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@ -414,6 +414,101 @@ As the e-e cusp originates from the divergence of the Coulomb operator at $r_{12
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Therefore, the impact of the incompleteness of $\Bas$ can be viewed as a removal of the divergence of the Coulomb interaction at $r_{12} = 0$.
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Therefore, the impact of the incompleteness of $\Bas$ can be viewed as a removal of the divergence of the Coulomb interaction at $r_{12} = 0$.
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The present paragraph briefly describes how to obtain an effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ which i) is finite at the e-e coalescence points as long as an incomplete basis set is used, and ii) tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb operator in the limit of a complete basis set.
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The present paragraph briefly describes how to obtain an effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ which i) is finite at the e-e coalescence points as long as an incomplete basis set is used, and ii) tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb operator in the limit of a complete basis set.
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%=================================================================
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\subsection{Complementary functional}
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%=================================================================
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\label{sec:ecmd}
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Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we propose here to approximate $\bE{}{\Bas}[\n{}{}]$ using a specific class of SR-DFT energy functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
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\begin{multline}
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\label{eq:ec_md_mu}
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\bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
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\\
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- \mel*{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]},
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\end{multline}
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where $\wf{}{\rsmu{}{}}[\n{}{}(\br{})]$ is defined by the constrained minimization
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\begin{equation}
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\label{eq:argmin}
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\wf{}{\rsmu{}{}}[\n{}{}(\br{})] = \arg \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
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\end{equation}
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with
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\begin{equation}
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\label{eq:weemu}
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\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})
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\end{equation}
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and
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\begin{equation}
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\label{eq:erf}
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\w{}{\lr,\rsmu{}{}}(r_{12}) = \frac{\erf(\rsmu{}{} r_{12})}{r_{12}}.
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\end{equation}
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is the long-range part of the Coulomb operator.
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The ECMD functionals admit, for any density $\n{}{}(\br{})$, the two following limiting forms:
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\begin{subequations}
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\begin{align}
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\label{eq:large_mu_ecmd}
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\lim_{\mu \to \infty} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] & = 0,
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\\
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\label{eq:small_mu_ecmd}
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\lim_{\mu \to 0} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] & = \Ec[\n{}{}(\br{})],
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\end{align}
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\end{subequations}
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where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT.
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These functionals differ from the standard RS-DFT correlation functional by the fact that the reference is not the KS Slater determinant but a multi-determinantal wave function, which makes them much more adapted in the present context where one aims at correcting a general multi-determinant WFT model.
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%--------------------------------------------
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%\subsubsection{Local density approximation}
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%--------------------------------------------
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Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we define the LDA version of ECMD as
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\begin{equation}
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\label{eq:def_lda_tot}
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\bE{\LDA}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \int \be{\UEG}{\sr}[\n{}{}(\br{}),\rsmu{}{}] \n{}{}(\br{}) \dbr{}
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\end{equation}
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where $\be{\UEG}{\sr}[\n{}{}(\br{}),\rsmu{}{}]$ is the multi-determinant short-range correlation energy per particle of the uniform electron gas (UEG) for which a parametrization can be found in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
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%In practice, for open-shell systems, we use the spin-resolved version of this functional (i.e., depending on both spin densities) but for simplicity we will continue to use only the notation of the spin-unpolarized case.
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%--------------------------------------------
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%\subsubsection{New PBE functional}
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%--------------------------------------------
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The short-range LDA correlation functional defined in Eq.~\eqref{eq:def_lda_tot} relies on the transferability of the physics of the UEG which is certainly valid for large $\mu$ but is known to over correlate for small $\mu$.
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In order to correct such a defect, we propose here a new ECMD functional inspired by the recently proposed functional of some of the present authors \cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional $\e{}{\PBE}(\n{}{},\nabla \n{}{})$ for small $\rsmu{}{}$ and the exact large-$\rsmu{}{}$ behaviour, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGori-PRB-06} yielding
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\begin{subequations}
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\begin{gather}
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\label{eq:epsilon_cmdpbe}
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\be{\PBE}{\sr}(\n{}{},\nabla \n{}{},\rsmu{}{}) = \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{1 + \beta(n,\nabla n, \mu)\mu^3 }
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\\
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\label{eq:epsilon_cmdpbe}
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\beta(n,\nabla n;\,\mu) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{\n{\UEG}{(2)}(\n{}{})}.
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\end{gather}
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\end{subequations}
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The difference between the ECMD PBE functional defined in Ref.~\cite{FerGinTou-JCP-18} and the present expression is that we approximate the \textit{exact} ground-state on-top pair density of the system $\n{}{(2)}(\br{})$ by its UEG version, i.e.
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\begin{equation}
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\label{eq:ueg_ontop}
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\n{}{(2)}(\br{}) \approx \n{\UEG}{(2)}(\br{}) = \n{}{}(\br{})^2 g_0[\n{}{}(\br{})]
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\end{equation}
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where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}.
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Therefore, the PBE complementary function reads
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\begin{equation}
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\label{eq:def_lda_tot}
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\bE{\PBE}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \int \be{\PBE}{\sr}[\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{}{}] \n{}{}(\br{}) \dbr{}
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\end{equation}
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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 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, 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_{\Bas \rightarrow \infty} \ecmuapproxmurmodel = 0 \quad ,
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\end{equation}
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for whatever choice of density $\denmodel$, wave function $\wf{}{\Bas}$ used to define the interaction, and ECMD functional used to approximate the exact ECMD.
