updated manuscript
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Emmanuel Giner
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@ -75,11 +75,12 @@
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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}}^{\basis,\psibasis}[\den]}
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\newcommand{\ecompmodellda}[0]{\bar{E}_{\text{LDA}}^{\basis,\psibasis}[\denmodel]}
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\newcommand{\ecompmodelldaval}[0]{\bar{E}_{\text{LDA, val}}^{\basis,\psibasis}[\den]}
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\newcommand{\ecompmodelpbe}[0]{\bar{E}_{\text{PBE}}^{\basis,\psibasis}[\den]}
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\newcommand{\ecompmodelpbeval}[0]{\bar{E}_{\text{PBE, val}}^{\basis,\psibasis}[\den]}
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\newcommand{\emulda}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denr;\mu({\bf r};\psibasis)\right)}
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\newcommand{\emuldamodel}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denmodelr;\mu({\bf r};\psibasis)\right)}
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\newcommand{\emuldaval}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denval ({\bf r});\murval;\psibasis)\right)}
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@ -323,13 +324,13 @@ Finally, we approximate complete basis set exFCI energy $\EexFCIinfty$ as
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\subsection{General scheme for the approximation of the unknown complementary functional $\efuncbasis$}
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The functional $\efuncbasis$ is not universal as it depends on the basis set $\basis$ used and a simple analytical form for such a functional is of course not known.
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Following the work of \onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\efuncbasis$ in two-steps using the RS-DFT formalism. First, we define a real-space representation of the coulomb interaction projected in $\basis$, which is then fitted with a long-range interaction thanks to a range-separation parameter $\mu(r)$ varying in space (see \ref{sec:weff}).
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Following the work of \onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\efuncbasis$ in two-steps which grantee the correct behaviour in the limit of a complete basis set (see \eqref{eq:limitfunc}). First, we define a real-space representation of the coulomb interaction projected in $\basis$, which is then fitted with a long-range interaction thanks to a range-separation parameter $\mu(r)$ varying in space (see \ref{sec:weff}).
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Then, we choose a specific class of short-range density functionals, namely the short-range correlation functionals with multi-determinantal reference (ECMD) introduced by Toulouse \textit{et al}\cite{TouGorSav-TCA-05}, that we evaluate at the density $\denmodel$ provided by the model (see \ref{sec:ecmd}) and with the range-separation parameter $\mu(r)$ varying in space.
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\subsection{Definition of a real-space representation of the coulomb operator truncated in a basis-set $\basis$}
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\label{sec:weff}
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One of the consequences of the use of an incomplete basis-set $\basis$ is that the wave function does not present a cusp near the electron coalescence point, which means that all derivatives of the wave function are continuous. As the exact electronic cusp originates from the divergence of the coulomb interaction at the electron coalescence point, a cusp-free wave function could also come from a non-divergent electron-electron interaction. Therefore, the impact of the incompleteness of a finite basis-set $\basis$ can be thought as a cutting of the divergence of the coulomb interaction at the electron coalescence point.
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One of the consequences of the use of an incomplete basis-set $\basis$ is that the wave function does not present a cusp near the electron coalescence point, which means that all derivatives of the wave function are continuous. As the exact electronic cusp originates from the divergence of the coulomb interaction at the electron coalescence point, a cusp-free wave function could also originate from an Hamiltonian with a non-divergent electron-electron interaction. Therefore, the impact of the incompleteness of a finite basis-set $\basis$ can be thought as a cutting of the divergence of the coulomb interaction at the electron coalescence point.
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The present paragraph briefly describes how to obtain an effective interaction $\wbasis$ which:
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\begin{itemize}
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@ -345,28 +346,28 @@ Consider the coulomb operator projected in the basis-set $\basis$
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\end{equation}
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where the indices run over all orthonormal spin-orbitals in $\basis$ and $\vijkl$ are the usual coulomb two-electron integrals.
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Consider now the expectation value of $\weeopbasis$ over a general wave function $\psibasis$ belonging to the $N-$electron Hilbert space spanned by the basis set $\basis$.
