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Manuscript/SI/srDFT_SC-SI.tex Normal file
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\documentclass[aip,jcp,reprint,noshowkeys]{revtex4-1}
%\documentclass[aip,jcp,noshowkeys]{revtex4-1}
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable}
\usepackage{mathpazo,libertine}
\usepackage[normalem]{ulem}
\newcommand{\alert}[1]{\textcolor{red}{#1}}
\definecolor{darkgreen}{RGB}{0, 180, 0}
\newcommand{\beurk}[1]{\textcolor{darkgreen}{#1}}
\newcommand{\trash}[1]{\textcolor{red}{\sout{#1}}}
\usepackage{xspace}
\usepackage{hyperref}
\hypersetup{
colorlinks=true,
linkcolor=blue,
filecolor=blue,
urlcolor=blue,
citecolor=blue
}
\newcommand{\cdash}{\multicolumn{1}{c}{---}}
\newcommand{\mc}{\multicolumn}
\newcommand{\fnm}{\footnotemark}
\newcommand{\fnt}{\footnotetext}
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
\newcommand{\mr}{\multirow}
\newcommand{\SI}{\textcolor{blue}{supporting information}}
% second quantized operators
\newcommand{\psix}[1]{\hat{\Psi}\left({\bf X}_{#1}\right)}
\newcommand{\psixc}[1]{\hat{\Psi}^{\dagger}\left({\bf X}_{#1}\right)}
\newcommand{\ai}[1]{\hat{a}_{#1}}
\newcommand{\aic}[1]{\hat{a}^{\dagger}_{#1}}
\newcommand{\vijkl}[0]{V_{ij}^{kl}}
\newcommand{\phix}[2]{\phi_{#1}(\bfr{#2})}
\newcommand{\phixprim}[2]{\phi_{#1}(\bfr{#2}')}
\newcommand{\CBS}{\text{CBS}}
%operators
\newcommand{\elemm}[3]{{\ensuremath{\bra{#1}{#2}\ket{#3}}\xspace}}
\newcommand{\ovrlp}[2]{{\ensuremath{\langle #1|#2\rangle}\xspace}}
%\newcommand{\ket}[1]{{\ensuremath{|#1\rangle}\xspace}}
%\newcommand{\bra}[1]{{\ensuremath{\langle #1|}\xspace}}
%
% energies
\newcommand{\Ec}{E_\text{c}}
\newcommand{\EPT}{E_\text{PT2}}
\newcommand{\EsCI}{E_\text{sCI}}
\newcommand{\EDMC}{E_\text{DMC}}
\newcommand{\EexFCI}{E_\text{exFCI}}
\newcommand{\EexFCIbasis}{E_\text{exFCI}^{\Bas}}
\newcommand{\EexFCIinfty}{E_\text{exFCI}^{\infty}}
\newcommand{\EexDMC}{E_\text{exDMC}}
\newcommand{\Ead}{\Delta E_\text{ad}}
\newcommand{\efci}[0]{E_{\text{FCI}}^{\Bas}}
\newcommand{\emodel}[0]{E_{\model}^{\Bas}}
\newcommand{\emodelcomplete}[0]{E_{\model}^{\infty}}
\newcommand{\efcicomplete}[0]{E_{\text{FCI}}^{\infty}}
\newcommand{\ecccomplete}[0]{E_{\text{CCSD(T)}}^{\infty}}
\newcommand{\ecc}[0]{E_{\text{CCSD(T)}}^{\Bas}}
\newcommand{\efuncbasisFCI}[0]{\bar{E}^\Bas[\denFCI]}
\newcommand{\efuncbasisfci}[0]{\bar{E}^\Bas[\denfci]}
\newcommand{\efuncbasis}[0]{\bar{E}^\Bas[\den]}
\newcommand{\efuncden}[1]{\bar{E}^\Bas[#1]}
\newcommand{\efuncdenpbe}[1]{\bar{E}_{\text{X}}^\Bas[#1]}
\newcommand{\ecompmodel}[0]{\bar{E}^\Bas[\denmodel]}
\newcommand{\ecmubis}[0]{\bar{E}_{\text{c,md}}^{\text{sr}}[\denr;\,\mu]}
\newcommand{\ecmubisldapbe}[0]{\bar{E}_{\text{c,md}\,\text{PBE}}^{\text{sr}}[\denr;\,\mu]}
\newcommand{\ecmuapprox}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mu]}
\newcommand{\ecmuapproxmur}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mur]}
