466 lines
24 KiB
TeX
466 lines
24 KiB
TeX
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\documentclass[aip,jcp,reprint,noshowkeys]{revtex4-1}
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%\documentclass[aip,jcp,noshowkeys]{revtex4-1}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amsmath,amssymb,amsfonts,physics,mhchem,xspace}
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\usepackage{mathpazo,libertine}
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\usepackage[normalem]{ulem}
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\newcommand{\alert}[1]{\textcolor{red}{#1}}
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\definecolor{darkgreen}{RGB}{0, 180, 0}
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\newcommand{\beurk}[1]{\textcolor{darkgreen}{#1}}
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\newcommand{\trash}[1]{\textcolor{red}{\sout{#1}}}
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\usepackage{xspace}
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\usepackage{hyperref}
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\hypersetup{
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colorlinks=true,
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linkcolor=blue,
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filecolor=blue,
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urlcolor=blue,
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citecolor=blue
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}
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\newcommand{\cdash}{\multicolumn{1}{c}{---}}
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\newcommand{\mc}{\multicolumn}
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\newcommand{\fnm}{\footnotemark}
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\newcommand{\fnt}{\footnotetext}
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\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
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\newcommand{\mr}{\multirow}
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\newcommand{\SI}{\textcolor{blue}{supporting information}}
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% second quantized operators
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\newcommand{\psix}[1]{\hat{\Psi}\left({\bf X}_{#1}\right)}
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\newcommand{\psixc}[1]{\hat{\Psi}^{\dagger}\left({\bf X}_{#1}\right)}
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\newcommand{\ai}[1]{\hat{a}_{#1}}
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\newcommand{\aic}[1]{\hat{a}^{\dagger}_{#1}}
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\newcommand{\vijkl}[0]{V_{ij}^{kl}}
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\newcommand{\phix}[2]{\phi_{#1}(\bfr{#2})}
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\newcommand{\phixprim}[2]{\phi_{#1}(\bfr{#2}')}
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\newcommand{\CBS}{\text{CBS}}
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%operators
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\newcommand{\elemm}[3]{{\ensuremath{\bra{#1}{#2}\ket{#3}}\xspace}}
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\newcommand{\ovrlp}[2]{{\ensuremath{\langle #1|#2\rangle}\xspace}}
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%\newcommand{\ket}[1]{{\ensuremath{|#1\rangle}\xspace}}
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%\newcommand{\bra}[1]{{\ensuremath{\langle #1|}\xspace}}
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%
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% energies
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\newcommand{\Ec}{E_\text{c}}
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\newcommand{\EPT}{E_\text{PT2}}
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\newcommand{\EsCI}{E_\text{sCI}}
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\newcommand{\EDMC}{E_\text{DMC}}
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\newcommand{\EexFCI}{E_\text{exFCI}}
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\newcommand{\EexFCIbasis}{E_\text{exFCI}^{\Bas}}
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\newcommand{\EexFCIinfty}{E_\text{exFCI}^{\infty}}
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\newcommand{\EexDMC}{E_\text{exDMC}}
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\newcommand{\Ead}{\Delta E_\text{ad}}
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\newcommand{\efci}[0]{E_{\text{FCI}}^{\Bas}}
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\newcommand{\emodel}[0]{E_{\model}^{\Bas}}
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\newcommand{\emodelcomplete}[0]{E_{\model}^{\infty}}
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\newcommand{\efcicomplete}[0]{E_{\text{FCI}}^{\infty}}
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\newcommand{\ecccomplete}[0]{E_{\text{CCSD(T)}}^{\infty}}
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\newcommand{\ecc}[0]{E_{\text{CCSD(T)}}^{\Bas}}
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\newcommand{\efuncbasisFCI}[0]{\bar{E}^\Bas[\denFCI]}
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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}}^{A+B}[#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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\newcommand{\ecmuapprox}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mu]}
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\newcommand{\ecmuapproxmur}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mur]}