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After a few mathematical work (see appendix A of \onlinecite{GinPraFerAssSavTou-JCP-18} for a detailed derivation), such an expectation value can be rewritten as an integral over $\rnum^6$:
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After a few mathematical work (see appendix A of \onlinecite{GinPraFerAssSavTou-JCP-18} for a detailed derivation), such an expectation value can be rewritten as an integral over the two-electron spin and space coordinates:
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\begin{equation}
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\label{eq:expectweeb}
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\elemm{\psibasis}{\weeopbasis}{\psibasis} = \frac{1}{2}\,\,\iint \dr{1}\,\dr{2} \,\, \fbasis,
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\elemm{\psibasis}{\weeopbasis}{\psibasis} = \frac{1}{2}\,\,\iint \dr{1}\,\dr{2} \,\, \fbasis\,\,,
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\end{equation}
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where the function $\fbasis$ is
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\begin{equation}
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\label{eq:fbasis}
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\begin{aligned}
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\fbasis = \sum_{ijklmn\,\,\in\,\,\basis} & \vijkl \,\, \gammaklmn{\psibasis} \\& \phix{n}{2} \phix{m}{1} \phix{i}{1} \phix{j}{2},
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\fbasis = \sum_{ijklmn\,\,\in\,\,\basis} & \vijkl \,\, \gammaklmn{\psibasis} \\& \phix{n}{2} \phix{m}{1} \phix{i}{1} \phix{j}{2}\,\,,
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\end{aligned}
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\end{equation}
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$\gammamnpq{\psibasis}$ is the two-body density tensor of $\psibasis$
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\begin{equation}
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\gammamnpq{\psibasis} = \elemm{\psibasis}{ \aic{p}\aic{q}\ai{n}\ai{m} }{\psibasis},
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\gammamnpq{\psibasis} = \elemm{\psibasis}{ \aic{p}\aic{q}\ai{n}\ai{m} }{\psibasis}\,\,,
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\end{equation}
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and $\bfr{}$ collects the space and spin variables,
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\begin{equation}
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\label{eq:define_x}
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\begin{aligned}
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&\bfr{} = \left({\bf r},\sigma \right)\qquad {\bf r} \in {\rm I\!R}^3,\,\, \sigma = \pm \frac{1}{2}\\
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&\int \, \dr{} = \sum_{\sigma = \pm \frac{1}{2}}\,\int_{{\rm I\!R}^3} \, \text{d}{\bf r}.
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&\int \, \dr{} = \sum_{\sigma = \pm \frac{1}{2}}\,\int_{{\rm I\!R}^3} \, \text{d}{\bf r}\,\,.
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\end{aligned}
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\end{equation}
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@ -395,11 +396,11 @@ where we introduced $\wbasis$
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which is the effective interaction in the basis set $\basis$.
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As already discussed in \onlinecite{GinPraFerAssSavTou-JCP-18}, such an effective interaction is symmetric, \textit{a priori} non translational nor rotational invariant if the basis set $\basis$ does not have such symmetries and is necessary \textit{finite} at the electron coalescence point for an incomplete basis set $\basis$.
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Also, as demonstrated in the appendix B of \onlinecite{GinPraFerAssSavTou-JCP-18}, $\wbasis$ tends to the regular coulomb interaction $1/r_{12}$ for all points in $(\bfr{1},\bfr{2})$ in the limit of a complete basis set $\basis$.
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Also, as demonstrated in the appendix B of \onlinecite{GinPraFerAssSavTou-JCP-18}, $\wbasis$ tends to the regular coulomb interaction $1/r_{12}$ for all points $(\bfr{1},\bfr{2})$ and any choice of $\psibasis$ in the limit of a complete basis set $\basis$.
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\subsubsection{Definition of a valence effective interaction}
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As the average inter electronic distances are very different between the valence electrons and the core electrons, it can be advantageous to define an effective interaction taking into account only for the valence electrons.
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As most of the WFT calculations are done using a frozen core approximation, it is important to define an effective interaction within a general subset of molecular orbitals that we refer as $\basisval$.
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According to \eqref{eq:def_weebasis} and \eqref{eq:expectweeb}, the effective interaction is defined by the expectation value of the coulomb operator over a wave function $\psibasis$. Therefore, to define an effective interaction accounting only for the valence electrons, one needs to define a function $\fbasisval$ satisfying
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\begin{equation}
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@ -412,7 +413,7 @@ where $\weeopbasisval$ is the valence coulomb operator defined as
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\weeopbasisval = \frac{1}{2}\,\, \sum_{ijkl\,\,\in\,\,\basisval} \,\, \vijkl \,\, \aic{k}\aic{l}\ai{j}\ai{i}\,\,\,,
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\end{aligned}
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\end{equation}
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and $\basisval$ is a given set of molecular orbitals associated to the valence space which will be defined later on.