\newcommand{\ecmuapproxmurfci}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denfci;\,\mur]}
\newcommand{\ecmuapproxmurmodel}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denmodel;\,\mur]}
\newcommand{\ecompmodellda}[0]{\bar{E}_{\text{LDA}}^{\Bas,\wf{}{\Bas}}[\denmodel]}
\newcommand{\ecompmodelldaval}[0]{\bar{E}_{\text{LDA, val}}^{\Bas,\wf{}{\Bas}}[\den]}
\newcommand{\ecompmodelpbe}[0]{\bar{E}_{\text{PBE}}^{\Bas,\wf{}{\Bas}}[\den]}
\newcommand{\ecompmodelpbeval}[0]{\bar{E}_{\text{PBE, val}}^{\Bas,\wf{}{\Bas}}[\den]}
\newcommand{\emulda}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denr;\mu({\bf r};\wf{}{\Bas})\right)}
\newcommand{\emuldamodel}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denmodelr;\mu({\bf r};\wf{}{\Bas})\right)}
\newcommand{\emuldaval}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denval ({\bf r});\murval;\wf{}{\Bas})\right)}
\newcommand{\ecmd}[0]{\varepsilon^{\text{c,md}}_{\text{PBE}}}
\newcommand{\psibasis}[0]{\Psi^{\basis}}
\newcommand{\BasFC}{\mathcal{A}}
%pbeuegxiHF
\newcommand{\pbeuegxihf}{\text{PBE-UEG-}\zeta\text{-HF}^\Bas}
\newcommand{\argpbeuegxihf}[0]{\den,\zeta,s,\ntwo_{\text{UEG}},\mu_{\text{HF}}^{\basis}}
\newcommand{\argrpbeuegxihf}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo_{\text{UEG}}(\br{}),\mu_{\text{HF}}^{\basis}(\br{})}
%pbeuegxiCAS
\newcommand{\pbeuegxi}{\text{PBE-UEG-}\zeta\text{-CAS}^\Bas}
\newcommand{\argpbeuegxicas}[0]{\den,\zeta,s,\ntwo_{\text{UEG}},\mu_{\text{CAS}}^{\basis}}
\newcommand{\argrpbeuegxicas}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo_{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
%pbeuegXiCAS
\newcommand{\pbeuegXi}{\text{PBE-UEG-}\tilde{\zeta}}
\newcommand{\argpbeuegXi}[0]{\den,\tilde{\zeta},s,\ntwo_{\text{UEG}},\mu_{\text{CAS}}^{\basis}}
\newcommand{\argrpbeuegXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),\ntwo_{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
%pbeontxiCAS
\newcommand{\pbeontxi}{\text{PBE-ot-}\zeta}
\newcommand{\argpbeontxi}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
\newcommand{\argrpbeontxi}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
%pbeontXiCAS
\newcommand{\pbeontXi}{\text{PBE-ot-}\tilde{\zeta}}
\newcommand{\argpbeontXi}[0]{\den,\tilde{\zeta},s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
\newcommand{\argrpbeontXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
%pbeont0xiCAS
\newcommand{\pbeontns}{\text{PBE-ot-}0\zeta}
\newcommand{\argpbeontns}[0]{\den,0,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
\newcommand{\argrpbeontns}[0]{\den(\br{}),0,s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
%%%%%% arguments
\newcommand{\argepbe}[0]{\den,\zeta,s}
\newcommand{\argebasis}[0]{\den,\zeta,s,\ntwo,\mu_{\Psi^{\basis}}}
\newcommand{\argecmd}[0]{\den,\zeta,s,\ntwo,\mu}
\newcommand{\argepbeueg}[0]{\den,\zeta,s,\ntwo_{\text{UEG}},\mu_{\Psi^{\basis}}}