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\newcommand{\ecmuapproxmurfci}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denfci;\,\mur]}
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\newcommand{\ecmuapproxmurmodel}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denmodel;\,\mur]}
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\newcommand{\ecompmodellda}[0]{\bar{E}_{\text{LDA}}^{\Bas,\wf{}{\Bas}}[\denmodel]}
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\newcommand{\ecompmodelldaval}[0]{\bar{E}_{\text{LDA, val}}^{\Bas,\wf{}{\Bas}}[\den]}
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\newcommand{\ecompmodelpbe}[0]{\bar{E}_{\text{PBE}}^{\Bas,\wf{}{\Bas}}[\den]}
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\newcommand{\ecompmodelpbeval}[0]{\bar{E}_{\text{PBE, val}}^{\Bas,\wf{}{\Bas}}[\den]}
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\newcommand{\emulda}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denr;\mu({\bf r};\wf{}{\Bas})\right)}
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\newcommand{\emuldamodel}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denmodelr;\mu({\bf r};\wf{}{\Bas})\right)}
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\newcommand{\emuldaval}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denval ({\bf r});\murval;\wf{}{\Bas})\right)}
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\newcommand{\ecmd}[0]{\varepsilon^{\text{c,md}}_{\text{PBE}}}
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\newcommand{\psibasis}[0]{\Psi^{\basis}}
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\newcommand{\BasFC}{\mathcal{A}}
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%pbeuegxiHF
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\newcommand{\pbeuegxihf}{\text{PBE-UEG-}\zeta\text{-HF}^\Bas}
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\newcommand{\argpbeuegxihf}[0]{\den,\zeta,s,\ntwo_{\text{UEG}},\mu_{\text{HF}}^{\basis}}
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\newcommand{\argrpbeuegxihf}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo_{\text{UEG}}(\br{}),\mu_{\text{HF}}^{\basis}(\br{})}
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%pbeuegxiCAS
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\newcommand{\pbeuegxi}{\text{PBE-UEG-}\zeta\text{-CAS}^\Bas}
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\newcommand{\argpbeuegxicas}[0]{\den,\zeta,s,\ntwo_{\text{UEG}},\mu_{\text{CAS}}^{\basis}}
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\newcommand{\argrpbeuegxicas}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo_{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
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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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%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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\newcommand{\argrpbeontxi}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
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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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%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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%%%%%% arguments
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\newcommand{\argepbe}[0]{\den,\zeta,s}
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\newcommand{\argebasis}[0]{\den,\zeta,s,\ntwo,\mu_{\Psi^{A+B}}}
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\newcommand{\argecmd}[0]{\den,\zeta,s,\ntwo,\mu}
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\newcommand{\argepbeueg}[0]{\den,\zeta,s,\ntwo_{\text{UEG}},\mu_{\Psi^{\basis}}}
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\newcommand{\argepbeontxicas}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
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\newcommand{\argepbeuegXihf}[0]{\den,\tilde{\zeta},s,\ntwo_{\text{UEG}},\mu_{\Psi^{\basis}}}
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\newcommand{\argrebasis}[0]{\denr,\zeta(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{A+B}}(\br{})}
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\newcommand{\argrebasisab}[0]{\denr,\zeta(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})}
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% numbers
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\newcommand{\rnum}[0]{{\rm I\!R}}
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\newcommand{\bfr}[1]{{\bf r}_{#1}}
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\newcommand{\dr}[1]{\text{d}\bfr{#1}}
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\newcommand{\rr}[2]{\bfr{#1}, \bfr{#2}}
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\newcommand{\rrrr}[4]{\bfr{#1}, \bfr{#2},\bfr{#3},\bfr{#4} }
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% effective interaction
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\newcommand{\twodm}[4]{\elemm{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}}
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\newcommand{\murpsibas}[0]{\mu_{\wf{}{\Bas}}({\bf r})}
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\newcommand{\murpsi}[0]{\mu({\bf r};\wf{}{A+B})}
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\newcommand{\murpsia}[0]{\mu({\bf r};\wf{}{A})}
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\newcommand{\murpsib}[0]{\mu({\bf r};\wf{}{B})}
