Bread on the board
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@ -161,6 +161,8 @@
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% methods
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\newcommand{\FCI}{\text{FCI}}
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\newcommand{\CCSDT}{\text{CCSD(T)}}
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\newcommand{\lr}{\text{lr}}
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\newcommand{\sr}{\text{sr}}
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\newcommand{\Nel}{N}
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@ -169,6 +171,9 @@
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\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}}
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\newcommand{\wf}[2]{\Psi_{#1}^{#2}}
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\newcommand{\W}[2]{W_{#1}^{#2}}
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\newcommand{\w}[2]{w_{#1}^{#2}}
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\newcommand{\hn}[2]{\Hat{n}_{#1}^{#2}}
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\newcommand{\rsmu}[2]{\mu_{#1}^{#2}}
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\newcommand{\SO}[2]{\phi_{#1}(\bx{#2})}
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\newcommand{\modX}{\text{X}}
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@ -389,8 +394,8 @@ Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only so
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%=================================================================
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However, in addition of being unknown, the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$.
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Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ following a two-step procedure which guarantees the correct behaviour in the limit $\Bas \to \infty$ [see Eq.~\eqref{eq:limitfunc}].
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First, we define a real-space representation of the Coulomb operator projected in $\Bas$, which is then fitted with a long-range interaction thanks to a range-separation parameter $\mu(\br{})$ varying in space. %(see Sec.~\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 $\n{\modX}{\Bas}$ with $\mu(\br{})$.% (see Sec.~\ref{sec:ecmd}) .
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First, we define a real-space representation of the Coulomb operator projected in $\Bas$, which is then fitted with a long-range interaction thanks to a range-separation \textit{function} $\mu(\br{})$ defined in real space. %(see Sec.~\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 $\n{\modX}{\Bas}$ alongside $\mu(\br{})$.% (see Sec.~\ref{sec:ecmd}) .
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%=================================================================
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@ -402,9 +407,9 @@ 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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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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%=================================================================
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\subsection{Effective Coulomb operator}
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%----------------------------------------------------------------
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%=================================================================
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In order to compute the effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ defined such that
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\begin{equation}
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\label{eq:int_eq_wee}
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@ -446,42 +451,11 @@ and $\Gam{mn}{pq}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{p}\aic{q}\ai{n}\ai{m}
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As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ is symmetric, \textit{a priori} non translational nor rotational invariant if $\Bas$ does not have such symmetries and is necessarily \textit{finite} at $r_{12} = 0$ for an incomplete basis set $\Bas$.
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Also, as demonstrated in Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\lim_{\Bas \to \infty}\W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = r_{12}^{-1}$.
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%----------------------------------------------------------------
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\subsubsection{Definition of a valence effective interaction}
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%----------------------------------------------------------------
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As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a general subset of molecular orbitals.
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We then split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor$ and $\Val$ are its core and valence parts, respectively, and $\Cor \bigcap \Val = \O$.
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According to Eqs.~\eqref{eq:expectweeb} and \eqref{eq:def_weebasis} , the effective interaction is defined by the expectation value of the coulomb operator over a wave function $\wf{}{\Bas}$. Therefore, to define an effective interaction accounting only for the valence electrons, one needs to define a function $\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ satisfying
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\begin{equation}
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\label{eq:expectweebval}
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\mel*{\wf{}{\Bas}}{\hWee{\Val}}{\wf{}{\Bas}} = \frac{1}{2} \iint \f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
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\end{equation}
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where $\hWee{\Val}$, the valence part of the Coulomb operator, has a similar expression as $\hWee{\Bas}$ in Eq.~\eqref{eq:WeeB}.
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%\begin{equation}
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% \hWee{\Val} = \frac{1}{2} \sum_{ijkl \in \Val} \vijkl \aic{k}\aic{l}\ai{j}\ai{i},
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%\end{equation}
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Following the spirit of Eq.~\eqref{eq:fbasis}, the function $\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ can be defined as
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\begin{multline}
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\label{eq:fbasisval}
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\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})
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\\
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= \sum_{ij \in \Bas} \sum_{klmn \in \Val} \SO{i}{1} \SO{j}{2} \vijkl \gammaklmn{\wf{}{\Bas}} \SO{n}{2} \SO{m}{1}.
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\end{multline}
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and, the valence part of the effective interaction is
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\begin{equation}
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\label{eq:def_weebasis}
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\W{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) = \frac{\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) }{\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2})},
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\end{equation}
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where $\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2})$ is the two body density associated to the valence electrons.
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%\begin{equation}
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% \twodmrdiagpsival = \sum_{klmn \in \Val} \SO{m}{1} \SO{n}{2} \gammamnkl[\wf{}{\Bas}] \SO{k}{1} \SO{l}{2} .
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%\end{equation}
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%It is worth noting that, in Eq.~\eqref{eq:fbasisval} the difference between the set of orbitals for the indices $(i,j)$, which span the full set of MOs within $\Bas$, and the $(k,l,m,n)$, which span only the valence space $\Basval$. 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 $\Bas$, whatever the choice of subset $\Basval$.
