getting there
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@ -394,25 +394,114 @@ Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only so
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% \EexFCIinfty \approx \EexFCIbasis + \efuncden{\dencipsi}
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%\end{equation}
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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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One of the consequences of the incompleteness of $\Bas$ is that $\wf{}{\Bas}$ does not have a cusp (i.e.~a discontinuous derivative) at the electron-electron (e-e) coalescence points.
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As the e-e cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could equivalently originate from a Hamiltonian with a non-divergent Coulomb interaction at $r_{12} = 0$.
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Therefore, it feels natural to evaluate $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals.
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Contrary to conventional RS-DFT schemes which require a range-separated parameter $\rsmu{}{}$, we must know the value of $\rsmu{}{}$ at any point in space due to the spatial inhomogeneity of $\Bas$, hence defining a range-separated \textit{function} $\rsmu{}{}(\br{})$.
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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 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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%Second, 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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%=================================================================
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%\subsection{General scheme for the approximation of the unknown complementary functional $\efuncbasis$}
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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 \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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%\subsection{Definition of a real-space representation of the coulomb operator truncated in a basis-set $\Bas$}
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%\label{sec:weff}
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%=================================================================
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One of the consequences of the incompleteness of $\Bas$ is that $\wf{}{\Bas}$ does not have a cusp (i.e.~a discontinuous derivative) at the electron-electron (e-e) coalescence points.
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As the e-e cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could also originate from a Hamiltonian with a non-divergent Coulomb interaction.
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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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\subsection{Effective Coulomb operator}
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%=================================================================
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%The present section 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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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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\mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}} = \iint \W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) \n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
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\end{equation}
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(where $\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})$ is the two-body density associated with $\wf{}{\Bas}$, $\bx{} = \qty(\br{},\sigma)$ collects space and spin variables, and $\int \dbx{} = \sum_{\sigma}\,\int_{\mathbb{R}^3} \dbr{}$), one must realise that
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\begin{equation}
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\mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}},
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\end{equation}
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which states that the expectation value of $\hWee{}$ over $\wf{}{\Bas}$ is equal to the expectation value of its projected version in $\Bas$
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\begin{equation}
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\label{eq:WeeB}
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\hWee{\Bas} = \frac{1}{2} \sum_{ijkl \in \Bas} \vijkl \aic{k}\aic{l}\ai{j}\ai{i}
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\end{equation}
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over the same wave function $\wf{}{\Bas}$, where the indices run over all spinorbitals $\SO{i}{}$ in $\Bas$ and $\vijkl$ are the usual two-electron Coulomb integrals.
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Because one can show (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that
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\begin{subequations}
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\begin{align}
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\label{eq:expectweeb}
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\mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}} & = \frac{1}{2} \iint \f{\wf{}{\Bas}}{}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
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\\
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\label{eq:expectwee}
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\mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} & = \frac{1}{2} \iint r_{12}^{-1} \n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
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\end{align}
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\end{subequations}
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where
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\begin{multline}
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\label{eq:fbasis}
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\f{\wf{}{\Bas}}{}(\bx{1},\bx{2})
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\\
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= \sum_{ijklmn \in \Bas} \SO{i}{1} \SO{j}{2} \vijkl \Gam{mn}{kl}[\wf{}{\Bas}] \SO{n}{2} \SO{m}{1},
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\end{multline}
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and $\Gam{mn}{pq}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{p}\aic{q}\ai{n}\ai{m} }{\wf{}{\Bas}}$ is the two-body density tensor of $\wf{}{\Bas}$, it comes naturally that
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\begin{equation}
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\label{eq:def_weebasis}
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\W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = \frac{\f{\wf{}{\Bas}}{}(\bx{1},\bx{2})}{\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})}.
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\end{equation}
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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.
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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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\subsection{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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\wbasiscoal{} = W_{\wf{}{\Bas}}(\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 $\wbasiscoal{ }$ with a long-range-like interaction:
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\begin{equation}
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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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\begin{equation}
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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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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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\rsmu{\wf{}{\Bas}}{}(\br{}) = \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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\label{eq:lim_W}
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\lim_{\Bas \rightarrow \infty}\wbasis = r_{12}^{-1} \quad \forall (\bx{1},\bx{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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\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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@ -509,93 +598,6 @@ 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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%=================================================================
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\subsection{Effective Coulomb operator}
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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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\mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}} = \iint \W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) \n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
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\end{equation}
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(where $\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})$ is the two-body density associated with $\wf{}{\Bas}$, $\bx{} = \qty(\br{},\sigma)$ collects space and spin variables, and $\int \dbx{} = \sum_{\sigma}\,\int_{\mathbb{R}^3} \dbr{}$), one must realise that
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\begin{equation}
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\mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}},
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\end{equation}
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which states that the expectation value of $\hWee{}$ over $\wf{}{\Bas}$ is equal to the expectation value of its projected version in $\Bas$
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\begin{equation}
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\label{eq:WeeB}
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\hWee{\Bas} = \frac{1}{2} \sum_{ijkl \in \Bas} \vijkl \aic{k}\aic{l}\ai{j}\ai{i}
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\end{equation}
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over the same wave function $\wf{}{\Bas}$, where the indices run over all spinorbitals $\SO{i}{}$ in $\Bas$ and $\vijkl$ are the usual two-electron Coulomb integrals.
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Because one can show (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that
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\begin{subequations}
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\begin{align}
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\label{eq:expectweeb}
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\mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}} & = \frac{1}{2} \iint \f{\wf{}{\Bas}}{}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
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\\
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\label{eq:expectwee}
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\mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} & = \frac{1}{2} \iint r_{12}^{-1} \n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
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\end{align}
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\end{subequations}
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where
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\begin{multline}
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\label{eq:fbasis}
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\f{\wf{}{\Bas}}{}(\bx{1},\bx{2})
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\\
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= \sum_{ijklmn \in \Bas} \SO{i}{1} \SO{j}{2} \vijkl \Gam{mn}{kl}[\wf{}{\Bas}] \SO{n}{2} \SO{m}{1},
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\end{multline}
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and $\Gam{mn}{pq}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{p}\aic{q}\ai{n}\ai{m} }{\wf{}{\Bas}}$ is the two-body density tensor of $\wf{}{\Bas}$, it comes naturally that
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\begin{equation}
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\label{eq:def_weebasis}
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\W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = \frac{\f{\wf{}{\Bas}}{}(\bx{1},\bx{2})}{\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})}.
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\end{equation}
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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.
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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{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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\wbasiscoal{} = W_{\wf{}{\Bas}}(\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 $\wbasiscoal{ }$ with a long-range-like interaction:
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\begin{equation}
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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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\begin{equation}
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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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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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\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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\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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\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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%=================================================================
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\subsection{Valence effective interaction}
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