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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)\,\basis}_{\text{CAS}}}
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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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\newcommand{\murpsival}[0]{\mu_{\text{val}}({\bf r};\wf{}{\Bas})}
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\newcommand{\murrval}[1]{\mu_{\text{val}}({\bf r}_{#1})}
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\newcommand{\weeopmu}[0]{\hat{W}_{\text{ee}}^{\text{lr},\mu}}
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\newcommand{\wbasis}[0]{W_{\wf{}{\Bas}}(\bfr{1},\bfr{2})}
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\newcommand{\wbasiscoal}[0]{W_{\wf{}{\Bas}}(\bfr{},\bfr{})}
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\newcommand{\wbasisval}[0]{W_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})}
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\newcommand{\fbasis}[0]{f_{\wf{}{\Bas}}(\bfr{1},\bfr{2})}
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\newcommand{\fbasisval}[0]{f_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})}
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\newcommand{\ontop}[2]{ n^{(2)}_{#1}({\bf #2}_1)}
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\newcommand{\twodmrpsi}[0]{ \ntwo_{\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})}
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\newcommand{\twodmrdiagpsi}[0]{ \ntwo_{\wf{}{\Bas}}(\rr{1}{2})}
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\newcommand{\twodmrdiagpsitot}[0]{ \ntwo_{\wf{}{A+B}}(\rr{1}{2})}
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\newcommand{\twodmrdiagpsiaa}[0]{ \ntwo_{\wf{}{AA}}(\rr{1}{2})}
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\newcommand{\twodmrdiagpsiaad}[0]{ \ntwo_{\wf{}{AA}}(\rr{}{})}
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\newcommand{\twodmrdiagpsibb}[0]{ \ntwo_{\wf{}{BB}}(\rr{1}{2})}
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\newcommand{\twodmrdiagpsibbd}[0]{ \ntwo_{\wf{}{BB}}(\rr{}{})}
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\newcommand{\twodmrdiagpsiab}[0]{ \ntwo_{\wf{}{AB}}(\rr{1}{2})}
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\newcommand{\twodmrdiagpsival}[0]{ \ntwo_{\wf{}{\Bas},\,\text{val}}(\rr{1}{2})}
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\newcommand{\gammamnpq}[1]{\Gamma_{mn}^{pq}[#1]}
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\newcommand{\gammamnkl}[0]{\Gamma_{mn}^{kl}}
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\newcommand{\gammaklmn}[1]{\Gamma_{kl}^{mn}[#1]}
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%\newcommand{\wbasiscoal}[1]{W_{\wf{}{\Bas}}({\bf r}_{#1})}
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\newcommand{\ontoppsi}[1]{ n^{(2)}_{\wf{}{\Bas}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})}
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\newcommand{\wbasiscoalval}[1]{W_{\wf{}{\Bas}}^{\text{val}}({\bf r}_{#1})}
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\newcommand{\ontoppsival}[1]{ n^{(2)}_{\wf{}{\Bas}}^{\text{val}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})}
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\newcommand{\ex}[4]{$^{#1}#2_{#3}^{#4}$}
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\newcommand{\ra}{\rightarrow}
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\newcommand{\De}{D_\text{e}}
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% MODEL
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\newcommand{\model}[0]{\mathcal{Y}}
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% densities
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\newcommand{\denmodel}[0]{\den_{\model}^\Bas}
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\newcommand{\denmodelr}[0]{\den_{\model}^\Bas ({\bf r})}
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\newcommand{\denfci}[0]{\den_{\psifci}}
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\newcommand{\denFCI}[0]{\den^{\Bas}_{\text{FCI}}}
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\newcommand{\denhf}[0]{\den_{\text{HF}}^\Bas}
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\newcommand{\denrfci}[0]{\denr_{\psifci}}
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\newcommand{\dencipsir}[0]{{n}_{\text{CIPSI}}^\Bas({\bf r})}
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\newcommand{\dencipsi}[0]{{n}_{\text{CIPSI}}^\Bas}
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\newcommand{\den}[0]{{n}}
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\newcommand{\denval}[0]{{n}^{\text{val}}}
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\newcommand{\denr}[0]{{n}({\bf r})}
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\newcommand{\onedmval}[0]{\rho_{ij,\sigma}^{\text{val}}}
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% wave functions
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\newcommand{\psifci}[0]{\Psi^{\Bas}_{\text{FCI}}}
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\newcommand{\psimu}[0]{\Psi^{\mu}}
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% operators
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\newcommand{\weeopbasis}[0]{\hat{W}_{\text{ee}}^\Bas}
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\newcommand{\kinop}[0]{\hat{T}}
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\newcommand{\weeopbasisval}[0]{\hat{W}_{\text{ee}}^{\Basval}}
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\newcommand{\weeop}[0]{\hat{W}_{\text{ee}}}
