Major change of notations

Manuscript/G2-srDFT.tex

@@ -339,36 +339,37 @@ In the final step, we employ short-range density functionals\cite{TouGorSav-TCA-
 We define the effective operator $\W{\Bas}{}(\br{1},\br{2})$ as 
 \begin{equation}
 	\label{eq:def_weebasis}
- \W{\Bas}{}(\br{1},\br{2})   = \left\{
-    \begin{array}{ll}
-       \f{\Bas}{}(\br{1},\br{2})/\n{2}{\wf{}{\Bas}}(\br{1},\br{2}) & \mbox{if } \n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \ne 0\\
-        \,\,\,\,+\infty & \mbox{otherwise.}
-    \end{array}
- \right.
+	\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.}
+    \end{cases}
 \end{equation}
-where  $\n{2}{\wf{}{\Bas}}(\br{1},\br{2})$ is the opposite-spin two-body density associated with $\wf{}{\Bas}$
+where
 \begin{equation}
 	\label{eq:n2basis}
-	\n{2}{\wf{}{\Bas}}(\br{1},\br{2})
-	= \sum_{pqrs \in \Bas}  \SO{p}{1} \SO{q}{2} \Gam{pq}{rs}[\wf{}{\Bas}] \SO{r}{1} \SO{s}{2}, 
+	\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}
-$\Gam{pq}{rs}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\wf{}{\Bas}}$ is the opposite-spin two-body density tensor of $\wf{}{\Bas}$, $\SO{i}{}$ are spinorbitals, $\f{\Bas}{}(\br{1},\br{2})$ is defined as
-\begin{multline}
+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
+\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}[\wf{}{\Bas}] \SO{t}{1} \SO{u}{2},
-\end{multline}
+	= \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. 
+\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})
 \begin{equation}
 	\label{eq:int_eq_wee}
-	\mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \dbr{1} \dbr{2},
+	\mel*{\wf{}{\Bas}}{\hWee{}}{\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
 \begin{equation}
-  \iint r_{12}^{-1} \n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \dbr{1} \dbr{2} =  \iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \dbr{1} \dbr{2},
+  \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. 
 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. 
@@ -384,7 +385,7 @@ Of course, there exists \textit{a priori} an infinite set of functions in $\math
 	\label{eq:lim_W}
 	\lim_{\Bas \to \infty}\W{\Bas}{}(\br{1},\br{2}) = r_{12}^{-1}\   
 \end{equation}
-for all points $(\br{1},\br{2})$ such that $\n{2}{\wf{}{\Bas}}(\br{1},\br{2}) \ne 0$ and for any choice of $\wf{}{\Bas}$, which therefore guarantees a physically satisfying limit. 
+for all points $(\br{1},\br{2})$ such that $\n{2}{}(\br{1},\br{2}) \ne 0$ and for any choice of $\wf{}{\Bas}$, which therefore 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.  
 As it has been shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} (see for instance Fig 1,2 and 3 therein), 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 the basis set $\Bas$ for weakly correlated systems.