notations alsmost OK

Manuscript/G2-srDFT.tex

@@ -94,14 +94,13 @@
 % operators
 \newcommand{\hT}{\Hat{T}}
 \newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}}
+\newcommand{\updw}{\uparrow\downarrow}
 \newcommand{\f}[2]{f_{#1}^{#2}}
 \newcommand{\Gam}[2]{\Gamma_{#1}^{#2}}
 
 % coordinates
 \newcommand{\br}[1]{\mathbf{r}_{#1}}
-%\newcommand{\br}[1]{\mathbf{x}_{#1}}
 \newcommand{\dbr}[1]{d\br{#1}}
-%\newcommand{\dbr}[1]{d\br{#1}}
 
 \newcommand{\ra}{\rightarrow}
 \newcommand{\De}{D_\text{e}}
@@ -215,99 +214,9 @@ Importantly, in the limit of a complete basis set (which we refer to as $\Bas \t
         \lim_{\Bas \to \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\modY}{\Bas}] ) = \E{\modX}{} \approx E,
 \end{equation}
 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 $\E{\FCI}{} = \E{}{}$.
+In the case $\modX = \FCI$, we have a 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 
-%\begin{equation}
-% \efci \approx \emodel
-%\end{equation}
-%and 
-%\begin{equation}
-% \denfci \approx \denmodel
-%\end{equation}
-%and by defining the energy provided by the model $\model$ in the complete basis set 
-%\begin{equation}
-% \emodelcomplete = \lim_{\Bas \rightarrow \infty} \emodel\,\, , 
-%\end{equation}
-%we can then write 
-%\begin{equation}
-% \emodelcomplete \approx \emodel + \ecompmodel
-%\end{equation}
-%which verifies the correct limit since 
-%\begin{equation}
-% \lim_{\Bas \rightarrow \infty} \ecompmodel = 0\,\, .
-%\end{equation}
-
-
-%=================================================================
-%\subsection{Basis set correction for the CIPSI algorithm and the CCSD(T) ansatz}
-%=================================================================
-%In this work we propose to apply the basis set correction to a selected CI algorithm, namely the CIPSI algorithm, and to the CCSD(T) ansatz in %order to speed-up the basis set convergence of these models. 
-
-%=================================================================
-%\subsubsection{Basis set correction for the CCSD(T) energy}
-%=================================================================
-%The CCSD(T) method is a very popular WFT approach which is known to provide very good estimation of the correlation energies for weakly correlated systems, whose wave function are dominated by the HF Slater determinant. 
-%Defining $\ecc$ as the CCSD(T) energy obtained in $\Bas$, in the present notations we have 
-%\begin{equation}
-% \emodel = \ecc \,\, .
-%\end{equation}
-%In the context of the basis set correction, one needs to choose a density as the density of the model $\denmodel$, and we chose here the HF density
-%\begin{equation}
-% \denmodel = \denhf \,\, .
-%\end{equation}
-%Such a choice can be motivated by the fact that the correction to the HF density brought by the excited Slater determinants are at least of second-order in perturbation theory. 
-%Therefore, we approximate the complete basis set CCSD(T) energy $\ecccomplete$ by
-%\begin{equation}
-% \ecccomplete \approx \ecc + \efuncden{\denhf} \,\, .
-%\end{equation}
-
-%=================================================================
-%\subsubsection{Correction of the CIPSI algorithm}
-%=================================================================
-%The CIPSI algorithm approximates the FCI wave function through an iterative selected CI procedure, and the FCI energy through a second-order multi-reference perturbation theory.
