Done with valence

Manuscript/G2-srDFT.tex

@@ -416,6 +416,7 @@ In order to compute the effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ de
 \end{equation}
 which states that the expectation value of $\hWee{}$ over $\wf{}{\Bas}$ is equal to the expectation value of its projected version in $\Bas$ 
 \begin{equation}
+\label{eq:WeeB}
 	\hWee{\Bas} =  \frac{1}{2} \sum_{ijkl \in \Bas} \vijkl \aic{k}\aic{l}\ai{j}\ai{i}
 \end{equation}
 over the same wave function $\wf{}{\Bas}$, where the indices run over all \alert{occupied} spinorbitals $\SO{i}{}$ in $\Bas$  and $\vijkl$ are the usual two-electron Coulomb integrals. 
@@ -434,7 +435,7 @@ where
 	\label{eq:fbasis}
 	\f{\wf{}{\Bas}}{}(\bx{1},\bx{2}) 
 	\\
-	=  \sum_{ijklmn \in \Bas} \vijkl \Gam{mn}{pq}[\wf{}{\Bas}] \SO{n}{2} \SO{m}{1} \SO{i}{1} \SO{j}{2},
+	=  \sum_{ijklmn \in \Bas}  \SO{i}{1} \SO{j}{2} \vijkl \Gam{mn}{pq}[\wf{}{\Bas}] \SO{n}{2} \SO{m}{1},
 \end{multline}
 and $\Gam{mn}{pq}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{p}\aic{q}\ai{n}\ai{m} }{\wf{}{\Bas}}$ is the two-body density tensor of $\wf{}{\Bas}$, it comes naturally that 
 \begin{equation}
@@ -445,43 +446,38 @@ and $\Gam{mn}{pq}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{p}\aic{q}\ai{n}\ai{m}
 As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ is symmetric, \textit{a priori} non translational nor rotational invariant if $\Bas$ does not have such symmetries and is necessarily \textit{finite} at $r_{12} = 0$ for an incomplete basis set $\Bas$. 
 Also, as demonstrated in Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\lim_{\Bas \to \infty}\W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = r_{12}^{-1}$. 
 
-
 %----------------------------------------------------------------
 \subsubsection{Definition of a 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 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 $\wf{}{\Bas}$. Therefore, to define an effective interaction accounting only for the valence electrons, one needs to define a function $\fbasisval$ satisfying 
+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 $\wf{}{\Bas}$. Therefore, to define an effective interaction accounting only for the valence electrons, one needs to define a function $\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ satisfying 
 \begin{equation}
    \label{eq:expectweebval}
-   \mel*{\wf{}{\Bas}}{\weeopbasisval}{\wf{}{\Bas}} =  \frac{1}{2}\,\,\iint \dr{1}\,\dr{2} \,\, \fbasisval,
+   \mel*{\wf{}{\Bas}}{\hWee{\Val}}{\wf{}{\Bas}} =  \frac{1}{2} \iint \f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
 \end{equation}
-where $\weeopbasisval$ is the valence coulomb operator defined as
- \begin{equation}
-    \begin{aligned}
-      \weeopbasisval =  \frac{1}{2}\,\, \sum_{ijkl\,\,\in\,\,\Basval} \,\, \vijkl \,\, \aic{k}\aic{l}\ai{j}\ai{i}\,\,\,,
-    \end{aligned}
- \end{equation}
-and $\Basval$ is the subset of molecular orbitals for which we want to define the expectation value, which will be typically the all MOs except those frozen. 
-Following the spirit of \eqref{eq:fbasis}, the function $\fbasisval$ can be defined as
- \begin{equation}
+where $\hWee{\Val}$, the valence part of the Coulomb operator, has a similar expression as $\hWee{\Bas}$ in Eq.~\eqref{eq:WeeB}.
+%\begin{equation}
+%      \hWee{\Val} =  \frac{1}{2} \sum_{ijkl \in \Val} \vijkl \aic{k}\aic{l}\ai{j}\ai{i},
+%\end{equation}
+Following the spirit of Eq.~\eqref{eq:fbasis}, the function $\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ can be defined as
+ \begin{multline}
 	\label{eq:fbasisval}
-   \begin{aligned}
-   \fbasisval =  \sum_{ij\,\,\in\,\,\Bas}  \,\, \sum_{klmn\,\,\in\,\,\Basval} & \vijkl \,\, \gammaklmn{\wf{}{\Bas}} \\& \phix{n}{2} \phix{m}{1}  \phix{i}{1} \phix{j}{2}.
-   \end{aligned}
- \end{equation}
-
-Then, the effective interaction associated to the valence $\wbasisval$ is simply defines as
+	\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) 
+	\\
+	=  \sum_{ij \in \Bas} \sum_{klmn \in \Val}  \SO{i}{1} \SO{j}{2} \vijkl \gammaklmn{\wf{}{\Bas}} \SO{n}{2} \SO{m}{1}.
+ \end{multline}
+and, the valence part of the effective interaction is
  \begin{equation}
    \label{eq:def_weebasis}
-   \wbasisval = \frac{\fbasisval}{\twodmrdiagpsival},
+   \W{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) = \frac{\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) }{\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2})},
  \end{equation}
-where $\twodmrdiagpsival$ is the two body density associated to the valence electrons:
-\begin{equation}
- \twodmrdiagpsival = \sum_{klmn\,\,\in\,\,\Basval} \gammamnkl[\wf{}{\Bas}] \,\, \phix{m}{1} \phix{n}{2}  \phix{k}{1} \phix{l}{2} .
-\end{equation}
-It is important to notice in \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 $\Basval$. 
+where $\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2})$ is the two body density associated to the valence electrons.
+%\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 $\Basval$. 
 
 
 \subsubsection{Definition of a range-separation parameter varying in space}