452 lines
23 KiB
TeX
452 lines
23 KiB
TeX
\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,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable}
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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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%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{\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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% 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{\murpsi}[0]{\mu({\bf r};\wf{}{\Bas})}
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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]{ n^{2}_{\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})}
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\newcommand{\twodmrdiagpsi}[0]{ n^{2}_{\wf{}{\Bas}}(\rr{1}{2})}
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\newcommand{\twodmrdiagpsival}[0]{ n^{2}_{\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{\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{\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{Mixing density functional theory and wave function theory for strong correlation: the best of both worlds}
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\begin{abstract}
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bla bla bla youpi tralala
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\end{abstract}
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\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}
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%%%%%%%%%%%%%%%%%%%%%%%%
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The main goal of quantum chemistry is to propose reliable theoretical tools to describe the rich area of chemistry.
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The accurate computation of the electronic structure of molecular systems plays a central role in the development of methods in quantum chemistry,
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but despite intense developments, no definitive solution to that problem have been found.
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The theoretical challenge to be overcome falls back in the category of the quantum many-body problem due the intrinsic quantum nature
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of the electrons and the coulomb repulsion between them, inducing the so-called electronic correlation problem.
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Tackling this problem translate to solving the Schroedinger equation for a $N$~-~electron system, and two roads have emerged to approximate the solution to this formidably complex mathematical problem: the wave function theory (WFT) and density functional theory (DFT).
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Although both WFT and DFT spring from the same problem, their formalisms are very different as the former deals with the complex
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$N$~-~body wave function whereas the latter handles the much simpler one~-~body density.
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The computational cost of DFT is very appealing as in its Kohn-Sham (KS) formulation it can be recast in a mean-field procedure.
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Therefore, although constant efforts are performed to reduce the computational cost of WFT, DFT remains still the workhorse of quantum chemistry.
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From the theoretician point of view, the complexity of description of a given chemical system can be roughly
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categorized by the strength of the electronic correlation appearing in its electronic structure.
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Weakly correlated systems, such as closed-shell organic molecules near their equilibrium geometry, are typically dominated by the avoidance effects when electron are near the electron coalescence point, which are often called short-range correlation effects,
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or far, typically dispersion forces. The theoretical description of weakly correlated systems is one of the more concrete achievement
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of quantum chemistry, and the main remaining issue for these systems is to push the limit in terms of the size of the chemical systems that can be treated.
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The case of the so-called strongly correlated systems, which are ubiquitous in chemistry, is much more problematic as they exhibits
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a much more exotic electronic structure.
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Transition metals containing systems, low-spin open shell systems, covalent bond breaking or excited states
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have all in common that they cannot be even qualitatively described by a single electronic configuration.
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It is now clear that the usual approximations in KS-DFT fails in giving an accurate description of these situations and WFT has become
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the standard for the treatment of strongly correlated systems.
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From the theoretical point of view, the complexity of the strong correlation problem is, at least, two-fold:
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i) the presence of near degeneracies and/or strong interactions among a primary set of electronic configurations
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(the size of which can potentially scale exponentially in some cases) determines the qualitative description of the wave function,
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ii) the quantitative description of the systems must take into account weak correlation effects which requires to take into account many
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other electronic configurations with typically much smaller weights in the wave function.
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Fulfilling these two objectives is a rather complicated task, specially if one adds the requirement of size-extensivity and additivity of the computed energy in the case of non interacting fragments, which is a very desirable property for any approximated method.
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To tackle this problem, many WFT methods have emerged which can be categorized in two branches: the single-reference (SR)
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and multi-reference (MR) methods.
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The SR methods rely on a single electronic configuration as a zeroth-order wave function, typically Hartree-Fock (HF).
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Then the electron correlation is introduced by increasing the rank of multiple hole-particle excitations,
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preferably treated in a coupled-cluster fashion for the sake of compactness of the wave function and extensivity of the computed energies.
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The advantage of these approaches rely on the rather straightforward way to improve the level of accuracy,
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which consists in increasing the rank of the excitation operators used to generate the CC wave function.
