diff --git a/JCTC_revision/GW-srDFT.tex b/JCTC_revision/GW-srDFT.tex
index aab3193..f08e038 100644
--- a/JCTC_revision/GW-srDFT.tex
+++ b/JCTC_revision/GW-srDFT.tex
@@ -436,14 +436,20 @@ where $\eGOWO{\HOMO}$ and $\eGOWO{\LUMO}$ are the HOMO and LUMO orbital energies
 \subsection{Basis set correction}
 \label{sec:BSC}
 %%%%%%%%%%%%%%%%%%%%%%%%
-\titou{The fundamental idea behind the present basis set correction is to recognize that the singular two-electron Coulomb interaction $\abs*{\br{} - \br{}'}^{-1}$ projected in a finite basis $\Bas$ is a finite, non-divergent quantity at $\abs*{\br{} - \br{}'} = 0$, which ``resembles'' the long-range interaction operator  $\abs*{\br{} - \br{}'}^{-1} \erf(\rsmu{}{} \abs*{\br{} - \br{}'})$ used within RS-DFT. \cite{Giner_2018}
-We can therefore define an effective two-electron operator which equals its Coulomb counterpart in an incomplete basis, \ie,
+\titou{The fundamental idea behind the present basis set correction is to recognize that the singular two-electron Coulomb interaction $\abs*{\br{} - \br{}'}^{-1}$ projected in a finite basis $\Bas$ is a finite, non-divergent quantity at $\abs*{\br{} - \br{}'} = 0$, which ``resembles'' the long-range interaction operator  $\abs*{\br{} - \br{}'}^{-1} \erf(\rsmu{}{} \abs*{\br{} - \br{}'})$ used within RS-DFT. \cite{Giner_2018}}
+%We can therefore define an effective two-electron operator $W^{\Bas}(\br{},\br{}')$ which equals its Coulomb counterpart in an incomplete basis, \ie,
+\manu{
+We can therefore define an effective non divergent two-electron interaction $W^{\Bas}(\br{},\br{}')$ which reproduces the expectation value of the Coulomb interaction over a given two-body density $\n{2}{\Bas}(\br{},\br{}')$, \ie,}
+\titou{
 \begin{equation}
 	\iint \frac{\n{2}{\Bas}(\br{},\br{}')}{\abs*{\br{} - \br{}'}} d\br{} d\br{}'
 	= 
 	\iint \n{2}{\Bas}(\br{},\br{}') W^{\Bas}(\br{},\br{}') d\br{} d\br{}'.
 \end{equation}
-Because the value of $W^{\Bas}(\br{},\br{}')$ at coalescence, $W^{\Bas}(\br{},\br{})$, is necessarily finite in a finite basis $\Bas$, one can map this operator to a non-divergent, long-range interaction of the form 
+}
+\manu{The derivation and properties of $W^{\Bas}(\br{},\br{}')$ can be found in details in Ref. \cite{Giner_2018}, but a key aspect is that }
+\titou{
+because the value of $W^{\Bas}(\br{},\br{}')$ at coalescence, $W^{\Bas}(\br{},\br{})$, is necessarily finite in a finite basis $\Bas$, one can} \manu{fit} \titou{$W^{\Bas}(\br{},\br{}')$ by a non-divergent, long-range interaction of the form 
 \begin{equation}
 	\label{eq:Wapprox}
 	W^{\Bas}(\br{},\br{}') 
@@ -459,32 +465,42 @@ Knowing that $\lim_{r \to 0} \erf(\mu r)/r = 2\mu/\sqrt{\pi}$, this yields
 \end{equation}
 }
 
-\titou{A convenient choice for the on-top value of the two-electron effective operator is, for example,
+
+\manu{In Refs. \cite{Giner_2018,Loos_2019} some of the present authors proposed the following definition for the $W^{\Bas}(\br{},\br{}')$}
+%\titou{A convenient choice for the on-top value of the two-electron effective operator is, for example,
+\titou{
 \begin{equation}
 \label{eq:W}
-	W^{\Bas}(\br{},\br{}) =  
+	W^{\Bas}(\br{},\br{}') =  
 	\begin{cases}
-		f^{\Bas}(\br{})/\n{2}{\Bas}(\br{}),		&	\n{2}{\Bas}(\br{})	\neq 0,		\\
+		f^{\Bas}(\br{},\br{}')/\n{2}{\Bas}(\br{},\br{}'),		&	\n{2}{\Bas}(\br{},\br{}')	\neq 0,		\\
 		\infty,									&	\text{otherwise},				\\
 	\end{cases}
 \end{equation}
-where, in the case of a single-determinant method (such as HF and KS-DFT),
+where, in the case of a single-determinant method (such as HF and KS-DFT),}
 \begin{equation}
-	f^{\Bas}(\br{}) = \sum_{pq}^{\Nbas} \sum_{ij}^{\Nocc} \MO{p}(\br{}) \MO{q}(\br{}) (pi|qj) \MO{i}(\br{}) \MO{j}(\br{}),
+	f^{\Bas}(\br{},\br{}') = \sum_{pq}^{\Nbas} \sum_{ij}^{\Nocc} \MO{p}(\br{})\MO{i}(\br{})  (pi|qj)\MO{q}(\br{}')  \MO{j}(\br{}'),
 \end{equation}
 and 
 \begin{equation}
-	\n{2}{\Bas}(\br{}) = \frac{1}{4} [\n{}{\Bas}(\br{})]^2
+	\n{2}{\Bas}(\br{}) = n_{\uparrow}^{\Bas}(\br{}) n_{\downarrow}^{\Bas}(\br{}')
 \end{equation}
-is the opposite-spin on-top pair density in a closed-shell system.
-}
+is the opposite-spin pair density in a closed-shell system.
 
-\titou{The effective operator $W^{\Bas}(\br{},\br{}')$ has some interesting properties.
+
+\manu{Because of this definition, }
+\titou{the effective interaction $W^{\Bas}(\br{},\br{}')$ has some interesting properties.
 For example, we have
 \begin{equation}
 	\lim_{\Bas \to \CBS}  W^{\Bas}(\br{},\br{}') = \abs*{\br{} - \br{}'}^{-1},
 \end{equation}
-which means that, in the limit of a complete basis, one recovers the genuine (divergent) Coulomb operator, and that the magnitude of the energetic correction tends to zero [see Eq.~\eqref{eq:limSig}].
+which means that, in the limit of a complete basis, one recovers the genuine (divergent) Coulomb interaction}. 
+\manu{
+Therefore, at the CBS limit the coalescence value $W^{\Bas}(\br{},\br{})$ goes to infinity and so does $\rsmu{}{\Bas}(\br{})$. 
+As the present basis set correction uses complementary short-range functionals derivatives from RS-DFT which go to zero when $\mu$ goes to infinity, 
+the present basis set correction vanishes at the CBS limit. 
+}
+\titou{
 Note also that the divergence condition of $W^{\Bas}(\br{},\br{}')$ in Eq.~\eqref{eq:W} ensures that one-electron systems are free of correction.
 }