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% units
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\newcommand{\IneV}[1]{#1 eV}
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\newcommand{\InAU}[1]{#1 a.u.}
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\newcommand{\InAA}[1]{#1 \AA}
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% methods
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\newcommand{\UEG}{\text{UEG}}
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\newcommand{\LDA}{\text{LDA}}
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\newcommand{\PBE}{\text{PBE}}
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\newcommand{\FCI}{\text{FCI}}
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\newcommand{\CCSDT}{\text{CCSD(T)}}
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\newcommand{\lr}{\text{lr}}
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\newcommand{\sr}{\text{sr}}
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\newcommand{\Nel}{N}
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\newcommand{\V}[2]{V_{#1}^{#2}}
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\newcommand{\n}[2]{n_{#1}^{#2}}
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\newcommand{\E}[2]{E_{#1}^{#2}}
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\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}}
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\newcommand{\bEc}[1]{\Bar{E}_\text{c}^{#1}}
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\newcommand{\e}[2]{\varepsilon_{#1}^{#2}}
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\newcommand{\be}[2]{\Bar{\varepsilon}_{#1}^{#2}}
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\newcommand{\bec}[1]{\Bar{e}^{#1}}
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\newcommand{\wf}[2]{\Psi_{#1}^{#2}}
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\newcommand{\W}[2]{W_{#1}^{#2}}
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\newcommand{\w}[2]{w_{#1}^{#2}}
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\newcommand{\hn}[2]{\Hat{n}_{#1}^{#2}}
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\newcommand{\rsmu}[2]{\mu_{#1}^{#2}}
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\newcommand{\SO}[2]{\phi_{#1}(\br{#2})}
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\newcommand{\modX}{\text{X}}
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\newcommand{\modY}{\text{Y}}
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% basis sets
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\newcommand{\setdenbasis}{\mathcal{N}_{\Bas}}
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\newcommand{\Bas}{\mathcal{B}}
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\newcommand{\basis}{\mathcal{B}}
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\newcommand{\Basval}{\mathcal{B}_\text{val}}
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\newcommand{\Val}{\mathcal{V}}
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\newcommand{\Cor}{\mathcal{C}}
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% operators
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\newcommand{\hT}{\Hat{T}}
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\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}}
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\newcommand{\f}[2]{f_{#1}^{#2}}
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\newcommand{\Gam}[2]{\Gamma_{#1}^{#2}}
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% coordinates
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\newcommand{\br}[1]{{\mathbf{r}_{#1}}}
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\newcommand{\bx}[1]{\mathbf{x}_{#1}}
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\newcommand{\dbr}[1]{d\br{#1}}
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\newcommand{\PBEspin}{PBEspin}
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\newcommand{\PBEueg}{PBE-UEG-{$\tilde{\zeta}$}}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, France}
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\begin{document}
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\title{A density-based basis-set correction for weak and strong correlation}
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\begin{abstract}
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\end{abstract}
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\maketitle
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\section{Extensivity of the basis set correction}
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\subsection{General considerations}
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The following paragraph proposes a demonstration of the size consistency of the basis set correction in the limit of dissociated fragments.
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The present basis set correction being an integral in real space,
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\begin{equation}
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\label{eq:def_ecmdpbebasis}
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\begin{aligned}
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& \efuncdenpbe{\argebasis} = \\ & \int \text{d}\br{} \,\denr \ecmd(\argrebasis),
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\end{aligned}
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\end{equation}
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where all the quantities $\argrebasis$ are obtained from the same wave function $\Psi^{A+B}$.
