diff --git a/Manuscript/G2-srDFT.tex b/Manuscript/G2-srDFT.tex
index 5fbb346..4d206ba 100644
--- a/Manuscript/G2-srDFT.tex
+++ b/Manuscript/G2-srDFT.tex
@@ -414,6 +414,101 @@ As the e-e cusp originates from the divergence of the Coulomb operator at $r_{12
 Therefore, the impact of the incompleteness of $\Bas$ can be viewed as a removal of the divergence of the Coulomb interaction at $r_{12} = 0$. 
 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. 
 
+
+
+%=================================================================
+\subsection{Complementary functional}
+%=================================================================
+\label{sec:ecmd}
+
+Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we propose here to approximate $\bE{}{\Bas}[\n{}{}]$ using a specific class of SR-DFT energy functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
+\begin{multline}
+  \label{eq:ec_md_mu}
+  \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
+  \\
+  - \mel*{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]},
+\end{multline}
+where $\wf{}{\rsmu{}{}}[\n{}{}(\br{})]$ is defined by the constrained minimization
+\begin{equation}
+\label{eq:argmin}
+	\wf{}{\rsmu{}{}}[\n{}{}(\br{})] = \arg \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
+\end{equation}
+with
+\begin{equation}
+\label{eq:weemu}
+	\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})
+\end{equation}
+and
+\begin{equation}
+\label{eq:erf}
+	\w{}{\lr,\rsmu{}{}}(r_{12}) = \frac{\erf(\rsmu{}{} r_{12})}{r_{12}}.
+\end{equation}
+is the long-range part of the Coulomb operator.
+The ECMD functionals admit, for any density $\n{}{}(\br{})$, the two following limiting forms:
+\begin{subequations}
+\begin{align}
+	\label{eq:large_mu_ecmd}
+	\lim_{\mu \to \infty}  \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] & = 0,
+	\\
+	\label{eq:small_mu_ecmd}
+	\lim_{\mu \to 0}  \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] & = \Ec[\n{}{}(\br{})],
+\end{align}
+\end{subequations}
+where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT. 
+These functionals differ from the standard RS-DFT correlation functional by the fact that the reference is not the KS Slater determinant but a multi-determinantal wave function, which makes them much more adapted in the present context where one aims at correcting a general multi-determinant WFT model. 
+
+%--------------------------------------------
+%\subsubsection{Local density approximation}
+%--------------------------------------------
+Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we define the LDA version of ECMD as 
+\begin{equation}
+	\label{eq:def_lda_tot}
+	\bE{\LDA}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \int \be{\UEG}{\sr}[\n{}{}(\br{}),\rsmu{}{}] \n{}{}(\br{}) \dbr{}
+\end{equation}
+where $\be{\UEG}{\sr}[\n{}{}(\br{}),\rsmu{}{}]$ is the multi-determinant short-range correlation energy per particle of the uniform electron gas (UEG) for which a parametrization can be found in Ref.~\onlinecite{PazMorGorBac-PRB-06}. 
+%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 recently proposed functional of some of the present authors \cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional $\e{}{\PBE}(\n{}{},\nabla \n{}{})$ for small $\rsmu{}{}$ and the exact large-$\rsmu{}{}$ behaviour, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGori-PRB-06} yielding
+\begin{subequations}
+\begin{gather}
+	\label{eq:epsilon_cmdpbe}
+	\be{\PBE}{\sr}(\n{}{},\nabla \n{}{},\rsmu{}{}) = \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{1 + \beta(n,\nabla n, \mu)\mu^3 }
+	\\
+	\label{eq:epsilon_cmdpbe}
+	\beta(n,\nabla n;\,\mu) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{\n{\UEG}{(2)}(\n{}{})}.
+\end{gather}
+\end{subequations}
+The difference between the ECMD PBE functional defined in Ref.~\cite{FerGinTou-JCP-18} and the present expression is that we approximate the \textit{exact} ground-state on-top pair density of the system $\n{}{(2)}(\br{})$ by its UEG version, i.e.
+\begin{equation}
+ \label{eq:ueg_ontop}
+	\n{}{(2)}(\br{}) \approx \n{\UEG}{(2)}(\br{}) =  \n{}{}(\br{})^2 g_0[\n{}{}(\br{})]
+\end{equation}
+where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. 