-%The CIPSI algorithm belongs to the general class of methods build upon selected CI\cite{bender,malrieu,buenker1,buenker-book,three_class_CIPSI,harrison,hbci}
-%which have been successfully used to converge to FCI correlation energies, one-body properties, and nodal surfaces.\cite{three_class_CIPSI,Rubio198698,cimiraglia_cipsi,cele_cipsi_zeroth_order,Angeli2000472,canadian,atoms_3d,f2_dmc,atoms_dmc_julien,GinTewGarAla-JCTC-18,SceBenJacCafLoo-JCP-18,LooSceBloGarCafJac-JCTC-18,LooBogSceCafJAc-JCTC-19}
-%The CIPSI algorithm used in this work uses iteratively enlarged selected CI spaces and
-%Epstein--Nesbet\cite{epstein,nesbet} multi-reference perturbation theory. Within a basis set $\Bas$, the CIPSI energy is
-%\begin{align}
-%  E_\mathrm{CIPSI}^{\Bas} &= E_\text{v} + E^{(2)} \,\,,
-%\end{align}
-%where $E_\text{v}$ is the variational energy
-%\begin{align}
-% E_\text{v} &= \min_{\{ c_{\rm I}\}} \frac{\elemm{\Psi^{(0)}}{\hat{H}}{\Psi^{(0)}}  }{\ovrlp{\Psi^{(0)}}{\Psi^{(0)}}}\,\,,
-%\end{align}
-%where the reference wave function $\ket{\Psi^{(0)}} = \sum_{{\rm I}\,\in\,\mathcal{R}} \,\,c_{\rm I} \,\,\ket{\rm I}$ is expanded in Slater determinants I within the CI reference space $\mathcal{R}$, and $E^{(2)}$ is the second-order energy correction
-%\begin{align}
-%  E^{(2)} &= \sum_{\kappa} \frac{|\elemm{\Psi^{(0)}}{\hat{H}}{\kappa}|^2}{E_\text{v} - \elemm{\kappa}{H}{\kappa}} = \sum_{\kappa} \,\, e_{\kappa}^{(2)} \,\, ,
-%\end{align}
-%where $\kappa$ denotes a determinant outside $\mathcal{R}$.
-%To reduce the cost of the evaluation of the second-order energy correction, the semi-stochastic multi-reference approach
-%of Garniron \textit{et al.} \cite{stochastic_pt_yan} was used, adopting the technical specifications recommended in that work.
-%The CIPSI energy is systematically refined by doubling the size of the CI reference space at each iteration, selecting
-%the determinants $\kappa$ with the largest $\vert e_{\kappa}^{(2)} \vert$. 
-%In order to reach a faster convergence of the estimation of the FCI energy, we use the extrapolated FCI energy (exFCI) proposed by Holmes \textit{et al}\cite{HolUmrSha-JCP-17} which we refer here as $\EexFCIbasis$. 
-%
-%In the context of the basis set correction, we use the following conventions
-%\begin{equation}
-% \emodel = \EexFCIbasis
-%\end{equation}
-%\begin{equation}
-% \denmodelr = \dencipsir
-%\end{equation}
-%where the density $\dencipsir$ is defined as
-%\begin{equation}
-% \dencipsi = \sum_{ij \in \Bas} \elemm{\Psi^{(0)}}{\aic{i}\ai{j}}{\Psi^{(0)}} \phi_i(\bfrb{} ) \phi_j(\bfrb{} ) \,\, ,
-%\end{equation}
-%and $\phi_i(\bfrb{} )$ are the spin orbitals in the MO basis evaluated at $\bfrb{}$. As it was shown in \onlinecite{GinPraFerAssSavTou-JCP-18} that the CIPSI density converges rapidly with the size of $\Psi^{(0)}$ for weakly correlated systems, $\dencipsir$ can be thought as a reasonable approximation of the FCI density $\denfci$. 
-%
-%Finally, we approximate complete basis set exFCI energy $\EexFCIinfty$ as
-%\begin{equation}
-% \EexFCIinfty \approx \EexFCIbasis + \efuncden{\dencipsi}
-%\end{equation}
-
 Rigorously speaking,  the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$. 
 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. 
@@ -318,32 +227,16 @@ Contrary to the conventional RS-DFT scheme which requires a range-separated \tex
 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. 
+In the final step, we employ short-range density functionals \cite{TouGorSav-TCA-05} with $\rsmu{\Bas}{}(\br{})$ 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}). 
 