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%=================================================================
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%=================================================================
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\subsection{Effective Coulomb operator}
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\subsection{Effective Coulomb operator}
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%=================================================================
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%=================================================================
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@ -502,99 +597,6 @@ and therefore
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%\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, .
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%\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, .
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\end{equation}
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\end{equation}
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%=================================================================
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\subsection{Complementary functional}
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%=================================================================
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\label{sec:ecmd}
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Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we propose here to approximate $\bE{}{\Bas}[\n{}{}]$ using a specific class of SR-DFT energy functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
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\begin{multline}
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\label{eq:ec_md_mu}
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\bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
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\\
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- \mel*{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]},
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\end{multline}
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where $\wf{}{\rsmu{}{}}[\n{}{}(\br{})]$ is defined by the constrained minimization
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\begin{equation}
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\label{eq:argmin}
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\wf{}{\rsmu{}{}}[\n{}{}(\br{})] = \arg \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
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\end{equation}
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with
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\begin{equation}
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\label{eq:weemu}
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\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})
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\end{equation}
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and
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\begin{equation}
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\label{eq:erf}
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\w{}{\lr,\rsmu{}{}}(r_{12}) = \frac{\erf(\rsmu{}{} r_{12})}{r_{12}}.
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\end{equation}
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is the long-range part of the Coulomb operator.
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The ECMD functionals admit, for any density $\n{}{}(\br{})$, the two following limiting forms:
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\begin{subequations}
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\begin{align}
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\label{eq:large_mu_ecmd}
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\lim_{\mu \to \infty} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] & = 0,
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\\
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\label{eq:small_mu_ecmd}
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\lim_{\mu \to 0} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] & = \Ec[\n{}{}(\br{})],
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\end{align}
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\end{subequations}
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where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT.
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These functionals differ from the standard RS-DFT correlation functional by the fact that the reference is not the KS Slater determinant but a multi-determinantal wave function, which makes them much more adapted in the present context where one aims at correcting a 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 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, 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_{\Bas \rightarrow \infty} \ecmuapproxmurmodel = 0 \quad ,
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\end{equation}
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for whatever choice of density $\denmodel$, wave function $\wf{}{\Bas}$ used to define the interaction, and ECMD functional used to approximate the exact ECMD.
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%--------------------------------------------
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%\subsubsection{Local density approximation}
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%--------------------------------------------
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Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we define the LDA version of ECMD as
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\begin{equation}
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\label{eq:def_lda_tot}
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\bE{\LDA}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \int \be{\UEG}{\sr}[\n{}{}(\br{}),\rsmu{}{}] \n{}{}(\br{}) \dbr{}
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\end{equation}
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where $\be{\UEG}{\sr}[\n{}{}(\br{}),\rsmu{}{}]$ is the multi-determinant short-range correlation energy per particle of the uniform electron gas (UEG) for which a parametrization can be found in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
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%In practice, for open-shell systems, we use the spin-resolved version of this functional (i.e., depending on both spin densities) but for simplicity we will continue to use only the notation of the spin-unpolarized case.
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%--------------------------------------------
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%\subsubsection{New PBE functional}
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%--------------------------------------------
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The short-range LDA correlation functional defined in Eq.~\eqref{eq:def_lda_tot} relies on the transferability of the physics of the UEG which is certainly valid for large $\mu$ but is known to over correlate for small $\mu$.
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In order to correct such a defect, we propose here a new ECMD functional inspired by the recently proposed functional of some of the present authors \cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional $\e{}{\PBE}(\n{}{},\nabla \n{}{})$ for small $\rsmu{}{}$ and the exact large-$\rsmu{}{}$ behaviour, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGori-PRB-06} yielding
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\begin{subequations}
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\begin{gather}
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\label{eq:epsilon_cmdpbe}
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\be{\PBE}{\sr}(\n{}{},\nabla \n{}{},\rsmu{}{}) = \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{1 + \beta(n,\nabla n, \mu)\mu^3 }
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\\
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\label{eq:epsilon_cmdpbe}
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\beta(n,\nabla n;\,\mu) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{\n{\UEG}{(2)}(\n{}{})}.
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\end{gather}
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\end{subequations}
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The difference between the ECMD PBE functional defined in Ref.~\cite{FerGinTou-JCP-18} and the present expression is that we approximate the \textit{exact} ground-state on-top pair density of the system $\n{}{(2)}(\br{})$ by its UEG version, i.e.
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\begin{equation}
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\label{eq:ueg_ontop}
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\n{}{(2)}(\br{}) \approx \n{\UEG}{(2)}(\br{}) = \n{}{}(\br{})^2 g_0[\n{}{}(\br{})]
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\end{equation}
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where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}.
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Therefore, the PBE complementary function reads
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\begin{equation}
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\label{eq:def_lda_tot}
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\bE{\PBE}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \int \be{\PBE}{\sr}[\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{}{}] \n{}{}(\br{}) \dbr{}
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\end{equation}
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%=================================================================
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%=================================================================
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\subsection{Valence effective interaction}
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\subsection{Valence effective interaction}
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%=================================================================
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%=================================================================
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