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Despite its appealing elegant simplicity, the computational cost of the CC methods increase drastically with the rank of the excitation
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operators, even if alternative approaches have been proposed using stochastic techniques\cite{alex_thom,piotr} or symmetry-broken approaches\cite{scuseria}.
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In the MR approaches, the zeroth order wave function consists in a linear combination of Slater determinants which are supposed to concentrate most of strong interactions and near degeneracies.
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On top of this zeroth-order wave function, weak correlation is introduced by the addition of other configurations
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A sensible advantage of WFT is its systematically improvable character to tend to the exact solution, which is the so-called full configuration interaction (FCI) in a complete basis set (CBS).
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Such a path can be expressed in two ways which are quite independent one from another: i) improving the description of the wave function in terms of multiple excitations expansion ii) improving the quality of the one-particle basis set.
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When the molecular system
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Theory}
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%%%%%%%%%%%%%%%%%%%%%%%%
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The theoretical framework of the basis set correction have been derived in details in \cite{GinPraFerAssSavTou-JCP-18}, so we recall briefly the main equations involved for the present study.
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\subsection{Basic formal equations}
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The exact ground state energy $E_0$ of a $N-$electron system can be obtained by the Levy-Lieb constrained search formalism which is an elegant mathematical framework connecting WFT and DFT
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\begin{equation}
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\label{eq:levy}
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E_0 = \min_{\denr} \bigg\{ F[\denr] + (v_{\text{ne}} (\br{}) |\denr) \bigg\},
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\end{equation}
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where $(v_{ne}(\br{})|\denr)$ is the nuclei-electron interaction for a given density $\denr$ and $F[\denr]$ is the so-called Levy-Liev universal density functional
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\begin{equation}
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\label{eq:levy_func}
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F[\denr] = \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop +\weeop }{\Psi}.
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\end{equation}
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The minimizing density $n_0$ of equation \eqref{eq:levy} is the exact ground state density.
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As in practical calculations the minimization is performed over the set $\setdenbasis$ which are the densities representable in a basis set $\Bas$, we assume from thereon that the densities used in the equations belong to $\setdenbasis$.
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Following equation (7) of \cite{GinPraFerAssSavTou-JCP-18}, we split $F[\denr]$ as
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\begin{equation}
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F[\denr] = \min_{\wf{}{\Bas} \rightarrow \denr} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\denr}
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\end{equation}
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where $\wf{}{\Bas}$ refer to $N-$electron wave functions expanded in $\Bas$, and
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where $\efuncden{\denr}$ is the density functional complementary to the basis set $\Bas$ defined as
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\begin{equation}
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\begin{aligned}
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\efuncden{\denr} =& \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop +\weeop }{\Psi} \\
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&- \min_{\Psi^{\Bas} \rightarrow \denr} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}}.
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\end{aligned}
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\end{equation}
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The functional $\efuncden{\denr}$ must therefore recover all physical effects not included in the basis set $\Bas$.
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Assuming that the FCI density $\denFCI$ in $\Bas$ is a good approximation of the exact density, one obtains the following approximation for the exact ground state density (see equations 12-15 of \cite{GinPraFerAssSavTou-JCP-18})
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\begin{equation}
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\label{eq:e0approx}
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E_0 = \efci + \efuncbasisFCI
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\end{equation}
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where $\efci$ is the ground state FCI energy within $\Bas$. As it was originally shown in \cite{GinPraFerAssSavTou-JCP-18} and further emphasized in \cite{G2,excited}, the main role of $\efuncbasisFCI$ is to correct for the basis set incompleteness errors, a large part of which originates from the lack of cusp in any wave function developed in an incomplete basis set.
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The whole purpose of this paper is to determine approximations for $\efuncbasisFCI$ which are suited for treating strong correlation regimes. The two requirement for such conditions are that i) it can be defined for multi-reference wave functions, ii) it must provide size extensive energies, iii) it is invariant of the $S_z$ component of a given spin multiplicity.
