diff --git a/Manuscript/G2-srDFT.pdf b/Manuscript/G2-srDFT.pdf
index d27e53b..9884ec4 100644
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diff --git a/Manuscript/G2-srDFT.tex b/Manuscript/G2-srDFT.tex
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--- a/Manuscript/G2-srDFT.tex
+++ b/Manuscript/G2-srDFT.tex
@@ -237,9 +237,9 @@ Also, as demonstrated in the appendix B of \cite{GinPraFerAssSavTou-JCP-18}, $\w
 \subsubsection{Definition of a valence effective interaction}
 As the average inter electronic distances are very different between the valence electrons and the core electrons, it can be advantageous to define an effective interaction taking into account only for the valence electrons which are the most important in most of the chemical processes. 
 
-According to \eqref{eq:def_weebasis} and \eqref{eq:expectweeb}, the effective interaction is defined by the expectation value of the coulomb operator over a wave function $\psibasis$. Therefore, to define an effective interaction accounting only for the valence electrons, one needs to define first the following function $\fbasisval$ satisfying 
+According to \eqref{eq:def_weebasis} and \eqref{eq:expectweeb}, the effective interaction is defined by the expectation value of the coulomb operator over a wave function $\psibasis$. Therefore, to define an effective interaction accounting only for the valence electrons, one needs to define a function $\fbasisval$ satisfying 
 \begin{equation}
-   \label{eq:expectweeb}
+   \label{eq:expectweebval}
    \elemm{\psibasis}{\weeopbasisval}{\psibasis} =  \frac{1}{2}\,\,\iint \dr{1}\,\dr{2} \,\, \fbasisval,
 \end{equation}
 where $\weeopbasisval$ is the valence coulomb operator defined as
@@ -248,15 +248,15 @@ where $\weeopbasisval$ is the valence coulomb operator defined as
       \weeopbasisval =  \frac{1}{2}\,\, \sum_{ijkl\,\,\in\,\,\basisval} \,\, \vijkl \,\, \aic{k}\aic{l}\ai{j}\ai{i},
     \end{aligned}
  \end{equation}
-$\basisval$ is a given set of molecular orbitals associated to the valence space which will be defined later on, 
-and the function $\fbasisval$ 
+$\basisval$ is a given set of molecular orbitals associated to the valence space which will be defined later on. 
+Following the spirit of \eqref{eq:fbasis}, the function $\fbasisval$ can be defined as
  \begin{equation}
    \label{eq:fbasis}
    \begin{aligned}
-   \fbasisval =  \sum_{ijklmn\,\,\in\,\,\basisval} & \vijkl \,\, \gammaklmn{\psibasis} \\& \phix{n}{2} \phix{m}{1}  \phix{i}{1} \phix{j}{2}.
+   \fbasisval =  \sum_{(ij)\,\,\in\,\,\basis}  \,\, \sum_{(klmn)\,\,\in\,\,\basisval} & \vijkl \,\, \gammaklmn{\psibasis} \\& \phix{n}{2} \phix{m}{1}  \phix{i}{1} \phix{j}{2}.
    \end{aligned}
  \end{equation}
-To define 
+It is important to notice the difference between the set of orbitals for the indices $(i,j)$, which span the full set of MOs within $\basis$,  and the $(k,l,m,n)$, which span only the valence space $\basisval$. 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 $\basis$, whatever the choice of subset $\basisval$. 
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 \section{Results}
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