\newcommand{\argepbeontxicas}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
\newcommand{\argepbeuegXihf}[0]{\den,\tilde{\zeta},s,\ntwo_{\text{UEG}},\mu_{\Psi^{\basis}}}
\newcommand{\argrebasis}[0]{\denr,\zeta(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})}
\newcommand{\argrebasisab}[0]{\denr,\zeta(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})}
% numbers
\newcommand{\rnum}[0]{{\rm I\!R}}
\newcommand{\bfr}[1]{{\bf r}_{#1}}
\newcommand{\dr}[1]{\text{d}\bfr{#1}}
\newcommand{\rr}[2]{\bfr{#1}, \bfr{#2}}
\newcommand{\rrrr}[4]{\bfr{#1}, \bfr{#2},\bfr{#3},\bfr{#4} }
% effective interaction
\newcommand{\twodm}[4]{\elemm{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}}
\newcommand{\murpsi}[0]{\mu({\bf r};\wf{}{A+B})}
\newcommand{\murpsia}[0]{\mu({\bf r};\wf{}{A})}
\newcommand{\murpsib}[0]{\mu({\bf r};\wf{}{B})}
\newcommand{\ntwo}[0]{n^{(2)}}
\newcommand{\ntwohf}[0]{n^{(2),\text{HF}}}
\newcommand{\ntwophi}[0]{n^{(2)}_{\phi}}
\newcommand{\ntwoextrap}[0]{\mathring{n}^{(2)}_{\psibasis}}
\newcommand{\ntwoextrapcas}[0]{\mathring{n}^{(2)\,\basis}_{\text{CAS}}}
\newcommand{\mur}[0]{\mu({\bf r})}
\newcommand{\murr}[1]{\mu({\bf r}_{#1})}
\newcommand{\murval}[0]{\mu_{\text{val}}({\bf r})}
\newcommand{\murpsival}[0]{\mu_{\text{val}}({\bf r};\wf{}{\Bas})}
\newcommand{\murrval}[1]{\mu_{\text{val}}({\bf r}_{#1})}
\newcommand{\weeopmu}[0]{\hat{W}_{\text{ee}}^{\text{lr},\mu}}
\newcommand{\wbasis}[0]{W_{\wf{}{\Bas}}(\bfr{1},\bfr{2})}
\newcommand{\wbasiscoal}[0]{W_{\wf{}{\Bas}}(\bfr{},\bfr{})}
\newcommand{\wbasisval}[0]{W_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})}
\newcommand{\fbasis}[0]{f_{\wf{}{\Bas}}(\bfr{1},\bfr{2})}
\newcommand{\fbasisval}[0]{f_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})}
\newcommand{\ontop}[2]{ n^{(2)}_{#1}({\bf #2}_1)}
\newcommand{\twodmrpsi}[0]{ \ntwo_{\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})}
\newcommand{\twodmrdiagpsi}[0]{ \ntwo_{\wf{}{\Bas}}(\rr{1}{2})}
\newcommand{\twodmrdiagpsitot}[0]{ \ntwo_{\wf{}{A+B}}(\rr{1}{2})}
\newcommand{\twodmrdiagpsiaa}[0]{ \ntwo_{\wf{}{AA}}(\rr{1}{2})}
\newcommand{\twodmrdiagpsibb}[0]{ \ntwo_{\wf{}{BB}}(\rr{1}{2})}
\newcommand{\twodmrdiagpsiab}[0]{ \ntwo_{\wf{}{AB}}(\rr{1}{2})}
\newcommand{\twodmrdiagpsival}[0]{ \ntwo_{\wf{}{\Bas},\,\text{val}}(\rr{1}{2})}
\newcommand{\gammamnpq}[1]{\Gamma_{mn}^{pq}[#1]}
\newcommand{\gammamnkl}[0]{\Gamma_{mn}^{kl}}
\newcommand{\gammaklmn}[1]{\Gamma_{kl}^{mn}[#1]}
%\newcommand{\wbasiscoal}[1]{W_{\wf{}{\Bas}}({\bf r}_{#1})}
\newcommand{\ontoppsi}[1]{ n^{(2)}_{\wf{}{\Bas}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})}
\newcommand{\wbasiscoalval}[1]{W_{\wf{}{\Bas}}^{\text{val}}({\bf r}_{#1})}
\newcommand{\ontoppsival}[1]{ n^{(2)}_{\wf{}{\Bas}}^{\text{val}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})}
\newcommand{\ex}[4]{$^{#1}#2_{#3}^{#4}$}
\newcommand{\ra}{\rightarrow}
\newcommand{\De}{D_\text{e}}