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Such an integral can be rewritten as the sum of the contribution on $A$ and $B$
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\begin{equation}
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\label{eq:def_ecmdpbebasis}
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\begin{aligned}
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& \efuncdenpbe{\argebasis} = \\ & \int_{ \br{} \in A} \text{d}\br{} \,\denr \ecmd(\argrebasis) \\ & + \int_{ \br{} \in B} \text{d}\br{} \,\denr \ecmd(\argrebasis),
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\end{aligned}
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\end{equation}
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Therefore, a sufficient condition to obtain size extensivity in the limit of dissociated fragments is that all arguments entering in the function $\ecmd(\argrebasis)$ are \textit{intensive}, which means that they \textit{locally} coincide in the system $A$ and in the sub system $A$ of the super system $A+B$.
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Regarding the density and its gradients, these are necessary intensive quantities. The remaining questions are therefore the local range-separation parameter $\murpsi$ and the on-top pair density.
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\subsection{Property of the on-top pair density}
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A crucial ingredient in the type of functionals used in the present paper together with the definition of the local-range separation parameter is the on-top pair density defined as
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\begin{equation}
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\label{eq:def_n2}
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\ntwo_{\wf{}{}}(\br{}) = \sum_{pqrs} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{},
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\end{equation}
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with $\Gam{pq}{rs} = 2 \mel*{\wf{}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{}}$.
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Assume now that the wave function $\wf{A+B}{}$ of the super system $A+B$ can be written as a product of two wave functions defined on two non-overlapping and non-interacting fragments $A$ and $B$
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\begin{equation}
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\ket{\wf{A+B}{}} = \ket{\wf{A}{}} \times \ket{\wf{B}{}}.
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\end{equation}
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Labelling the orbitals of fragment $A$ as $p_A,q_A,r_A,s_A$ and of fragment $B$ as $p_B,q_B,r_B,s_B$ and assuming that they don't overlap, one can split the two-body density operator as
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\begin{equation}
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\begin{aligned}
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\hat{\Gamma}(\br{1},\br{2}) = \hat{\Gamma}_{AA}{}(\br{1},\br{2}) + \hat{\Gamma}_{BB}{}(\br{1},\br{2}) + \hat{\Gamma}_{AB}{}(\br{1},\br{2})
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\end{aligned}
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\end{equation}
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with
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\begin{equation}
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\begin{aligned}
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\hat{\Gamma}_{AA}(\br{1},\br{2}) = \sum_{p_A,q_A,r_A,s_A}& \SO{r_A}{1} \SO{s_A}{2} \SO{p_A}{1} \SO{q_A}{2} \\ & \aic{r_{A,\downarrow}}\aic{s_{A,\uparrow}}\ai{q_{A,\uparrow}}\ai{p_{A,\downarrow}} ,
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\end{aligned}
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\end{equation}
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(and equivalently for $B$),
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%\begin{equation}
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% \begin{aligned}
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% \hat{\Gamma}_{BB} = \sum_{p_B,q_B,r_B,s_B} \aic{r_{B,\downarrow}}\aic{s_{B,\uparrow}}\ai{q_{B,\uparrow}}\ai{p_{B,\downarrow}},
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% \end{aligned}
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%\end{equation}
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and
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\begin{equation}
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\begin{aligned}
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\hat{\Gamma}_{AB} = \sum_{p_A,q_B,r_A,s_B} & \SO{r_A}{1} \SO{s_B}{2} \SO{p_A}{1} \SO{q_B}{2} \\ & \left( \aic{r_{A,\downarrow}}\ai{p_{A,\downarrow}} \aic{s_{B,\uparrow}}\ai{q_{B,\uparrow}} + \aic{s_{B,\uparrow}}\ai{q_{B,\uparrow}} \aic{r_{A,\downarrow}}\ai{p_{A,\downarrow}} \right) .
