\documentclass[aip,jcp,reprint,noshowkeys]{revtex4-1} %\documentclass[aip,jcp,noshowkeys]{revtex4-1} \usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amsmath,amssymb,amsfonts,physics,mhchem,xspace} \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{}}^\Bas[#1]} \newcommand{\efuncdenpbeAB}[1]{\bar{E}_{\text{A}+\text{B}}^\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]{\bar{\varepsilon}_{\text{c,md}}^{\text{sr},\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,n_{2}^{\text{UEG}},\mu_{\text{HF}}^{\basis}} \newcommand{\argrpbeuegxihf}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),n_{2}^{\text{UEG}}(\br{}),\mu_{\text{HF}}^{\basis}(\br{})} %pbeuegxiCAS \newcommand{\pbeuegxi}{\text{PBE-UEG-}\zeta\text{-CAS}^\Bas} \newcommand{\argpbeuegxicas}[0]{\den,\zeta,s,n_{2}^{\text{UEG}},\mu_{\text{CAS}}^{\basis}} \newcommand{\argrpbeuegxicas}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),n_{2}^{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})} %pbeuegXiCAS \newcommand{\pbeuegXi}{\text{PBE-UEG-}\tilde{\zeta}} \newcommand{\argpbeuegXi}[0]{\den,\tilde{\zeta},s,n_{2}^{\text{UEG}},\mu_{\text{CAS}}^{\basis}} \newcommand{\argrpbeuegXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),n_{2}^{\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,\ntwo,\mu} \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(\br{}),\ntwo(\br{}),\mu(\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{\murpsibas}[0]{\mu_{\wf{}{\Bas}}({\bf r})} \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]{ n^{2,\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})} \newcommand{\twodmrdiagpsi}[0]{ n_{2,{\wf{}{\Bas}}}(\rr{1}{2})} \newcommand{\twodmrdiagpsitot}[0]{ n_{2,\wf{}{A+B}}(\rr{1}{2})} \newcommand{\twodmrdiagpsiaa}[0]{ n_{2,\wf{}{AA}}(\rr{1}{2})} \newcommand{\twodmrdiagpsiaad}[0]{ n_{2,\wf{}{AA}}(\rr{}{})} \newcommand{\twodmrdiagpsibb}[0]{ n_{2,\wf{}{BB}}(\rr{1}{2})} \newcommand{\twodmrdiagpsibbd}[0]{ n_{2,\wf{}{BB}}(\rr{}{})} \newcommand{\twodmrdiagpsiab}[0]{ n_{2\wf{}{AB}}(\rr{1}{2})} \newcommand{\twodmrdiagpsival}[0]{ n_{2\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{A density-based basis-set correction for weak and strong correlation} \begin{abstract} \end{abstract} \maketitle \section{Size consistency of the basis-set correction} \subsection{Sufficient condition for size consistency} The basis-set correction is expressed as an integral in real space \begin{equation} \label{eq:def_ecmdpbebasis} \begin{aligned} & \efuncdenpbe{\argebasis} = \\ & \int \text{d}\br{} \,\denr \ecmd(\argrebasis), \end{aligned} \end{equation} where all the local quantities $\argrebasis$ are obtained from the same wave function $\Psi$. In the limit of two non-overlapping and non-interacting dissociated fragments $\text{A}+\text{B}$, this integral can be rewritten as the sum of the integral over the region $\Omega_\text{A}$ and the integral over the region $\Omega_\text{B}$ \begin{equation} \label{eq:def_ecmdpbebasis} \begin{aligned} & \efuncdenpbeAB{\argebasis} = \\ & \int_{\Omega_\text{A}} \text{d}\br{} \,\denr \ecmd(\argrebasis) \\ & + \int_{\Omega_\text{B}} \text{d}\br{} \,\denr \ecmd(\argrebasis). \end{aligned} \end{equation} Therefore, a sufficient condition to obtain size consistency is that all the local quantities $\argrebasis$ are \textit{intensive}, i.e. that they \textit{locally} coincide in the supersystem $\text{A}+\text{B}$ and in each isolated fragment $\text{X}=\text{A}$ or $\text{B}$. Hence, for $\br{} \in \Omega_\text{X}$, we should have \begin{subequations} \begin{equation} n_\text{A+B}(\br{}) = n_\text{X}(\br{}), \label{nAB} \end{equation} \begin{equation} \zeta_\text{A+B}(\br{}) = \zeta_\text{X}(\br{}), \label{zAB} \end{equation} \begin{equation} s_\text{A+B}(\br{}) = s_\text{X}(\br{}), \label{sAB} \end{equation} \begin{equation} n_{2,\text{A+B}}(\br{}) = n_{2,\text{X}}(\br{}), \label{n2AB} \end{equation} \begin{equation} \mu_{\text{A+B}}(\br{}) = \mu_{\text{X}}(\br{}), \label{muAB} \end{equation} \end{subequations} where the left-hand-side quantities are for the supersystem and the right-hand-side quantities for an isolated fragment. Such conditions can be difficult to fulfil in the presence of degeneracies since the system X can be in a different mixed state (i.e. ensemble) in the supersystem $\text{A}+\text{B}$ and in the isolated fragment~\cite{Sav-CP-09}. Here, we will consider the simple case where the wave function of the supersystem is multiplicatively separable, i.e. \begin{equation} \ket{\wf{\text{A}+\text{B}}{}} = \ket{\wf{\text{A}}{}} \otimes \ket{\wf{\text{B}}{}}, \end{equation} where $\otimes$ is the antisymmetric tensor product. In this case, it is easy to shown that Eqs.