From f5c820a402f67b91ff712a32a89945dd21b9d98d Mon Sep 17 00:00:00 2001
From: Pierre-Francois Loos <pierrefrancois.loos@gmail.com>
Date: Thu, 11 Apr 2019 16:48:24 +0200
Subject: [PATCH] Major change of notations

---
 Manuscript/G2-srDFT.tex | 96 ++++++++++++++++++++---------------------
 1 file changed, 47 insertions(+), 49 deletions(-)

diff --git a/Manuscript/G2-srDFT.tex b/Manuscript/G2-srDFT.tex
index 8ecadca..8e3a030 100644
--- a/Manuscript/G2-srDFT.tex
+++ b/Manuscript/G2-srDFT.tex
@@ -189,7 +189,7 @@ We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} fo
 %\subsection{Correcting the basis set error of a general WFT model}
 %=================================================================
 Let us assume we have both the energy $\E{\modX}{\Bas}$ and density $\n{\modY}{\Bas}$ of a $\Ne$-electron system described by two methods $\modX$ and $\modY$ (potentially identical) in an incomplete basis set $\Bas$. 
-According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modX}{\Bas}$ and $\n{\modY}{\Bas}$ are reasonable approximations of the FCI energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be written as
+According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modX}{\Bas}$ and $\n{\modY}{\Bas}$ are reasonable approximations of the \alert{FCI} energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be written as
 \begin{equation}
 	\label{eq:e0basis}
 	\E{}{} 
@@ -204,19 +204,19 @@ where
 	- \min_{\wf{}{\Bas} \to \n{}{}}  \mel*{\wf{}{\Bas}}{\hT + \hWee{}}{\wf{}{\Bas}}
 \end{equation}
 is the basis-dependent complementary density functional, $\hT$ is the kinetic operator and $\hWee{} = \sum_{i<j} r_{ij}^{-1}$ is the interelectronic repulsion operator.
-In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron wave functions belonging to the Hilbert space spanned by $\Bas$ and a complete basis, respectively.
+In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis, respectively.
 Both wave functions yield the same target density $\n{}{}$. 
 
 %\alert{Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only approximation performed in \eqref{eq:e0basis} is that the FCI density $\n{\FCI}{\Bas}$ coincides with the exact ground state density, which in general is a reasonable approximation as the density converges rapidly with the basis set.}
 
-Importantly, in the limit of a complete basis set $\Bas$ (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$, which implies that
+Importantly, in the limit of a complete basis set (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$, which implies that
 \begin{equation}
   \label{eq:limitfunc}
-        \lim_{\Bas \to \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\modY}{\Bas}] ) = \E{\modX}{\infty} \approx E,
+        \lim_{\Bas \to \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\modY}{\Bas}] ) = \E{\modX}{} \approx E,
 \end{equation}
-where  $\E{\modX}{\infty}$ is the energy associated with the method $\modX$ in the complete basis set.
-In the case $\modX = \FCI$, we have as strict equality as $E_{\FCI}^\infty = E$.
-Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only source of error at this stage lies in the potential approximate nature of the methods $\modX$ and $\modY$ for the FCI energy and density within $\Bas$, respectively.
+where  $\E{\modX}{}$ is the energy associated with the method $\modX$ in the complete basis set.
+In the case $\modX = \FCI$, we have as strict equality as $\E{\FCI}{} = \E{}{}$.
+Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only source of error at this stage lies in the potential approximate nature of the methods $\modX$ and $\modY$ for the \titou{FCI} energy and density within $\Bas$, respectively.
 