|
||
\end{aligned}
|
||
\end{equation}
|
||
Therefore, one can express the two-body density as
|
||
\begin{equation}
|
||
\twodmrdiagpsitot = \twodmrdiagpsiaa + \twodmrdiagpsibb + \twodmrdiagpsiab
|
||
\end{equation}
|
||
where $\twodmrdiagpsiaa$ and $\twodmrdiagpsibb$ are the two-body densities of the isolated fragments
|
||
\begin{equation}
|
||
\begin{aligned}
|
||
& \twodmrdiagpsiaa = \bra{\wf{A}{}} \hat{\Gamma}_{AA}(\br{1},\br{2}) \ket{\wf{A}{}}
|
||
\end{aligned}
|
||
\end{equation}
|
||
(and equivalently for $B$),
|
||
and $\twodmrdiagpsiab$ is simply the product of the one body densities of the sub systems
|
||
\begin{equation}
|
||
\begin{aligned}
|
||
& \twodmrdiagpsiab = n_{A}(\br{1}) n_B(\br{2}) + n_{B}(\br{1}) n_A(\br{2}),
|
||
\end{aligned}
|
||
\end{equation}
|
||
\begin{equation}
|
||
\begin{aligned}
|
||
& n_{A}(\br{}) = \sum_{p_A r_A} \SO{p_A}{} \bra{\wf{A}{}}\aic{s_{A,\uparrow}}\ai{q_{A,\uparrow}}\ket{\wf{A}{}} \SO{r_A}{} ,
|
||
\end{aligned}
|
||
\end{equation}
|
||
(and equivalently for $B$).
|
||
As the densities of $A$ and $B$ are by definition non overlapping, one can express the on-top pair density as the sum of the on-top pair densities of the isolated systems
|
||
\begin{equation}
|
||
\begin{aligned}
|
||
\ntwo_{\wf{A+B}{}}(\br{}) = \twodmrdiagpsiaad + \twodmrdiagpsibbd
|
||
\end{aligned}
|
||
\end{equation}
|
||
As $\ntwo_{\wf{}{A/A}}(\br{}) = 0 \text{ if }\br{} \in B$ (and equivalently for $\ntwo_{\wf{}{B/B}}(\br{}) $ on $A$), one can conclude that provided that the wave function is multiplicative, the on-top pair density is a local intensive quantity.
|
||
\subsection{Property of the local-range separation parameter}
|
||
The local range separation parameter depends on the on-top pair density at a given point $\br{}$ and on the numerator
|
||
\begin{equation}
|
||
\label{eq:def_f}
|
||
f_{\wf{}{}}(\bfr{},\bfr{}) = \sum_{pqrstu\in \Bas} \SO{p}{ } \SO{q}{ } \V{pq}{rs} \Gam{rs}{tu} \SO{t}{ } \SO{u}{ }.
|
||
\end{equation}
|
||
As the summations run over all orbitals in the basis set $\Bas$, the quantity $f_{\wf{}{\Bas}}(\bfr{},\bfr{})$ is orbital invariant and therefore can be expressed in terms of localized orbitals.
|
||
In the limit of dissociated fragments, the coulomb interaction is vanishing between $A$ and $B$ and therefore any two-electron integral involving orbitals on both the system $A$ and $B$ vanishes.
|
||
Therefore, one can rewrite eq. \eqref{eq:def_f} as
|
||
\begin{equation}
|
||
\label{eq:def_fa+b}
|
||
f_{\wf{A+B}{}}(\bfr{},\bfr{}) = f_{\wf{AA}{}}(\bfr{},\bfr{}) + f_{\wf{BB}{}}(\bfr{},\bfr{}),
|
||
\end{equation}
|
||
with
|
||
\begin{equation}
|
||
\begin{aligned}
|
||
\label{eq:def_faa}
|
||
& f_{\wf{AA}{}}(\bfr{},\bfr{}) = \\ & \sum_{p_A q_A r_A s_A t_A u_A} \SO{p_A }{ } \SO{q_A}{ } \V{p_A q_A}{r_A s_A} \Gam{r_A s_A}{t_A u_A} \SO{t_A}{ } \SO{u_A}{ },
|
||
\end{aligned}
|
||
\end{equation}
|
||
(and equivalently for $B$).
|
||
%\begin{equation}
|
||
% \begin{aligned}
|
||
% \label{eq:def_faa}
|
||
% & f_{\wf{BB}{}}(\bfr{},\bfr{}) = \\ &\sum_{p_B q_B r_B s_B t_B u_B} \SO{p_B }{ } \SO{q_B}{ } \V{p_B q_B}{r_B s_B} \Gam{r_B s_B}{t_B u_B} \SO{t_B}{ } \SO{u_B}{ }.