~(\ref{nAB})-(\ref{sAB}) are valid, as well known, and it remains to show that Eqs.~(\ref{n2AB}) and~(\ref{muAB}) are also valid. Before showing this, we note that even though we do not explicity consider the case of degeneracies, the lack of size consistency which could arise from spin-multiplet degeneracies can be avoided by the same strategy used for imposing the energy independence on $S_z$, i.e. by using the effective spin polarization $\tilde{\zeta}(n(\br{}),n_{2}(\br{}))$ or a zero spin polarization $\zeta(\br{}) = 0$. Moreover, the lack of size consistency which could arise from spatial degeneracies (e.g., coming from atomic p states) can also be avoided by selecting the same member of the ensemble in the supersystem and in the isolated fragement. This applies to the systems treated in this work. \subsection{Intensivity of the on-top pair density and of the local range-separation parameter} The on-top pair density can be written in an orthonormal spatial orbital basis $\{\SO{p}{}\}$ as \begin{equation} \label{eq:def_n2} n_{2{}}(\br{}) = \sum_{pqrs \in \Bas} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{}, \end{equation} with $\Gam{pq}{rs} = 2 \mel*{\wf{}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{}}$. As the summations run over all orbitals in the basis set $\Bas$, $n_{2{}}(\br{})$ is invariant to orbital rotations and can thus be expressed in terms of localized orbitals. For two non-overlapping fragments $\text{A}+\text{B}$, the basis set can then partitioned into orbitals localized on the fragment A and orbitals localized on B, i.e. $\Bas=\Bas_\text{A}\cup \Bas_\text{B}$. Therefore, we see that the on-top pair density of the supersystem $\text{A}+\text{B}$ is additively separable \begin{equation} \label{eq:def_n2} n_{2,\text{A}+\text{B}}(\br{}) = n_{2,\text{A}}(\br{}) + n_{2,\text{B}}(\br{}), \end{equation} where $n_{2,\text{X}}(\br{})$ is the on-top pair density of the fragment X \begin{equation} \label{eq:def_n2} n_{2,\text{X}}(\br{}) = \sum_{pqrs \in \Bas_\text{X}} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{}. \end{equation} This shows that the on-top pair density is a local intensive quantity. The local range-separation parameter is defined by \begin{equation} \label{eq:def_murAnnex} \mur = \frac{\sqrt{\pi}}{2} \frac{f(\bfr{},\bfr{})}{n_{2}(\br{})}, \end{equation} where \begin{equation} \label{eq:def_f} f(\bfr{},\bfr{}) = \sum_{pqrstu\in \Bas} \SO{p}{ } \SO{q}{ } \V{pq}{rs} \Gam{rs}{tu} \SO{t}{ } \SO{u}{ }. \end{equation} Again, $f(\bfr{},\bfr{})$ is invariant to orbital rotations and can be expressed in terms of orbitals localized on the fragments A and B. In the limit of infinitely separated fragments, the Coulomb interaction vanishes between A and B and therefore any two-electron integral $\V{pq}{rs}$ involving orbitals on both $A$ and $B$ vanishes. We thus see that the quantity $f(\bfr{},\bfr{})$ of the supersystem $\text{A}+\text{B}$ is additively separable \begin{equation} \label{eq:def_fa+b} f_{\text{A}+\text{B}}(\bfr{},\bfr{}) = f_{\text{A}}(\bfr{},\bfr{}) + f_{\text{B}}(\bfr{},\bfr{}), \end{equation} with \begin{equation} \label{eq:def_fX} f_\text{X}(\bfr{},\bfr{}) = \sum_{pqrstu\in \Bas_\text{X}} \SO{p}{ } \SO{q}{ } \V{pq}{rs} \Gam{rs}{tu} \SO{t}{ } \SO{u}{ }. \end{equation} So, $f(\bfr{},\bfr{})$ is a local intensive quantity. As a consequence, the local range-separation parameter of the supersystem $\text{A}+\text{B}$ is \begin{equation} \label{eq:def_murAB} \mu_{\text{A}+\text{B}}(\bfr{}) = \frac{\sqrt{\pi}}{2} \frac{f_{\text{A}}(\bfr{},\bfr{}) + f_{\text{B}}(\bfr{},\bfr{})}{n_{2,\text{A}}(\br{}) + n_{2,\text{B}}(\br{})}, \end{equation} which gives \begin{equation} \label{eq:def_murABsum} \mu_{\text{A}+\text{B}}(\bfr{}) = \mu_{\text{A}}(\bfr{}) + \mu_{\text{B}}(\bfr{}), \end{equation} with $\mu_{\text{X}}(\bfr{}) = (\sqrt{\pi}/2) f_{\text{X}}(\bfr{},\bfr{})/n_{2,\text{X}}(\br{})$. The local range-separation parameter is thus a local intensive quantity. In conclusion, if the wave function of the supersystem $\text{A}+\text{B}$ is multiplicative separable, all local quantities used in the basis-set correction functional 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 $n_{2,\wf{\Bas}{}}(\br{})$ 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} n_{2,\wf{\Bas}{}}(\br{}) = n_{2,\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} n_{2,\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 $n_{2,\mathcal{A}}(\br{})$ 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}