 %Here we propose to generalize such approach to a general WFT model, referred here as $\model$, projected in a basis set $\Bas$ which must provides a density $\denmodel$ and an energy $\emodel$. 
 %As any wave function model is necessary an approximation to the FCI model, one can write 
@@ -309,18 +309,16 @@ Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only so
 %\end{equation}
 
 Rigorously speaking,  the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$. 
-Nevertheless, as $\bE{}{\Bas}[\n{}{}]$ aims at fixing the incompleteness of $\Bas$, its main role is to correct 
-for the lack of electronic cups in $\wf{}{\Bas}$ (i.e.~a discontinuous derivative) at the electron-electron (e-e) 
-coalescence points. 
-Therefore, the physical role of $\bE{}{\Bas}[\n{}{}]$ is to account for a universal condition of exact wave functions. 
+Moreover, as $\bE{}{\Bas}[\n{}{}]$ aims at fixing the incompleteness of $\Bas$, its main role is to correct 
+for the lack of cusp in $\wf{}{\Bas}$ (i.e.~a discontinuous derivative) at the e-e coalescence points, a universal condition of exact wave functions. 
 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 $r_{12} = 0$. 
-Therefore, as we shall do later on, it feels natural to  approximate $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals which deal with a smooth non divergent two-electron interaction.
-Contrary to the conventional RS-DFT scheme which requires a range-separated \textit{parameter} $\rsmu{}{}$, here we use a range-separated \textit{function} $\rsmu{}{}(\br{})$ which automatically adapts to quantify the incompleteness of a basis set $\Bas$ for each point in ${\rm I\!R}^3$.  
+Therefore, as we shall do later on, it feels natural to  approximate $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals which deal with a smooth long-range electron interaction.
+Contrary to the conventional RS-DFT scheme which requires a range-separated \textit{parameter} $\rsmu{}{}$, here we use a range-separated \textit{function} $\rsmu{\Bas}{}(\br{})$ which automatically adapts to quantify the incompleteness of $\Bas$ in $\mathbb{R}^3$.  
 