|
||
% \end{aligned}
|
||
%\end{equation}
|
||
As a consequence, the local range-separation parameter in the super system $A+B$
|
||
\begin{equation}
|
||
\label{eq:def_mur}
|
||
\murpsi = \frac{\sqrt{\pi}}{2} \frac{f_{\wf{A+B}{}}(\bfr{},\bfr{})}{\ntwo_{\wf{A+B}{}}(\br{})}
|
||
\end{equation}
|
||
which, in the case of a multiplicative wave function is nothing but
|
||
\begin{equation}
|
||
\label{eq:def_mur}
|
||
\murpsi = \murpsia + \murpsib.
|
||
\end{equation}
|
||
As $\murpsia = 0 \text{ if }\br{} \in B$ (and equivalently for $\murpsib $ on $B$), $\murpsi$ is an intensive quantity. The conclusion of this paragraph is that, provided that the wave function for the system $A+B$ is multiplicative in the limit of the dissociated fragments, all quantities used for the basis set correction are intensive and therefore the basis set correction is size consistent.
|
||
|
||
\section{Computational considerations}
|
||
The computational cost of the present approach is driven by two quantities: the computation of the on-top pair density and the $\murpsibas$ on the real-space grid. Within a blind approach, for each grid point the computational cost is of order $n_{\Bas}^4$ and $n_{\Bas}^6$ for the on-top pair density $\ntwo_{\wf{\Bas}{}}(\br{})$ and the local range separation parameter $\murpsibas$, respectively.
|
||
Nevertheless, using CASSCF wave functions to compute these quantities leads to significant simplifications which can substantially reduce the CPU time.
|
||
\subsection{Computation of the on-top pair density for a CASSCF wave function}
|
||
Given a generic wave function developed on a basis set with $n_{\Bas}$ basis functions, the evaluation of the on-top pair density is of order $\left(n_{\Bas}\right)^4$.
|
||
Nevertheless, assuming that the wave function $\Psi^{\Bas}$ is of CASSCF type, a lot of simplifications happen.
|
||
If the active space is referred as the set of spatial orbitals $\mathcal{A}$ which are labelled by the indices $t,u,v,w$, and the doubly occupied orbitals are the set of spatial orbitals $\mathcal{C}$ labeled by the indices $i,j$, one can write the on-top pair density of a CASSCF wave function as
|
||
\begin{equation}
|
||
\label{def_n2_good}
|
||
\ntwo_{\wf{\Bas}{}}(\br{}) = \ntwo_{\mathcal{A}}(\br{}) + n_{\mathcal{C}}(\br{}) n_{\mathcal{A}}(\br{}) + \left( n_{\mathcal{C}}(\br{})\right)^2
|
||
\end{equation}
|
||
where
|
||
\begin{equation}
|
||
\label{def_n2_act}
|
||
\ntwo_{\mathcal{A}}(\br{}) = \sum_{t,u,v,w \, \in \mathcal{A}} 2 \mel*{\wf{}{\Bas}}{ \aic{t_\downarrow}\aic{u_\uparrow}\ai{v_\uparrow}\ai{w_\downarrow}}{\wf{}{\Bas}} \phi_t (\br{}) \phi_u (\br{}) \phi_v (\br{}) \phi_w (\br{})
|
||
\end{equation}
|
||
is the purely active part of the on-top pair density,
|
||
\begin{equation}
|
||
n_{\mathcal{C}}(\br{}) = \sum_{i\, \in \mathcal{C}} \left(\phi_i (\br{}) \right)^2,
|
||
\end{equation}
|
||
and
|
||
\begin{equation}
|
||
n_{\mathcal{A}}(\br{}) = \sum_{t,u\, \in \mathcal{A}} \phi_t (\br{}) \phi_u (\br{})
|
||
\mel*{\wf{}{\Bas}}{ \aic{t_\downarrow}\ai{u_\downarrow} + \aic{t_\uparrow}\ai{u_\uparrow}}{\wf{}{\Bas}}
|
||
\end{equation}
|
||
is the purely active one-body density.