-
+We define the effective operator as 
-%=================================================================
-%\subsection{General scheme for the approximation of the unknown complementary functional $\efuncbasis$}
-%=================================================================
-
-
-%=================================================================
-%\subsection{Definition of a real-space representation of the coulomb operator truncated in a basis set $\Bas$}
-%\label{sec:weff}
-%=================================================================
-%=================================================================
-%\subsection{Effective Coulomb operator}
-%=================================================================
-We define the effective operator $\W{\Bas}{}(\br{1},\br{2})$ as 
 \begin{equation}
 	\label{eq:def_weebasis}
 	\W{\Bas}{}(\br{1},\br{2})   = 
  	\begin{cases}
        \f{\Bas}{}(\br{1},\br{2})/\n{2}{}(\br{1},\br{2}), 	& \text{if $\n{2}{}(\br{1},\br{2}) \ne 0$,}
        \\
-       \infty, 															& \text{otherwise.}
+       \infty, 												& \text{otherwise,}
     \end{cases}
 \end{equation}
 where
@@ -352,108 +245,56 @@ where
 	\n{2}{}(\br{1},\br{2})
 	= \sum_{pqrs \in \Bas}  \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}
 \end{equation}
-is the opposite-spin two-body density associated with $\wf{}{\Bas}$, $\Gam{pq}{rs} = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\wf{}{\Bas}}$ is the opposite-spin two-body density tensor of $\wf{}{\Bas}$, $\SO{p}{}$ is a spinorbital, $\f{\Bas}{}(\br{1},\br{2})$ is defined as
+and $\Gam{pq}{rs} = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\wf{}{\Bas}}$ are the opposite-spin two-body density and density tensor (respectively) associated with $\wf{}{\Bas}$, $\SO{p}{}$ is a spinorbital,
 \begin{equation}
 	\label{eq:fbasis}
 	\f{\Bas}{}(\br{1},\br{2}) 
 	= \sum_{pqrstu \in \Bas}  \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
 \end{equation}
 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 $\W{\Bas}{}(\br{1},\br{2})$ diverges when the two-body density vanishes, which ensures that one-electron systems do not have any basis set correction. 
+Note that the divergence condition of $\W{\Bas}{}(\br{1},\br{2})$ in Eq.~\eqref{eq:def_weebasis} ensures that one-electron systems do not have any basis set correction. 
 \PFL{I don't agree with this. There must be a correction for one-electron system.
 However, it does not come from the e-e cusp but from the e-n cusp.}
-With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ verifies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
+With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ satisfies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
 \begin{equation}
 	\label{eq:int_eq_wee}
-	\mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{}(\br{1},\br{2}) \dbr{1} \dbr{2},
+	\mel*{\wf{}{\Bas}}{\hWee{\updw}}{\wf{}{\Bas}} = \iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{}(\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
+where $\hWee{\updw}$ contains only the opposite-spin component of $\hWee{}$.
+Because Eq.~\eqref{eq:int_eq_wee} can be rewritten as
 \begin{equation}
   \iint r_{12}^{-1} \n{2}{}(\br{1},\br{2}) \dbr{1} \dbr{2} =  \iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{}(\br{1},\br{2}) \dbr{1} \dbr{2},
 \end{equation}
-which intuitively motivates $\W{\Bas}{}(\br{1},\br{2})$ as a potential candidate for an effective interaction. 
+it 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
+An important quantity to define in the present context is the value of the effective interaction at coalescence of opposite-spin electrons
 \begin{equation}
 	\label{eq:wcoal}
-  \W{\Bas}{}(\br{}) =  \W{\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. 
+and which is necessarily \textit{finite} for an 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 $\mathbb{R}^6$ 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 
+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 definition one can show that (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) 
 \begin{equation}
 	\label{eq:lim_W}
 	\lim_{\Bas \to \infty}\W{\Bas}{}(\br{1},\br{2}) = r_{12}^{-1}\   
 \end{equation}
-for any $(\br{1},\br{2})$ such that $\n{2}{}(\br{1},\br{2}) \ne 0$ and for any $\wf{}{\Bas}$, which therefore guarantees a physically satisfying limit. 
+for any $(\br{1},\br{2})$ such that $\n{2}{}(\br{1},\br{2}) \ne 0$ and for any $\wf{}{\Bas}$, which guarantees a physically satisfying limit. 
-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.  
+%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 shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, 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 $\Bas$ for weakly correlated systems. 
+%As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, 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 $\Bas$ for weakly correlated systems. 
 