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\subsection{Definition of an effective interaction within $\Bas$}
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As it was originally shown by Kato\cite{kato}, the cusp in the exact wave function originates from the divergence of the coulomb interaction at the coalescence point. Therefore, the lack of cusp in any wave function $\wf{}{\Bas}$ could also originate from an effective non-divergent electron-electron interaction. In other words, the incompleteness of a finite basis set can be understood as the removal of the divergence at the electron coalescence point.
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As it was originally derived in \cite{GinPraFerAssSavTou-JCP-18} (see section D and annexes), one can obtain an effective non divergent interaction, here referred as $\wbasis$, which reproduces the expectation value of the coulomb operator over a given wave function $\wf{}{\Bas}$. As we are interested in the behaviour at the coalescence point, we focus on the opposite spin part of the electron-electron interaction.
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More specifically, we define the effective interaction associated to a given wave function $\wf{}{\Bas}$ as
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\begin{equation}
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\label{eq:wbasis}
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\wbasis =
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\begin{cases}
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\fbasis /\twodmrdiagpsi, & \text{if $\twodmrdiagpsi \ne 0$,}
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\\
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\infty, & \text{otherwise,}
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\end{cases}
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\end{equation}
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where $\twodmrdiagpsi$ is the opposite spin two-body density associated to $\wf{}{\Bas}$
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\begin{equation}
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\twodmrdiagpsi = \sum_{pqrs} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
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\end{equation}
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$\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$ its associated two-body tensor, $\SO{p}{}$ are the spatial orthonormal orbitals,
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\begin{equation}
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\label{eq:fbasis}
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\fbasis
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= \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
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\end{equation}
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and $\V{pq}{rs}=\langle pq | rs \rangle$ are the usual two-electron Coulomb integrals.
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With such a definition, one can show that $\wbasis$ satisfies
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\begin{equation}
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\int \int \dr{1} \dr{2} \wbasis \twodmrdiagpsi = \int \int \dr{1} \dr{2} \frac{\twodmrdiagpsi}{|\br{1}-\br{2}|}.
|
||
\end{equation}
|
||
As it was shown in \cite{GinPraFerAssSavTou-JCP-18}, the effective interaction $\wbasis$ is necessary finite at coalescence for an incomplete basis set, and tends to the regular coulomb interaction in the limit of a complete basis set, that is
|
||
\begin{equation}
|
||
\label{eq:cbs_wbasis}
|
||
\lim_{\Bas \rightarrow \text{CBS}} \wbasis = \frac{1}{|\br{1}-\br{2}|}.
|
||
\end{equation}
|
||
The condition of equation \eqref{eq:cbs_wbasis} is fundamental as it guarantees the good behaviour of all the theory in the limit of a complete basis set.
|
||
\subsection{Definition of an range-separation parameter varying in real space}
|
||
As the effective interaction within a basis set $\wbasis$ is non divergent, one can fit such a function with a long-range interaction defined in the framework of RSDFT which depends on the range-separation parameter $\mu$
|
||
\begin{equation}
|
||
\label{eq:weelr}
|
||
w_{ee}^{\lr}(\mu;\br{1},\br{2}) = \frac{\text{erf}\big(\mu \,|\br{1}-\br{2}| \big)}{|\br{1}-\br{2}|}.
|
||
\end{equation}
|
||
As originally proposed in \cite{GinPraFerAssSavTou-JCP-18}, we introduce a range-separation parameter $\murpsi$ varying in real space
|
||
\begin{equation}
|
||
\murpsi = \frac{\sqrt{\pi}}{2} \wbasiscoal
|
||
\end{equation}
|
||
such that
|
||
\begin{equation}
|
||
w_{ee}^{\lr}(\murpsi;\br{ },\br{ }) = \wbasiscoal.
|
||
\end{equation}
|
||
Because of the very definition of $\wbasis$, one has the following properties at the CBS limit (see \eqref{eq:cbs_wbasis})
|
||
\begin{equation}
|
||
\label{eq:cbs_mu}
|
||
\lim_{\Bas \rightarrow \text{CBS}} \murpsi = \infty,
|
||
\end{equation}
|
||
which is fundamental to guarantee the good behaviour of the theory at the CBS limit.