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and $\basisval$ is the subset of molecular orbitals for which we want to define the expectation value, which will be typically the all MOs except those frozen.
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Following the spirit of \eqref{eq:fbasis}, the function $\fbasisval$ can be defined as
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\begin{equation}
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\label{eq:fbasisval}
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@ -430,7 +431,7 @@ where $\twodmrdiagpsival$ is the two body density associated to the valence elec
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\begin{equation}
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\twodmrdiagpsival = \sum_{klmn\,\,\in\,\,\basisval} \gammamnkl[\psibasis] \,\, \phix{m}{1} \phix{n}{2} \phix{k}{1} \phix{l}{2} .
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\end{equation}
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It is important to notice in \eqref{eq:fbasisval} the difference between the set of orbitals for the indices $(i,j)$, which span the full set of MOs within $\basis$, and the $(k,l,m,n)$, which span only the valence space $\basisval$. With such a definition, one can show (see annex) that $\fbasisval$ fulfills \eqref{eq:expectweebval} and tends to the exact interaction $1/r_{12}$ in the limit of a complete basis set $\basis$, whatever the choice of subset $\basisval$.
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It is important to notice in \eqref{eq:fbasisval} the difference between the set of orbitals for the indices $(i,j)$, which span the full set of MOs within $\basis$, and the $(k,l,m,n)$, which span only the valence space $\basisval$. Only with such a definition, one can show (see annex) that $\fbasisval$ fulfills \eqref{eq:expectweebval} and tends to the exact interaction $1/r_{12}$ in the limit of a complete basis set $\basis$, whatever the choice of subset $\basisval$.
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\subsubsection{Definition of a range-separation parameter varying in space}
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@ -441,7 +442,7 @@ More precisely, if we define the value of the interaction at coalescence as
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\wbasiscoal{} = W_{\psibasis}(\bfr{},\bar{{\bf X}}_{}).
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\end{equation}
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where $(\bfr{},\bar{{\bf X}}_{})$ means a couple of anti-parallel spins at the same point in $\bfrb{}$,
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we propose a fit for each point in $\rnum^3$ of $\wbasiscoal{1}$ with a long-range-like interaction:
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we propose a fit for each point in $\rnum^3$ of $\wbasiscoal{ }$ with a long-range-like interaction:
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\begin{equation}
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\wbasiscoal{} = w^{\text{lr},\murpsi}(\bfrb{},\bfrb{})
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\end{equation}
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@ -459,11 +460,33 @@ As we defined an effective interaction for the valence electrons, we also introd
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\label{eq:mu_of_r_val}
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\murpsival = \frac{\sqrt{\pi}}{2} \, \wbasiscoalval{} \, .
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\end{equation}
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An important point to notice is that, in the limit of a complete basis set $\basis$, as
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\begin{equation}
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\begin{aligned}
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&\lim_{\basis \rightarrow \infty}\wbasis = 1/r_{12} \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\\
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&\lim_{\basis \rightarrow \infty}\wbasisval = 1/r_{12} \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\,\, ,
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\end{aligned}
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\end{equation}
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one has
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\begin{equation}
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\begin{aligned}
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&\lim_{\basis \rightarrow \infty} \wbasiscoal{} = +\infty\,\,, \\
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&\lim_{\basis \rightarrow \infty} \wbasiscoalval{} = +\infty\,\,,
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\end{aligned}
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\end{equation}
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and therefore
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\begin{equation}
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\label{eq:lim_mur}
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\begin{aligned}
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&\lim_{\basis \rightarrow \infty} \murpsi = +\infty \,\, \\
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&\lim_{\basis \rightarrow \infty} \murpsival = +\infty \,\, .
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\end{aligned}
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\end{equation}
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\subsection{Approximations for the complementary functional $\ecompmodel$}
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\subsubsection{General scheme}
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\label{sec:ecmd}
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In \onlinecite{GinPraFerAssSavTou-JCP-18} the authors have proposed to approximate the complementary functional $\ecompmodel$ by using a specific class of SRDFT energy functionals, namely the ECMD whose general definition is\cite{TouGorSav-TCA-05}:
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In \onlinecite{GinPraFerAssSavTou-JCP-18} the authors have proposed to approximate the complementary functional $\efuncbasis$ by using a specific class of SRDFT energy functionals, namely the ECMD whose general definition is\cite{TouGorSav-TCA-05}:
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\begin{equation}
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\begin{aligned}
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\label{eq:ec_md_mu}
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@ -486,33 +509,37 @@ with
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w^{\text{lr},\mu}(|{\bf r}_1 - {\bf r}_2|) = \frac{\text{erf}(\mu |{\bf r}_1 - {\bf r}_2|)}{|{\bf r}_1 - {\bf r}_2|},
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\end{equation}
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and the pair-density operator $\hat{n}^{(2)}({\bf r}_1,{\bf r}_2) =\hat{n}({\bf r}_1) \hat{n}({\bf r}_2) - \delta ({\bf r}_1-{\bf r}_2) \hat{n}({\bf r}_1)$.