+Therefore, the PBE complementary function reads
+\begin{equation}
+	\label{eq:def_lda_tot}
+	\bE{\PBE}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \int \be{\PBE}{\sr}[\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{}{}] \n{}{}(\br{}) \dbr{}
+\end{equation}
+
+
+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{Effective Coulomb operator}
 %=================================================================
@@ -502,99 +597,6 @@ and therefore
 %\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, .
 \end{equation}
 
-
-%=================================================================
-\subsection{Complementary functional}
-%=================================================================
-\label{sec:ecmd}
-
-Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we propose here to approximate $\bE{}{\Bas}[\n{}{}]$ using a specific class of SR-DFT energy functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
-\begin{multline}
-  \label{eq:ec_md_mu}
-  \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
-  \\
-  - \mel*{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]},
-\end{multline}
-where $\wf{}{\rsmu{}{}}[\n{}{}(\br{})]$ is defined by the constrained minimization
-\begin{equation}
-\label{eq:argmin}
-	\wf{}{\rsmu{}{}}[\n{}{}(\br{})] = \arg \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
-\end{equation}
-with
-\begin{equation}
-\label{eq:weemu}
-	\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})
-\end{equation}
-and
-\begin{equation}
-\label{eq:erf}
-	\w{}{\lr,\rsmu{}{}}(r_{12}) = \frac{\erf(\rsmu{}{} r_{12})}{r_{12}}.
-\end{equation}
-is the long-range part of the Coulomb operator.
-The ECMD functionals admit, for any density $\n{}{}(\br{})$, the two following limiting forms:
-\begin{subequations}
-\begin{align}
-	\label{eq:large_mu_ecmd}
-	\lim_{\mu \to \infty}  \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] & = 0,
-	\\
-	\label{eq:small_mu_ecmd}
-	\lim_{\mu \to 0}  \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] & = \Ec[\n{}{}(\br{})],
-\end{align}
-\end{subequations}
-where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT. 
-These functionals differ from the standard RS-DFT correlation functional by the fact that the reference is not the KS Slater determinant but a multi-determinantal wave function, which makes them much more adapted in the present context where one aims at correcting a general multi-determinant WFT model. 
-
-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. 
-
-%--------------------------------------------
-%\subsubsection{Local density approximation}
-%--------------------------------------------
-Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we define the LDA version of ECMD as 
-\begin{equation}
-	\label{eq:def_lda_tot}
-	\bE{\LDA}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \int \be{\UEG}{\sr}[\n{}{}(\br{}),\rsmu{}{}] \n{}{}(\br{}) \dbr{}
-\end{equation}
-where $\be{\UEG}{\sr}[\n{}{}(\br{}),\rsmu{}{}]$ is the multi-determinant short-range correlation energy per particle of the uniform electron gas (UEG) for which a parametrization can be found in Ref.~\onlinecite{PazMorGorBac-PRB-06}. 
-%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 recently proposed functional of some of the present authors \cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional $\e{}{\PBE}(\n{}{},\nabla \n{}{})$ for small $\rsmu{}{}$ and the exact large-$\rsmu{}{}$ behaviour, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGori-PRB-06} yielding
-\begin{subequations}
-\begin{gather}
-	\label{eq:epsilon_cmdpbe}
-	\be{\PBE}{\sr}(\n{}{},\nabla \n{}{},\rsmu{}{}) = \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{1 + \beta(n,\nabla n, \mu)\mu^3 }
-	\\
-	\label{eq:epsilon_cmdpbe}
-	\beta(n,\nabla n;\,\mu) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{\n{\UEG}{(2)}(\n{}{})}.
-\end{gather}
-\end{subequations}
-The difference between the ECMD PBE functional defined in Ref.~\cite{FerGinTou-JCP-18} and the present expression is that we approximate the \textit{exact} ground-state on-top pair density of the system $\n{}{(2)}(\br{})$ by its UEG version, i.e.
-\begin{equation}
- \label{eq:ueg_ontop}
-	\n{}{(2)}(\br{}) \approx \n{\UEG}{(2)}(\br{}) =  \n{}{}(\br{})^2 g_0[\n{}{}(\br{})]
-\end{equation}
-where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. 
-Therefore, the PBE complementary function reads
-\begin{equation}
-	\label{eq:def_lda_tot}
-	\bE{\PBE}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \int \be{\PBE}{\sr}[\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{}{}] \n{}{}(\br{}) \dbr{}
-\end{equation}
-
 %=================================================================
 \subsection{Valence effective interaction}
 %=================================================================