|
||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||
\section{Results}
|
||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||
\begin{figure}
|
||
\includegraphics[width=\linewidth]{data/N2/DFT_avdzE_relat.eps}
|
||
\includegraphics[width=\linewidth]{data/N2/DFT_avdzE_relat_zoom.eps}
|
||
\includegraphics[width=\linewidth]{data/N2/DFT_avdzE_error.eps}
|
||
\caption{
|
||
N$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one.
|
||
\label{fig:N2_avdz}}
|
||
\end{figure}
|
||
|
||
\begin{figure}
|
||
\includegraphics[width=\linewidth]{data/N2/DFT_avtzE_relat.eps}
|
||
\includegraphics[width=\linewidth]{data/N2/DFT_avtzE_relat_zoom.eps}\\
|
||
\includegraphics[width=\linewidth]{data/N2/DFT_avtzE_error.eps}\\
|
||
% \includegraphics[width=\linewidth]{fig2c}
|
||
\caption{
|
||
N$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one.
|
||
\label{fig:N2_avtz}}
|
||
\end{figure}
|
||
|
||
\begin{figure}
|
||
\includegraphics[width=\linewidth]{data/F2/DFT_avdzE_relat.eps}
|
||
\includegraphics[width=\linewidth]{data/F2/DFT_avdzE_relat_zoom.eps}\\
|
||
\includegraphics[width=\linewidth]{data/F2/DFT_avdzE_error.eps}\\
|
||
% \includegraphics[width=\linewidth]{fig2c}
|
||
\caption{
|
||
F$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one.
|
||
\label{fig:F2_avdz}}
|
||
\end{figure}
|
||
|
||
\begin{figure}
|
||
\includegraphics[width=\linewidth]{data/F2/DFT_avtzE_relat.eps}
|
||
\includegraphics[width=\linewidth]{data/F2/DFT_avtzE_relat_zoom.eps}\\
|
||
\includegraphics[width=\linewidth]{data/F2/DFT_avtzE_error.eps}\\
|
||
% \includegraphics[width=\linewidth]{fig2c}
|
||
\caption{
|
||
F$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one.
|
||
\label{fig:F2_avtz}}
|
||
\end{figure}
|
||
|
||
|
||
\begin{figure}
|
||
% \includegraphics[width=\linewidth]{data/H10/DFT_avdzE_relat.eps}
|
||
\includegraphics[width=\linewidth]{data/H10/DFT_vdzE_relat_zoom.eps}\\
|
||
\includegraphics[width=\linewidth]{data/H10/DFT_vdzE_error.eps}\\
|
||
% \includegraphics[width=\linewidth]{fig2c}
|
||
\caption{
|
||
H$_{10}$, cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one.
|
||
\label{fig:H10_vdz}}
|
||
\end{figure}
|
||
|
||
|
||
\begin{figure}
|
||
\includegraphics[width=\linewidth]{data/H10/DFT_vtzE_relat_zoom.eps}\\
|
||
\includegraphics[width=\linewidth]{data/H10/DFT_vtzE_error.eps}\\
|
||
% \includegraphics[width=\linewidth]{fig2c}
|
||
\caption{
|
||
H$_{10}$, cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one.
|
||
\label{fig:H10_vtz}}
|
||
\end{figure}
|
||
|
||
|
||
\begin{figure}
|
||
\includegraphics[width=\linewidth]{data/H10/DFT_vqzE_relat_zoom.eps}\\
|
||
\includegraphics[width=\linewidth]{data/H10/DFT_vqzE_error.eps}\\
|
||
% \includegraphics[width=\linewidth]{fig2c}
|
||
\caption{
|
||
H$_{10}$, cc-pvqz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one.
|
||
\label{fig:H10_vqz}}
|
||
\end{figure}
|
||
|
||
|
||
|
||
|
||
|
||
\bibliography{srDFT_SC}
|
||
|
||
\end{document}
|