-%=================================================================
+Because the Coulomb operator within a basis set $\Bas$ is a non divergent two-electron interaction, we can straightforwardly link the present theory with the RS-DFT which uses the so-called long-range interaction which are smooth bounded two-electron operators. 
-%\subsection{Range-separation function}
+To do so, we choose a range-separation \textit{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} $\rsmu{\Bas}{}(\br{})$
 \begin{equation}
 	\label{eq:mu_of_r}
  \rsmu{\Bas}{}(\br{})  = \frac{\sqrt{\pi}}{2} \W{\Bas}{}(\br{}) 
 \end{equation}
-such that the long-range interaction $\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2})$
+such that the long-range interaction
 \begin{equation}
         \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}
 \PFL{This expression looks like a cheap spherical average.}
-coincides with the effective interaction $\W{\Bas}{}(\br{})$ for all points in $\mathbb{R}^3$  
+coincides with the effective interaction $\W{\Bas}{}(\br{})$ at coalescence, i.e.~$\w{}{\lr,\rsmu{\Bas}{}}(\br{},\br{}) = \W{\Bas}{}(\br{})$.
-\begin{equation}
-        \w{}{\lr,\rsmu{\Bas}{}}(\br{},\br{}) = \W{\Bas}{}(\br{}). 
-\end{equation}
-
-
-%More precisely, if we define the value of the interaction at coalescence as 
-%\begin{equation}
-% \label{eq:def_wcoal}
-% \wbasiscoal{} = W_{\wf{}{\Bas}}(\bfr{},\bar{{\bf x}}_{}).
-%\end{equation}
-%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{\Bas}{}}(\bfrb{},\bfrb{})
-%\end{equation}
-%where the long-range-like interaction is defined as
-%\begin{equation}
-%	\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}} }.  
-%\end{equation}
-%Equation \eqref{eq:def_wcoal} is equivalent to the following condition
-%\begin{equation}
-% \label{eq:mu_of_r}
-% 	\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}
-% \label{eq:mu_of_r_val}
-% \murpsival = \frac{\sqrt{\pi}}{2} \, \wbasiscoalval{} \, .
-%\end{equation}
-%An important point to notice is that, in the limit of a complete basis set $\Bas$, as 
-%\begin{equation}
-%\label{eq:lim_W}
-%	\lim_{\Bas \rightarrow \infty}\wbasis = r_{12}^{-1}  \quad \forall  (\br{1},\br{2})
-%% &\lim_{\Bas \rightarrow \infty}\wbasisval = 1/r_{12} \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\,\, , 
-%\end{equation}
-%one has $\lim_{\Bas \rightarrow \infty} \wbasiscoal{} = \infty$
-%% &\lim_{\Bas \rightarrow \infty} \wbasiscoalval{} = +\infty\,\,, 
-%and therefore 
-%\begin{equation}
-%\label{eq:lim_mur}
-%	\lim_{\Bas \rightarrow \infty} \rsmu{\Bas}{}(\br{}) = \infty
-%%\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, .
-%\end{equation}
-
-
-
-%=================================================================
-%\subsection{Complementary functional}
-%=================================================================
-%\label{sec:ecmd}
 
 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}
@@ -486,24 +327,17 @@ 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 $\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) \approx \W{\Bas}{}(\br{1},\br{2})$, then the wave function $\wf{}{\rsmu{\Bas}{}(\br{})}$ 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 $\wf{}{\rsmu{\Bas}{}}$ 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 $\rsmu{\Bas}{}(\br{})$. 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{},
 \end{equation}
 where $\be{\LDA}{\sr}(\n{}{},\rsmu{}{})$ is the short-range ECMD per particle of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} for which a parametrization can be found in Ref.~\onlinecite{PazMorGorBac-PRB-06}. 
-%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.
 