% MODEL
\newcommand{\model}[0]{\mathcal{Y}}
% densities
\newcommand{\denmodel}[0]{\den_{\model}^\Bas}
\newcommand{\denmodelr}[0]{\den_{\model}^\Bas ({\bf r})}
\newcommand{\denfci}[0]{\den_{\psifci}}
\newcommand{\denFCI}[0]{\den^{\Bas}_{\text{FCI}}}
\newcommand{\denhf}[0]{\den_{\text{HF}}^\Bas}
\newcommand{\denrfci}[0]{\denr_{\psifci}}
\newcommand{\dencipsir}[0]{{n}_{\text{CIPSI}}^\Bas({\bf r})}
\newcommand{\dencipsi}[0]{{n}_{\text{CIPSI}}^\Bas}
\newcommand{\den}[0]{{n}}
\newcommand{\denval}[0]{{n}^{\text{val}}}
\newcommand{\denr}[0]{{n}({\bf r})}
\newcommand{\onedmval}[0]{\rho_{ij,\sigma}^{\text{val}}}
% wave functions
\newcommand{\psifci}[0]{\Psi^{\Bas}_{\text{FCI}}}
\newcommand{\psimu}[0]{\Psi^{\mu}}
% operators
\newcommand{\weeopbasis}[0]{\hat{W}_{\text{ee}}^\Bas}
\newcommand{\kinop}[0]{\hat{T}}
\newcommand{\weeopbasisval}[0]{\hat{W}_{\text{ee}}^{\Basval}}
\newcommand{\weeop}[0]{\hat{W}_{\text{ee}}}
% units
\newcommand{\IneV}[1]{#1 eV}
\newcommand{\InAU}[1]{#1 a.u.}
\newcommand{\InAA}[1]{#1 \AA}
% methods
\newcommand{\UEG}{\text{UEG}}
\newcommand{\LDA}{\text{LDA}}
\newcommand{\PBE}{\text{PBE}}
\newcommand{\FCI}{\text{FCI}}
\newcommand{\CCSDT}{\text{CCSD(T)}}
\newcommand{\lr}{\text{lr}}
\newcommand{\sr}{\text{sr}}
\newcommand{\Nel}{N}
\newcommand{\V}[2]{V_{#1}^{#2}}
\newcommand{\n}[2]{n_{#1}^{#2}}
\newcommand{\E}[2]{E_{#1}^{#2}}
\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}}
\newcommand{\bEc}[1]{\Bar{E}_\text{c}^{#1}}
\newcommand{\e}[2]{\varepsilon_{#1}^{#2}}
\newcommand{\be}[2]{\Bar{\varepsilon}_{#1}^{#2}}
\newcommand{\bec}[1]{\Bar{e}^{#1}}
\newcommand{\wf}[2]{\Psi_{#1}^{#2}}
\newcommand{\W}[2]{W_{#1}^{#2}}
\newcommand{\w}[2]{w_{#1}^{#2}}
\newcommand{\hn}[2]{\Hat{n}_{#1}^{#2}}
\newcommand{\rsmu}[2]{\mu_{#1}^{#2}}
\newcommand{\SO}[2]{\phi_{#1}(\br{#2})}
\newcommand{\modX}{\text{X}}
\newcommand{\modY}{\text{Y}}
% basis sets
\newcommand{\setdenbasis}{\mathcal{N}_{\Bas}}
\newcommand{\Bas}{\mathcal{B}}
\newcommand{\basis}{\mathcal{B}}
\newcommand{\Basval}{\mathcal{B}_\text{val}}
\newcommand{\Val}{\mathcal{V}}
\newcommand{\Cor}{\mathcal{C}}
% operators
\newcommand{\hT}{\Hat{T}}
\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}}
\newcommand{\f}[2]{f_{#1}^{#2}}
\newcommand{\Gam}[2]{\Gamma_{#1}^{#2}}
% coordinates
\newcommand{\br}[1]{{\mathbf{r}_{#1}}}
\newcommand{\bx}[1]{\mathbf{x}_{#1}}
\newcommand{\dbr}[1]{d\br{#1}}
\newcommand{\PBEspin}{PBEspin}
\newcommand{\PBEueg}{PBE-UEG-{$\tilde{\zeta}$}}
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, France}
\begin{document}
\title{Mixing density functional theory and wave function theory for strong correlation: the best of both worlds}
\begin{abstract}
bla bla bla youpi tralala
\end{abstract}
\maketitle
\section{Extensivity of the basis set correction}
The following paragraph proposes a demonstration of the size consistency of the basis set correction in the limit of dissociated fragments.