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\subsubsection{Definition of a range-separation parameter varying in space}
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To be able to approximate the complementary functional $\efuncbasis$ thanks to functionals developed in the field of RSDFT, we fit the effective interaction with a long-range interaction having a range-separation parameter \textit{varying in space}.
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%=================================================================
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\subsubsection{Range-separation function}
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%=================================================================
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To be able to approximate the complementary functional $\bE{}{\Bas}[\n{}{}]$ thanks to functionals developed in the field of RS-DFT, we fit the effective interaction with a long-range interaction having a range-separation parameter \textit{varying in space}.
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More precisely, if we define the value of the interaction at coalescence as
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\begin{equation}
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\label{eq:def_wcoal}
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@ -490,71 +464,101 @@ More precisely, if we define the value of the interaction at coalescence as
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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 $\wbasiscoal{ }$ 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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\wbasiscoal{} = \w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\bfrb{},\bfrb{})
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\end{equation}
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where the long-range-like interaction is defined as:
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where the long-range-like interaction is defined as
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\begin{equation}
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w^{\text{lr},\mur}(\bfrb{1},\bfrb{2}) = \frac{ 1 }{2} \bigg( \frac{\text{erf}\big( \murr{1} \, r_{12}\big)}{r_{12}} + \frac{\text{erf}\big( \murr{2} \, r_{12}\big)}{ r_{12}}\bigg).
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\w{}{\lr,\rsmu{}{}}(\br{1},\br{2}) = \frac{1}{2} \qty{ \frac{\erf[ \murr{1} \, r_{12}]}{r_{12}} + \frac{\erf[ \murr{2} r_{12}]}{ r_{12}} }.
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\end{equation}
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The equation \eqref{eq:def_wcoal} is equivalent to the following condition for $\murpsi$:
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Equation \eqref{eq:def_wcoal} is equivalent to the following condition
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\begin{equation}
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\label{eq:mu_of_r}
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\murpsi = \frac{\sqrt{\pi}}{2} \, \wbasiscoal{} \, .
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\end{equation}
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As we defined an effective interaction for the valence electrons, we also introduce a valence range-separation parameter as
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\begin{equation}
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\label{eq:mu_of_r_val}
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\murpsival = \frac{\sqrt{\pi}}{2} \, \wbasiscoalval{} \, .
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\rsmu{\wf{}{\Bas}}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{}(\br{})
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\end{equation}
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%As we defined an effective interaction for the valence electrons, we also introduce a valence range-separation parameter as
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%\begin{equation}
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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 $\Bas$, as
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\begin{equation}
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\begin{aligned}
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&\lim_{\Bas \rightarrow \infty}\wbasis = 1/r_{12} \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\\
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&\lim_{\Bas \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_{\Bas \rightarrow \infty} \wbasiscoal{} = +\infty\,\,, \\
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&\lim_{\Bas \rightarrow \infty} \wbasiscoalval{} = +\infty\,\,,
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\end{aligned}
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\label{eq:lim_W}
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\lim_{\Bas \rightarrow \infty}\wbasis = r_{12}^{-1} \quad \forall (\br{1},\br{2})
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% &\lim_{\Bas \rightarrow \infty}\wbasisval = 1/r_{12} \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\,\, ,
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\end{equation}
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one has $\lim_{\Bas \rightarrow \infty} \wbasiscoal{} = \infty$
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% &\lim_{\Bas \rightarrow \infty} \wbasiscoalval{} = +\infty\,\,,
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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_{\Bas \rightarrow \infty} \murpsi = +\infty \,\, \\
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&\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, .
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\end{aligned}
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\label{eq:lim_mur}
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\lim_{\Bas \rightarrow \infty} \rsmu{\wf{}{\Bas}}{}(\br{}) = \infty
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%\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, .
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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 $\efuncbasis$ by using a specific class of SRDFT energy functionals, namely the ECMD whose general definition is\cite{TouGorSav-TCA-05}:
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%=================================================================
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\subsection{Valence effective interaction}
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%=================================================================
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As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a general subset of molecular orbitals.
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We then split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor$ and $\Val$ are its core and valence parts, respectively, and $\Cor \bigcap \Val = \O$.
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According to Eqs.~\eqref{eq:expectweeb} and \eqref{eq:def_weebasis} , the effective interaction is defined by the expectation value of the coulomb operator over a wave function $\wf{}{\Bas}$. Therefore, to define an effective interaction accounting only for the valence electrons, one needs to define a function $\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ satisfying
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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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\ecmubis = & \min_{\Psi \rightarrow \denr}\elemm{\Psi}{\kinop +\weeop}{\Psi}\\-\;&\elemm{\psimu[\denr]}{\kinop+\weeop}{\psimu[\denr]},
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\end{aligned}
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\label{eq:expectweebval}
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\mel*{\wf{}{\Bas}}{\hWee{\Val}}{\wf{}{\Bas}} = \frac{1}{2} \iint \f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
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\end{equation}
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where the wave function $\psimu[\denr]$ is defined by the constrained minimization
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where $\hWee{\Val}$, the valence part of the Coulomb operator, has a similar expression as $\hWee{\Bas}$ in Eq.~\eqref{eq:WeeB}.