-%--------------------------------------------
-%\subsubsection{New PBE functional}
-%--------------------------------------------
 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 $\mu$. 
 In order to correct such a defect, we propose here a new ECMD functional inspired by the recent functional proposed by some of the present authors \cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional $\e{\PBE}{}(\n{}{},\nabla \n{}{})$ for $\rsmu{}{}=0$ and the exact large-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGorBac-PRB-06} yielding
 \begin{subequations}
@@ -515,8 +349,8 @@ In order to correct such a defect, we propose here a new ECMD functional inspire
 	\beta(n,\nabla n,\rsmu{}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{\n{2}{\UEG}(\n{}{})}.
 \end{gather}
 \end{subequations}
-The difference between the ECMD PBE functional defined in Ref.~\onlinecite{FerGinTou-JCP-18} and the present expression \eqref{eq:epsilon_cmdpbe} is that we approximate here the \textit{exact} ground-state on-top pair density of the system $\n{2}{}(\br{})$ by its UEG version, i.e.~$\n{2}{}(\br{})  \approx \n{2}{\UEG}(\br{}) =  \left(\n{}{}(\br{})\right)^2  g_0(\n{}{}(\br{}) )$, where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. 
+The difference between the ECMD PBE functional defined in Ref.~\onlinecite{FerGinTou-JCP-18} and the present expression \eqref{eq:epsilon_cmdpbe} is that we approximate here the \textit{exact} ground-state on-top pair density of the system $\n{2}{}(\br{})$ by its UEG version, i.e.~$\n{2}{}(\br{})  \approx \n{2}{\UEG}(\br{}) =  \n{}{}(\br{})^2  g_0(\n{}{}(\br{}) )$, where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. 
-This represents a major computational saving without loss of performance as we eschew the computation of $\n{2}{}$.
+This represents a major computational saving without loss of performance as we eschew the computation of $\n{2}{}(\br{})$.
 Therefore, the PBE complementary functional reads
 \begin{equation}
 	\label{eq:def_pbe_tot}
@@ -524,27 +358,9 @@ Therefore, the PBE complementary functional reads
 \end{equation}
 Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modY}{}]$ is then equal to $\bE{\LDA}{\sr}[\n{\modY}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\modY}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ 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 $\mu$.  
-%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 $\denmodel$
-%\begin{equation}
-% \label{eq:approx_ecfuncbasis}
-% \ecompmodel \approx \ecmuapproxmurmodel
-%\end{equation}
-%Therefore, any approximated ECMD can be used to estimate $\ecompmodel$. 
-%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 
-%\begin{equation}
-% \lim_{\Bas \rightarrow \infty} \ecmuapproxmurmodel = 0 \quad ,
-%\end{equation}
-%for whatever choice of density $\denmodel$, wave function $\wf{}{\Bas}$ used to define the interaction, and ECMD functional used to approximate the exact ECMD. 
-
-%=================================================================
-%\subsection{Valence effective interaction}
-%=================================================================
 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. 
 We then naturally split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor$ and $\Val$ are its core and valence parts, respectively.% and $\Cor \bigcap \Val = \O$.
 
-%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 $\wf{}{\Bas}$. 
 We therefore define the valence-only effective interaction 
  \begin{equation}
 	\W{\Bas}{\Val}(\br{1},\br{2})  = 
@@ -559,44 +375,19 @@ with
 \begin{gather}
 	\label{eq:fbasisval}
 	\f{\Bas}{\Val}(\br{1},\br{2})
-	=  \sum_{pq \in \Bas} \sum_{rstu \in \Val}  \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu}[{\wf{}{\Bas}}] \SO{t}{1} \SO{u}{2},
+	=  \sum_{pq \in \Bas} \sum_{rstu \in \Val}  \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
 	\\
 	\n{2}{\Val}(\br{1},\br{2})
 	= \sum_{pqrs \in \Val}  \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}, 
 \end{gather}
 \end{subequations}
-and the corresponding valence range separation function $\rsmu{\Bas}{\Val}(\br{})$
+and the corresponding valence range separation function
 \begin{equation}
 	\label{eq:muval}
 	\rsmu{\Bas}{\Val}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{\Val}(\br{},\br{}).
 \end{equation}
-%\begin{equation}
-% \twodmrdiagpsival = \sum_{klmn \in \Val} \SO{m}{1} \SO{n}{2} \gammamnkl[\wf{}{\Bas}] \SO{k}{1} \SO{l}{2} .
-%\end{equation}
-%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 $\Basval$. 
 It is worth noting that, within the present definition, $\W{\Bas}{\Val}(\br{1},\br{2})$ still satisfies Eq.~\eqref{eq:lim_W}.
 