The present basis set correction being an integral in real space, in the limit of two dissociated fragments $A\ldots B$, one can split the contribution in
\begin{equation}
\begin{aligned}
\label{eq:def_ecmdpbebasis}
\efuncdenpbe{\argebasis} = &\int d\br{} \,\denr \\ & \ecmd(\argrebasis)
\end{aligned}
\end{equation}
\subsection{Property of the on-top pair density}
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
\begin{equation}
\ntwo_{\wf{}{}}(\br{}) = \sum_{pqrs} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{},
\end{equation}
with $\Gam{pq}{rs} = 2 \mel*{\wf{}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$.
Assume now that the wave function $\wf{}{\Bas}$ can be written as a product of two wave functions defined on two non-overlapping and non-interacting fragments $A$ and $B$
\begin{equation}
\ket{\wf{A+B}{}} = \ket{\wf{A}{}} \times \ket{\wf{B}{}}.
\end{equation}
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 operator as
\begin{equation}
\begin{aligned}
\hat{\Gamma} = \hat{\Gamma}_{AA}{} + \hat{\Gamma}_{BB}{} + \hat{\Gamma}_{AB}{}
\end{aligned}
\end{equation}
with
\begin{equation}
\begin{aligned}
\hat{\Gamma}_{AA} = \sum_{p_A,q_A,r_A,s_A} \aic{r_{A,\downarrow}}\aic{s_{A,\uparrow}}\ai{q_{A,\uparrow}}\ai{p_{A,\downarrow}},
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\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}},
\end{aligned}
\end{equation}
and
\begin{equation}
\begin{aligned}
\hat{\Gamma}_{AB} = \sum_{p_A,q_B,r_A,s_B} \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
\begin{equation}
\begin{aligned}
& \twodmrdiagpsiaa = \\ & \sum_{p_A q_A r_A s_A} \SO{p_A}{1} \SO{q_A}{2} \bra{\wf{A}{}}\hat{\Gamma}_{AA}\ket{\wf{A}{}} \SO{r_A}{1} \SO{s_A}{2},
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
& \twodmrdiagpsibb = \\ & \sum_{p_B q_B r_B s_B} \SO{p_B}{1} \SO{q_B}{2} \bra{\wf{B}{}}\hat{\Gamma}_{BB}\ket{\wf{B}{}} \SO{r_B}{1} \SO{s_B}{2},
\end{aligned}
\end{equation}
and
\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}
where $n_{A}(\br{})$ and $n_{A}(\br{})$ are the one body density of the sub systems
\begin{equation}
\begin{aligned}
& n_{B}(\br{}) = \sum_{p_B r_B} \SO{p_B}{} \bra{\wf{B}{}}\aic{s_{B,\uparrow}}\ai{q_{B,\uparrow}}\ket{\wf{B}{}} \SO{r_B}{}
\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}
Based on these considerations, one can express the on-top pair density is simply
\begin{equation}
\begin{aligned}
\ntwo_{\wf{A+B}{}}(\br{}) = \ntwo_{\wf{A/A}{}}(\br{}) + \ntwo_{\wf{B/B}{}}(\br{})
\end{aligned}
\end{equation}
with
\begin{equation}
\ntwo_{\wf{}{A/A}}(\br{}) = \sum_{p_A q_A r_A s_A} \SO{p_A}{} \SO{q_A}{} \Gam{p_A q_A}{r_A s_A} \SO{r_A}{} \SO{s_A}{}
\end{equation}
and
\begin{equation}
\ntwo_{\wf{}{B/B}}(\br{}) = \sum_{p_B q_B r_B s_B} \SO{p_B}{} \SO{q_B}{} \Gam{p_B q_B}{r_B s_B} \SO{r_B}{} \SO{s_B}{}
\end{equation}
being the on-top pair densities of the local fragments associated to $\ket{\wf{A}{}}$ and $\ket{\wf{B}{}}$. Therefore, 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{A+B}{}}(\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}
\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 $A+B$ with a multiplicative function is simply
\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 is nothing but
\begin{equation}
\label{eq:def_mur}
\murpsi = \murpsia + \murpsib
\end{equation}
and therefore 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.
\bibliography{../srDFT_SC}
\end{document}

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