick,dashed] (2.1,6.2) -- (6.1,6.2);
|
||||
\draw[thick,dashed] (2.1,0) -- (6.1,0);
|
||||
|
||||
% Draw excitations
|
||||
|
||||
\node [double1, align=center] at (0.45,0.6) (p1) {};
|
||||
\node [double1, align=center] at (0.45,6.0) (h1) {};
|
||||
\node [double1, align=center] at (0.55,0.6) (p2) {};
|
||||
\node [double1, align=center] at (0.55,5.0) (h2) {};
|
||||
\node [double2, align=center] at (1.55,0.6) (p3) {};
|
||||
\node [double2, align=center] at (1.55,4.0) (h3) {};
|
||||
\node [double2, align=center] at (1.45,0.6) (p4) {};
|
||||
\node [double2, align=center] at (1.45,1.95)(h4) {};
|
||||
\node [single, align=center] at (1.05,0.8) (p5) {};
|
||||
\node [single, align=center] at (1.05,3.5) (h5) {};
|
||||
\path
|
||||
(p1) edge [->, green, bend left, in=170, out=10] (h1)
|
||||
(p2) edge [->, green, bend left, in=170, out=10] (h2)
|
||||
(p3) edge [->, magenta, bend right, in=190, out=-10] (h3)
|
||||
(p4) edge [->, magenta, bend right, in=190, out=-10] (h4)
|
||||
(p5) edge [->, orange, bend left, in=170, out=10] (h5)
|
||||
;
|
||||
|
||||
|
||||
\end{tikzpicture}
|
||||
|
||||
\end{document}
|
||||
Loading…
Reference in New Issue
Block a user