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These functionals differ from the standard RSDFT correlation functional by the fact that the reference is not the Konh-Sham determinant but a multi determinant wave function, which makes them much more adapted in the present context where one aims at correcting the FCI energy.
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The ECMD functionals admit two limits as function of $\mu$
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\begin{equation}
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\label{eq:large_mu_ecmd}
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\lim_{\mu \rightarrow \infty} \ecmubis = 0 \quad \forall\,\,\denr
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\end{equation}
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\begin{equation}
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\label{eq:small_mu_ecmd}
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\lim_{\mu \rightarrow 0} \ecmubis = E_{\text{c}}[\denr]\quad \forall\,\,\denr
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\end{equation}
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where $E_{\text{c}}[\denr]$ is the usual universal correlation functional defined in the Kohn-Sham DFT.
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These functionals differ from the standard RSDFT correlation functional by the fact that the reference is not the Kohn-Sham Slater determinant but a multi determinant wave function, which makes them much more adapted in the present context where one aims at correcting the general multi-determinant WFT model.
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The general scheme for estimating $\ecompmodel$ is the following. Consider a given approximated ECMD functional $\ecmuapprox$ labelled by ECMD-$\mathcal{X}$.
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Such a functional which might depend on the density $\denr$ (and potentially its derivatives $\nabla \denr$) is defined for any value of the range-separation parameter $\mu$.
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A general scheme to approximate $\ecompmodel$ is to use $\ecmuapprox$ with the $\mur$ defined in \eqref{eq:def_weebasis} and to evaluate it at the FCI density $\denfci$
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Such a functional of the density $\denr$ (and potentially its derivatives $\nabla \denr$) is defined for any value of the range-separation parameter $\mu$.
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A general scheme to approximate $\ecompmodel$ is to use $\ecmuapprox$ with the $\mur$ defined in \eqref{eq:def_weebasis} and to evaluate it at the density defined by the model $\denmodel$
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\begin{equation}
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\label{eq:approx_ecfuncbasis}
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\ecompmodel \approx \ecmuapproxmurmodel
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\end{equation}
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Therefore, any approximated ECMD can be used to estimate $\ecompmodel$.
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It is important to notice that in the limit of a complete basis set, as
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\begin{equation}
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\lim_{\basis \rightarrow \infty} \wbasiscoalval{} = +\infty \quad \forall\,\, \psibasis\,\,\text{and}\,\,\,{\bf r}\,,
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\end{equation}
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the local range separation parameter $\murpsi$ (or $\murpsival$) tends to infinity and therefore
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It is important to notice that in the limit of a complete basis set, according to equations \eqref{eq:lim_mur} and \eqref{eq:large_mu_ecmd} one has
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\begin{equation}
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\lim_{\basis \rightarrow \infty} \ecmuapproxmurmodel = 0 \quad ,
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\end{equation}
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which is a condition required by the exact theory (see \eqref{eq:limitfunc}).
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Also, it means that one recovers a WFT model in the limit of a complete basis set, whatever the choice of $\psibasis$, functional ECMD or density used.
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for whatever choice of density $\denmodel$, wave function $\psibasis$ used to define the interaction, and ECMD functional used to approximate the exact ECMD.
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\subsubsection{LDA approximation for the complementary functional}
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Therefore, one can define an LDA-like functional for $\ecompmodel$ as
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As done in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, one can define an LDA-like approximation for $\ecompmodel$ as
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\begin{equation}
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\label{eq:def_lda_tot}
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\ecompmodellda = \int \, \text{d}{\bf r} \,\, \denr \,\, \emulda\,,
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\ecompmodellda = \int \, \text{d}{\bf r} \,\, \denmodelr \,\, \emuldamodel\,,
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\end{equation}
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where $\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}(n,\mu)$ is the multi-determinant short-range correlation energy per particle of the uniform electron gas for which a parametrization can be found in Ref.~\onlinecite{PazMorGorBac-PRB-06}. In practice, for open-shell systems, we use the spin-polarized version of this functional (i.e., depending on the spin densities) but for simplicity we will continue to use only the notation of the spin-unpolarized case.