|
||
Written as in eq. \eqref{def_n2_good}, the leading computational cost is the evaluation of $\ntwo_{\mathcal{A}}(\br{})$ which, according to eq. \eqref{def_n2_act}, scales as $\left( n_{\mathcal{A}}\right) ^4$ where $n_{\mathcal{A}}$ is the number of active orbitals which is much smaller than the number of basis functions $n_{\Bas}$. Therefore, the final computational scaling of the on-top pair density for a CASSCF wave function over the whole real-space grid is of $\left( n_{\mathcal{A}}\right) ^4 n_G$, where $n_G$ is the number of grid points.
|
||
\subsection{Computation of $\murpsibas$}
|
||
At a given grid point, the computation of $\murpsibas$ needs the computation of $f_{\wf{}{}}(\bfr{},\bfr{}) $ defined in eq. \eqref{eq:def_f} and the on-top pair density defined in eq. \eqref{eq:def_n2}. In the previous paragraph we gave an explicit form of the on-top pair density in the case of a CASSCF wave function with a computational scaling of $\left( n_{\mathcal{A}}\right)^4$. In the present paragraph we focus on simplifications that one can obtain for the computation of $f_{\wf{}{}}(\bfr{},\bfr{}) $ in the case of a CASSCF wave function.
|
||
|
||
One can rewrite $f_{\wf{}{}}(\bfr{},\bfr{}) $ as
|
||
\begin{equation}
|
||
\label{eq:f_good}
|
||
f_{\wf{}{}}(\bfr{},\bfr{}) = \sum_{r,s \in \Bas} \mathcal{V}_r^s(\bfr{}) \, \mathcal{N}_{r}^s(\bfr{})
|
||
\end{equation}
|
||
where
|
||
\begin{equation}
|
||
\mathcal{V}_r^s(\bfr{}) = \sum_{p,q \in \Bas} V_{pq}^{rs} \phi_p(\br{}) \phi_q(\br{})
|
||
\end{equation}
|
||
and
|
||
\begin{equation}
|
||
\mathcal{N}_{r}^s(\bfr{}) = \sum_{p,q \in \Bas} \Gam{pq}{rs} \phi_p(\br{}) \phi_q(\br{}) .
|
||
\end{equation}
|
||
\textit{A priori}, for a given grid point, the computational scaling of eq. \eqref{eq:f_good} is of $\left(n_{\Bas}\right)^4$ and the total computational cost over the whole grid scales therefore as $\left(n_{\Bas}\right)^4 n_G$.
|
||
|
||
In the case of a CASSCF wave function, it is interesting to notice that $\Gam{pq}{rs}$ vanishes if one index $p,q,r,s$ does not belong
|
||
to the set of of doubly occupied or active orbitals $\mathcal{C}\cup \mathcal{A}$. Therefore, at a given grid point, the matrix $\mathcal{N}_{r}^s(\bfr{})$ has only at most $\left(n_{\mathcal{A}}+n_{\mathcal{C}}\right)^2$ non-zero elements, which is usually much smaller than $\left(n_{\Bas}\right)^2$.
|
||
As a consequence, in the case of a CASSCF wave function one can rewrite $f_{\wf{}{}}(\bfr{},\bfr{})$ as
|
||
\begin{equation}
|
||
f_{\wf{}{}}(\bfr{},\bfr{}) = \sum_{r,s \in \mathcal{C}\cup\mathcal{A}} \mathcal{V}_r^s(\bfr{}) \, \mathcal{N}_{r}^s(\bfr{}).
|
||
\end{equation}
|
||
Therefore the final computational cost of $f_{\wf{}{}}(\bfr{},\bfr{})$ is dominated by that of $\mathcal{V}_r^s(\bfr{})$, which scales as $\left(n_{\mathcal{A}}+n_{\mathcal{C}}\right)^2 \left( n_{\Bas} \right)^2 n_G$, which is much weaker than $\left(n_{\Bas}\right)^4 n_G$.
|
||
\bibliography{../srDFT_SC}
|
||
|
||
\end{document}
|