-The first step of the basis set correction consists in obtaining an effective two-electron interaction $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ which represents the effect of the projection in an incomplete basis set $\Bas$ of the Coulomb operator. 
-We use a definition for which $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$  i) is finite at the e-e coalescence point as long as an incomplete basis set is used, and ii) tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb interaction in the limit of a complete basis set. 
-In a second step, we shall link $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ to $\rsmu{}{}(\br{})$.
-In the final step, we employ short-range density functionals\cite{TouGorSav-TCA-05} with $\rsmu{}{}(\br{})$ as the range separation. 
+The first step of the basis set correction consists in obtaining an effective two-electron interaction $\W{\Bas}{}(\br{1},\br{2})$ which represents the effect of the projection of the Coulomb operator in an incomplete basis set $\Bas$. 
+The present definition ensures that $\W{\Bas}{}(\br{1},\br{2})$ is finite at the e-e coalescence point as long as an incomplete basis set is used, and tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb interaction as $\Bas \to \infty$. 
+In a second step, we shall link $\W{\Bas}{}(\br{1},\br{2})$ to $\rsmu{\Bas}{}(\br{})$.
+In the final step, we employ short-range density functionals\cite{TouGorSav-TCA-05} alongside $\rsmu{\Bas}{}(\br{})$ as range separation. 
 %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}]. 
 %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}) .   
 %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}). 
@@ -338,12 +336,12 @@ In the final step, we employ short-range density functionals\cite{TouGorSav-TCA-
 %=================================================================
 %\subsection{Effective Coulomb operator}
 %=================================================================
-We define the effective operator $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ as 
+We define the effective operator $\W{\Bas}{}(\br{1},\br{2})$ as 
 \begin{equation}
 	\label{eq:def_weebasis}
- \W{\wf{}{\Bas}}{}(\br{1},\br{2})   = \left\{
+ \W{\Bas}{}(\br{1},\br{2})   = \left\{
     \begin{array}{ll}
-       \f{\wf{}{\Bas}}{}(\br{1},\br{2})/\n{2}{\wf{}{\Bas}}(\br{1},\br{2}) & \mbox{if } \n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \ne 0\\
+       \f{\Bas}{}(\br{1},\br{2})/\n{2}{\wf{}{\Bas}}(\br{1},\br{2}) & \mbox{if } \n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \ne 0\\
         \,\,\,\,+\infty & \mbox{otherwise.}
     \end{array}
  \right.
@@ -354,58 +352,58 @@ where  $\n{2}{\wf{}{\Bas}}(\br{1},\br{2})$ is the opposite-spin two-body density
 	\n{2}{\wf{}{\Bas}}(\br{1},\br{2})
 	= \sum_{pqrs \in \Bas}  \SO{p}{1} \SO{q}{2} \Gam{pq}{rs}[\wf{}{\Bas}] \SO{r}{1} \SO{s}{2}, 
 \end{equation}
-$\Gam{pq}{rs}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\wf{}{\Bas}}$ is the opposite-spin two-body density tensor of $\wf{}{\Bas}$, $\SO{i}{}$ are spinorbitals, $\f{\wf{}{\Bas}}{}(\br{1},\br{2})$ is defined as
+$\Gam{pq}{rs}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\wf{}{\Bas}}$ is the opposite-spin two-body density tensor of $\wf{}{\Bas}$, $\SO{i}{}$ are spinorbitals, $\f{\Bas}{}(\br{1},\br{2})$ is defined as
 \begin{multline}
 	\label{eq:fbasis}
-	\f{\wf{}{\Bas}}{}(\br{1},\br{2}) 
+	\f{\Bas}{}(\br{1},\br{2}) 
 	\\
 	=  \sum_{pqrstu \in \Bas}  \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu}[\wf{}{\Bas}] \SO{t}{1} \SO{u}{2},
 \end{multline}
 and $\V{pq}{rs}$ are the usual Coulomb two-electron integrals. 
-The definition of equation \eqref{eq:def_weebasis} is the same of equation (27) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, but here we add the extra condition that $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ diverges when the two-body density vanishes, which ensures that one-electron systems do not have any basis set correction. 
-With such a definition, $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ verifies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
+The definition of equation \eqref{eq:def_weebasis} is the same of equation (27) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, but here we add the extra condition that $\W{\Bas}{}(\br{1},\br{2})$ diverges when the two-body density vanishes, which ensures that one-electron systems do not have any basis set correction. 
+With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ verifies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
 \begin{equation}
 	\label{eq:int_eq_wee}
-	\mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \iint \W{\wf{}{\Bas}}{}(\br{1},\br{2}) \n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \dbr{1} \dbr{2},
+	\mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \dbr{1} \dbr{2},
 \end{equation}
 where here $\hWee{}$ contains only the opposite-spins component of the two-electron interaction, and \eqref{eq:int_eq_wee} can be rewritten as
 \begin{equation}
-  \iint r_{12}^{-1} \n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \dbr{1} \dbr{2} =  \iint \W{\wf{}{\Bas}}{}(\br{1},\br{2}) \n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \dbr{1} \dbr{2},
+  \iint r_{12}^{-1} \n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \dbr{1} \dbr{2} =  \iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \dbr{1} \dbr{2},
 \end{equation}
-which intuitively motivates $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ as a potential candidate for an effective interaction. 
-As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ is symmetric, \textit{a priori} non translational nor rotational invariant if $\Bas$ does not have such symmetries. 
-An important quantity to define in the present context is $\W{\wf{}{\Bas}}{}(\br{})$ which is the value of the effective interaction at $\br{}$ for opposite spins at coalescence
+which intuitively motivates $\W{\Bas}{}(\br{1},\br{2})$ as a potential candidate for an effective interaction. 
+As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\Bas}{}(\br{1},\br{2})$ is symmetric, \textit{a priori} non translational nor rotational invariant if $\Bas$ does not have such symmetries. 
+An important quantity to define in the present context is $\W{\Bas}{}(\br{})$ which is the value of the effective interaction at $\br{}$ for opposite spins at coalescence
 \begin{equation}
   \label{eq:wcoal}
-  \W{\wf{}{\Bas}}{}(\br{}) =  \W{\wf{}{\Bas}}{}(\br{},{\br{}})
+  \W{\Bas}{}(\br{}) =  \W{\Bas}{}(\br{},{\br{}})
 \end{equation}
 and which is necessarily \textit{finite} at for an \textit{incomplete} basis set as long as the on-top two-body density is non vanishing. 
 