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@ -538,7 +565,7 @@ where $n_{\uparrow}({\bf} r)$ and $ n_{\downarrow}({\bf} r)$ are, respectively,
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\end{equation}
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and $g_0(n_{\uparrow} ,\, n_{\downarrow})$ is the correlation factor of the UEG whose parametrization can be found in \cite{ueg_ontop}.
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As the form in \eqref{eq:ecmd_large_mu} diverges for small values of $\mu$ as $1/\mu^3$, we follow the work proposed in \cite{FerGinTou-JCP-18} and interpolate between the large-$\mu$ limit and the $\mu = 0$ limit where the $\ecmubis$ reduces to the Kohn-Sham correlation functional, for which we take the PBE approximation as in \cite{FerGinTou-JCP-18}.
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As the form in \eqref{eq:ecmd_large_mu} diverges for small values of $\mu$ as $1/\mu^3$, we follow the work proposed in \cite{FerGinTou-JCP-18} and interpolate between the large-$\mu$ limit and the $\mu = 0$ limit where the $\ecmubis$ reduces to the Kohn-Sham correlation functional (see equation \eqref{eq:small_mu_ecmd}), for which we take the PBE approximation as in \cite{FerGinTou-JCP-18}.
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More precisely, we propose the following expression for the
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\begin{equation}
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\label{eq:ecmd_large_mu}
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@ -561,7 +588,7 @@ Therefore, we propose this approximation for the complementary functional $\ecom
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\end{equation}
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\subsection{Valence-only approximation for the complementary functional}
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We now introduce a valence-only approximation for the complementary functional, which, as we shall see, performs much better than the usual approximations in the context of atomization energies.
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We now introduce a valence-only approximation for the complementary functional which is needed to correct for frozen core WFT models.
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Defining the valence one-body spin density matrix as
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\begin{equation}
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\begin{aligned}
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@ -583,7 +610,6 @@ Therefore, we propose the following valence-only approximations for the compleme
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\ecompmodelpbeval = \int \, \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(\denval({\bf r}),\nabla \denval({\bf r});\,\murval)
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\end{equation}
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\subsection{Generalization of the basis-set correction to any model of WFT}
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Results}
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%%%%%%%%%%%%%%%%%%%%%%%%
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@ -592,15 +618,7 @@ Therefore, we propose the following valence-only approximations for the compleme
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We begin the investigation of the behavior of the basis-set correction by the study of the atomization energies of the C$_2$, N$_2$, O$_2$, F$_2$ homo-nuclear diatomic molecules in the Dunning cc-pVXZ and cc-pCVXZ (X=D,T,Q,5) using both the CIPSI algorithm and the CCSD(T). All through this work, we follow the frozen core (FC) convention of Klopper \textit{et. al}\cite{HauKlo-JCP-12} which consists in all-electron calculations for Li-Be, a He core for B-Na atoms and a Ne core for the Al-Cl series. In the context of the DFT correction for the basis-set, this implies that, for a given system in a given basis set $\basis$, the set of valence orbitals $\basisval$ involved in the definition of the valence interaction $\wbasisval$ and density $\onedmval$ refers to all MOs except the core.
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\subsubsection{CIPSI calculations and the basis-set correction}
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All CIPSI calculations were performed in two steps. First, a CIPSI calculation was performed until the zeroth-order wave function reaches $10^6$ Slater determinants, from which we extracted the natural orbitals. From this set of natural orbitals, we performed CIPSI calculations until the $\EexFCIbasis$ reaches about $0.1$ mH convergence for each systems. Such convergence criterion is more than sufficient for the CIPSI densities $\dencipsi$.
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Therefore, from now on, we assume that
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\begin{equation}
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\efci \approx \EexFCIbasis
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\end{equation}
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and that
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\begin{equation}
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\denrfci \approx \dencipsi.
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\end{equation}
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Regarding the wave function chosen to define the local range-separation parameter $\mur$, we take a single Slater determinant built with the natural orbitals of the first CIPSI calculation.
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Regarding the wave function $\psibasis$ chosen to define the local range-separation parameter $\mur$, we take a single Slater determinant built with the natural orbitals of the first CIPSI calculation.
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\subsubsection{CCSD(T) calculations and the basis-set correction}
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\begin{table*}
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