-%We now introduce a valence-only approximation for the complementary functional which is needed to correct for frozen core WFT models. 
-%Defining the valence one-body spin density matrix as
-%\begin{equation}
-% \begin{aligned}
-% \onedmval[\wf{}{\Bas}] & = \elemm{\wf{}{\Bas}}{a^{\dagger}_{i,\sigma} a_{j,\sigma}}{\wf{}{\Bas}} \qquad \text{if }(i,j)\in \Basval \\
-%                      & = 0 \qquad \text{in other cases}
-% \end{aligned}
-%\end{equation}
-%then one can define the valence density as:
-%\begin{equation}
-%  \denval_{\sigma}({\bf r}) = \sum_{i,j} \onedmval[\wf{}{\Bas}] \phi_i({\bf r}) \phi_j({\bf r}) 
-%\end{equation}
-%Therefore, we propose the following valence-only approximations for the complementary functional 
-%\begin{equation}
-% \label{eq:def_lda_tot}
-%  \ecompmodelldaval = \int \, \text{d}{\bf r} \,\, \denval({\bf r}) \,\, \emuldaval\,,
-%\end{equation}
-%\begin{equation}
-% \label{eq:def_lda_tot}
-%  \ecompmodelpbeval = \int \, \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(\denval({\bf r}),\nabla \denval({\bf r});\,\murval)
-%\end{equation}
 Defining $\n{\modY}{\Val}$ as the valence one-electron density obtained with the model $\modY$, the valence part of the complementary functional $\bE{}{\Val}[\n{\modY}{\Val}]$ is then evaluated as $\bE{\LDA}{\sr}[\n{\modY}{\Val}(\br{}),\rsmu{\Bas}{\Val}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\modY}{\Val}(\br{}),\rsmu{\Bas}{\Val}(\br{})]$.
 
 Regarding now the main computational source of the present approach, it consists in the evaluation 
@@ -606,15 +397,13 @@ for $\Gam{rs}{tu}$ and therefore the computational bottleneck reduces to the eva
 at each quadrature grid point of 
 \begin{equation}
 	\label{eq:fcoal}
-        \f{\Bas}{\HF}(\br{}) = \sum_{p,q\in\Bas} \sum_{i\in \nocca} \sum_{j\in \noccb}  \SO{p}{} \SO{q}{} \V{pq}{ij} \SO{i}{} \SO{j}{}
+        \f{\Bas}{\HF}(\br{}) = \sum_{pq \in \Bas} \sum_{i \in \nocca} \sum_{j\in \noccb}  \SO{p}{} \SO{q}{} \V{pq}{ij} \SO{i}{} \SO{j}{}
 \end{equation}
 which scales as $\Nb^2\times N_{elec}^2 \times \Ng$ and is embarassingly parallel. Within the present formulation, the bottleneck is the four-index transformation to obtain the two-electron integrals on the MO basis which appear in \eqref{eq:fcoal}. Nevertheless, this step has in general to be performed before a correlated WFT calculations and therefore it represent a minor limitation. 
 When the four-index transformation become prohibitive, by performing successive matrix multiplications, one could rewrite the equations directly in the AO basis where it scales formally as $\order{\Ng \Nb^4}$  but where one can take advantage of the sparsity atomic-orbital-based algorithms to significantly speed up the calculations.  
 
-
 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 $\rsmu{\Bas}{}(\br{})$ 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. 
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 \section{Results}