-Of course, there exists \textit{a priori} an infinite set of functions in ${\rm I\!R}^6$ satisfying \eqref{eq:int_eq_wee}, but thanks to its very definition one can show (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that 
+Of course, there exists \textit{a priori} an infinite set of functions in $\mathbb{R}^6$ satisfying \eqref{eq:int_eq_wee}, but thanks to its very definition one can show (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that 
 \begin{equation}
 	\label{eq:lim_W}
-	\lim_{\Bas \to \infty}\W{\wf{}{\Bas}}{}(\br{1},\br{2}) = r_{12}^{-1}\   
+	\lim_{\Bas \to \infty}\W{\Bas}{}(\br{1},\br{2}) = r_{12}^{-1}\   
 \end{equation}
 for all points $(\br{1},\br{2})$ such that $\n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \ne 0$ and for any choice of $\wf{}{\Bas}$, which therefore guarantees a physically satisfying limit. 
-An important point here is that, with the present definition of $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$, one can quantify the effect of the incompleteness of $\Bas$ on the Coulomb operator itself as a removal of the divergence of the two-electron interaction near the electron coalescence.  
-As it has been shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} (see for instance Fig 1,2 and 3 therein), choosing a HF wave function as $\wf{}{\Bas}$ to define the effective interaction $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ already provides a quantitative representation of the incompleteness of the basis set $\Bas$ for weakly correlated systems. 
+An important point here is that, with the present definition of $\W{\Bas}{}(\br{1},\br{2})$, one can quantify the effect of the incompleteness of $\Bas$ on the Coulomb operator itself as a removal of the divergence of the two-electron interaction near the electron coalescence.  
+As it has been shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} (see for instance Fig 1,2 and 3 therein), choosing a HF wave function as $\wf{}{\Bas}$ to define the effective interaction $\W{\Bas}{}(\br{1},\br{2})$ already provides a quantitative representation of the incompleteness of the basis set $\Bas$ for weakly correlated systems. 
 
 %=================================================================
 %\subsection{Range-separation function}
 %=================================================================
 As we can map the Coulomb operator within a basis set $\Bas$ with a non divergent two-electron interaction, we can link the present theory with the RS-DFT which uses the so-called long-range interaction which are smooth bounded two-electron operators. 
-To do so, we choose a range-separation \textit{function} $\rsmu{\wf{}{\Bas}}{}(\br{})$
+To do so, we choose a range-separation \textit{function} $\rsmu{\Bas}{}(\br{})$
 \begin{equation}
 	\label{eq:mu_of_r}
- \rsmu{\wf{}{\Bas}}{}(\br{})  = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{}(\br{}) 
+ \rsmu{\Bas}{}(\br{})  = \frac{\sqrt{\pi}}{2} \W{\Bas}{}(\br{}) 
 \end{equation}
-such that the long-range interaction $\w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\br{1},\br{2})$
+such that the long-range interaction $\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2})$
 \begin{equation}
-        \w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\br{1},\br{2}) = \frac{1}{2} \qty{ \frac{\erf[ \rsmu{\wf{}{\Bas}}{}(\br{1}) r_{12}]}{r_{12}} + \frac{\erf[ \rsmu{\wf{}{\Bas}}{}(\br{2}) r_{12}]}{ r_{12}} }
+        \w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) = \frac{1}{2} \qty{ \frac{\erf[ \rsmu{\Bas}{}(\br{1}) r_{12}]}{r_{12}} + \frac{\erf[ \rsmu{\Bas}{}(\br{2}) r_{12}]}{ r_{12}} }
 \end{equation}
-coincides with the effective interaction $\W{\wf{}{\Bas}}{}(\br{})$ for all points in ${\rm I\!R}^3$  
+coincides with the effective interaction $\W{\Bas}{}(\br{})$ for all points in $\mathbb{R}^3$  
 \begin{equation}
-        \w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\br{},\br{}) = \W{\wf{}{\Bas}}{}(\br{})\quad \forall \,\, \br{}. 
+        \w{}{\lr,\rsmu{\Bas}{}}(\br{},\br{}) = \W{\Bas}{}(\br{}). 
 \end{equation}
 