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%\begin{equation}
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% \hWee{\Val} = \frac{1}{2} \sum_{ijkl \in \Val} \vijkl \aic{k}\aic{l}\ai{j}\ai{i},
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%\end{equation}
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Following the spirit of Eq.~\eqref{eq:fbasis}, the function $\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ can be defined as
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\begin{multline}
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\label{eq:fbasisval}
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\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})
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\\
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= \sum_{ij \in \Bas} \sum_{klmn \in \Val} \SO{i}{1} \SO{j}{2} \vijkl \gammaklmn{\wf{}{\Bas}} \SO{n}{2} \SO{m}{1}.
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\end{multline}
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and, the valence part of the effective interaction is
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\begin{equation}
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\label{eq:def_weebasis}
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\W{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) = \frac{\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) }{\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2})},
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\end{equation}
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where $\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2})$ is the two body density associated to the valence electrons.
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%\begin{equation}
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% \twodmrdiagpsival = \sum_{klmn \in \Val} \SO{m}{1} \SO{n}{2} \gammamnkl[\wf{}{\Bas}] \SO{k}{1} \SO{l}{2} .
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%\end{equation}
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%It is worth noting that, in Eq.~\eqref{eq:fbasisval} the difference between the set of orbitals for the indices $(i,j)$, which span the full set of MOs within $\Bas$, and the $(k,l,m,n)$, which span only the valence space $\Basval$. 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 $\Bas$, whatever the choice of subset $\Basval$.
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It is worth noting that, within the present definition, $\W{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ and $\murpsival$ fulfils Eqs.~\eqref{eq:lim_W} and \eqref{eq:lim_mur}.
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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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In Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} the authors have proposed to approximate the complementary functional $\bE{}{\Bas}[\n{}{}]$ using a specific class of SR-DFT energy functionals, namely the ECMD whose general definition is \cite{TouGorSav-TCA-05}
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\begin{multline}
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\label{eq:ec_md_mu}
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\ecmubis = \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 the wave function $\wf{}{\rsmu{}{}}[\n{}{}(\br{})]$ is defined by the constrained minimization
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\begin{equation}
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\label{eq:argmin}
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\psimu[\denr] = \arg \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop + \weeopmu}{\Psi},
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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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where $\weeopmu$ is the long-range electron-electron interaction operator
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where
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\begin{equation}
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\label{eq:weemu}
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\weeopmu = \frac{1}{2} \iint \text{d}{\bf r}_1 \text{d}{\bf r}_2 \; w^{\text{lr},\mu}(|{\bf r}_1 - {\bf r}_2|) \hat{n}^{(2)}({\bf r}_1,{\bf r}_2),
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\label{eq:weemu}
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\hWee{\lr,\rsmu{}{}} = \frac{1}{2} \iint \w{}{\lr,\rsmu{}{}}(r_{12}) \hn{}{(2)}(\br{1},\br{2}) \dbr{1} \dbr{2},
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\end{equation}
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with
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is the long-range electron-electron interaction operator with
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\begin{equation}
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\label{eq:erf}
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w^{\text{lr},\mu}(r_{12}) = \frac{\text{erf}(\mu r_{12})}{r_{12}},
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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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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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and the pair-density operator $\hn{}{(2)}(\br{1},\br{2}) =\hn{}{}(\br{1}) \hn{}{}(\br{2}) - \delta (\br{1}-\br{2}) \hn{}{}(\br{1})$.
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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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@ -581,7 +585,9 @@ It is important to notice that in the limit of a complete basis set, according t
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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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\subsubsection{LDA approximation for the complementary functional}
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%--------------------------------------------
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\subsubsection{Local density approximation}
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%--------------------------------------------
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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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@ -589,7 +595,9 @@ As done in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, one can define an LDA-li
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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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\subsubsection{New PBE interpolated ECMD functional}
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%--------------------------------------------
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\subsubsection{New PBE functional}
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%--------------------------------------------
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The LDA-like functional defined in \eqref{eq:def_lda_tot} relies only on the transferability of the physics of UEG which is certainly valid for large values of $\mu$ but which is known to over correlate for small values of $\mu$.
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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 when $\mu \rightarrow 0$ and the exact behaviour which is known when $\mu \rightarrow \infty$.
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@ -633,7 +641,9 @@ Therefore, we propose this approximation for the complementary functional $\ecom
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\ecompmodelpbe = \int \, \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(n({\bf r}),\nabla n({\bf r});\,\mur)
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\end{equation}
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%--------------------------------------------
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\subsection{Valence-only approximation for the complementary functional}
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%--------------------------------------------
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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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