 
@@ -417,7 +415,7 @@ coincides with the effective interaction $\W{\wf{}{\Bas}}{}(\br{})$ for all poin
 %where $(\br{},\Bar{\br{}})$ means a couple of anti-parallel spins at the same position $\br{}$, 
 %we propose a fit for each point in $\rnum^3$ of $\wbasiscoal{ }$ with a long-range-like interaction:
 %\begin{equation}
-%  \wbasiscoal{} = \w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\bfrb{},\bfrb{})
+%  \wbasiscoal{} = \w{}{\lr,\rsmu{\Bas}{}}(\bfrb{},\bfrb{})
 %\end{equation}
 %where the long-range-like interaction is defined as
 %\begin{equation}
@@ -426,7 +424,7 @@ coincides with the effective interaction $\W{\wf{}{\Bas}}{}(\br{})$ for all poin
 %Equation \eqref{eq:def_wcoal} is equivalent to the following condition
 %\begin{equation}
 % \label{eq:mu_of_r}
-% 	\rsmu{\wf{}{\Bas}}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{}(\br{})
+% 	\rsmu{\Bas}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{}(\br{})
 %\end{equation}
 %As we defined an effective interaction for the valence electrons, we also introduce a valence range-separation parameter as
 %\begin{equation}
@@ -444,7 +442,7 @@ coincides with the effective interaction $\W{\wf{}{\Bas}}{}(\br{})$ for all poin
 %and therefore 
 %\begin{equation}
 %\label{eq:lim_mur}
-%	\lim_{\Bas \rightarrow \infty} \rsmu{\wf{}{\Bas}}{}(\br{}) = \infty
+%	\lim_{\Bas \rightarrow \infty} \rsmu{\Bas}{}(\br{}) = \infty
 %%\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, .
 %\end{equation}
 
@@ -455,7 +453,7 @@ coincides with the effective interaction $\W{\wf{}{\Bas}}{}(\br{})$ for all poin
 %=================================================================
 %\label{sec:ecmd}
 
-Once defined the range-separation function $\rsmu{\wf{}{\Bas}}{}(\br{})$, we can use the functionals defined in the field of RS-DFT to approximate $\bE{}{\Bas}[\n{}{}]$. As in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ using a specific class of short-range correlation functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
+Once defined the range-separation function $\rsmu{\Bas}{}(\br{})$, we can use the functionals defined in the field of RS-DFT to approximate $\bE{}{\Bas}[\n{}{}]$. As in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ using a specific class of short-range correlation functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
 \begin{multline}
   \label{eq:ec_md_mu}
   \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
@@ -486,14 +484,14 @@ The ECMD functionals admit, for any density $\n{}{}(\br{})$, the two following l
 \end{subequations}
 where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT. 
 The choice of the ECMD as the functionals to be used in this scheme is motivated by the analogy between the definition of $\bE{}{\Bas}[\n{}{}]$ [see equation \eqref{eq:E_funcbasis}] and that of the ECMD functionals [see equation \eqref{eq:ec_md_mu}]. 
-Indeed, provided that $\w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\br{1},\br{2}) \approx \W{\wf{}{\Bas}}{}(\br{1},\br{2})$, then the wave function $\wf{}{\rsmu{\wf{}{\Bas}}{}(\br{})}$ coincides with $\wf{}{\Bas}$. 
+Indeed, provided that $\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) \approx \W{\Bas}{}(\br{1},\br{2})$, then the wave function $\wf{}{\rsmu{\Bas}{}(\br{})}$ coincides with $\wf{}{\Bas}$. 
 %The ECMD 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.
 %This makes them particularly well adapted to the present context where one aims at correcting a general WFT method. 
 
 %--------------------------------------------
 %\subsubsection{Local density approximation}
 %--------------------------------------------
-Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ by the ECMD functionals evaluated with the range separation function $\rsmu{\wf{}{\Bas}}{}(\br{})$. Therefore, we define the LDA version of $\bE{}{\Bas}[\n{}{}]$ as 
+Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ by the ECMD functionals evaluated with the range separation function $\rsmu{\Bas}{}(\br{})$. Therefore, we define the LDA version of $\bE{}{\Bas}[\n{}{}]$ as 
 \begin{equation}
 	\label{eq:def_lda_tot}
         \bE{\LDA}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\LDA}{\sr}\big(\n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{},
@@ -522,7 +520,7 @@ Therefore, the PBE complementary functional reads
 	\label{eq:def_pbe_tot}
         \bE{\PBE}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\PBE}{\sr}\big(\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{}.
 \end{equation}
-Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\wf{}{\Bas}}{}]$ is then equal to $\bE{\LDA}{\sr}[\n{\wf{}{\Bas}}{}(\br{}),\rsmu{\wf{}{\Bas}}{}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\wf{}{\Bas}}{}(\br{}),\rsmu{\wf{}{\Bas}}{}(\br{})]$ where $\rsmu{\wf{}{\Bas}}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
+Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\wf{}{\Bas}}{}]$ is then equal to $\bE{\LDA}{\sr}[\n{\wf{}{\Bas}}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\wf{}{\Bas}}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ where $\rsmu{\Bas}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
 
 %The general scheme for estimating $\ecompmodel$ is the following. Consider a given approximated ECMD functional $\ecmuapprox$  labelled by ECMD-$\mathcal{X}$. 
 %Such a functional of the density $\denr$ (and potentially its derivatives $\nabla \denr$) is defined for any value of the range-separation parameter $\mu$.  
@@ -601,7 +599,7 @@ It is worth noting that, within the present definition, $\W{\wf{}{\Bas}}{\Val}(\
 Defining $\n{\wf{}{\Bas}}{\Val}$ as the valence one-electron density, the valence part of the complementary functional $\bE{}{\Val}[\n{\wf{}{\Bas}}{\Val}]$ is then evaluated as $\bE{\LDA}{\sr}[\n{\wf{}{\Bas}}{\Val}(\br{}),\rsmu{\wf{}{\Bas}}{\Val}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\wf{}{\Bas}}{\Val}(\br{}),\rsmu{\wf{}{\Bas}}{\Val}(\br{})]$.
 
 Regarding now the main computational source of the present approach, it consists in the evaluation 
-of $\W{\wf{}{\Bas}}{}(\br{})$ [See Eqs.~\eqref{eq:wcoal}] at each quadrature grid point. 
+of $\W{\Bas}{}(\br{})$ [See Eqs.~\eqref{eq:wcoal}] at each quadrature grid point. 
 All through this paper, we use two-body density matrix of a single Slater determinant (typically HF) 
 for $\Gam{rs}{tu}[\wf{}{\Bas}]$ and therefore the computational bottleneck reduces to the evaluation 
 at each quadrature grid point of 
@@ -615,7 +613,7 @@ When the four-index transformation become prohibitive, by performing successive
 
 To conclude this theory session, it is important to notice that the basis set correction proposed here has the folowing properties whatever the approximations made in the DFT part: i) it can be applied to any WFT model that provides an energy and a density, ii) it vanishes for one-electron systems, 
 iii) it vanishes in the limit of a complete basis set and thus garentees the correct CBS limit of the WFT model. 
-%, because of the behaviour of the effective interaction [see \eqref{eq:lim_W}], the range separation function $\rsmu{\wf{}{\Bas}}{}(\br{})$ tends to infinity. Therefore, in the limit of a complete basis set, according to equation \eqref{eq:large_mu_ecmd}, the complementary functional tends to zero \textit{whatever the approximated functional} used for the DFT part. 
+%, because of the behaviour of the effective interaction [see \eqref{eq:lim_W}], the range separation function $\rsmu{\Bas}{}(\br{})$ tends to infinity. Therefore, in the limit of a complete basis set, according to equation \eqref{eq:large_mu_ecmd}, the complementary functional tends to zero \textit{whatever the approximated functional} used for the DFT part. 
 
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 \section{Results}