Sec IIB

Manuscript/srDFT_SC.tex

@@ -386,9 +386,9 @@ where $\efci$ and $n_\text{FCI}^\Bas$ are the ground-state FCI energy and densit
 
 \subsection{Definition of an effective interaction within $\Bas$}
 \label{sec:wee}
-As originally shown by Kato\cite{Kat-CPAM-57}, the cusp in the exact wave function originates from the divergence of the Coulomb interaction at the coalescence point. Therefore, a cuspless wave function $\wf{}{\Bas}$ could also be obtained from a Hamiltonian with a non-divergent electron-electron interaction. In other words, the impact of the incompleteness of a finite basis set can be understood as the removal of the divergence of the usual Coulomb electron-electron interaction at the coalescence point. 
+As originally shown by Kato, \cite{Kat-CPAM-57} the electron-electron cusp of the exact wave function originates from the divergence of the Coulomb interaction at the coalescence point. Therefore, a cuspless wave function $\wf{}{\Bas}$ could also be obtained from a Hamiltonian with a non-divergent electron-electron interaction. \titou{In other words, the impact of the basis set incompleteness can be understood as the removal of the divergence of the usual Coulomb electron-electron interaction.}
 
-As originally derived in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} (Section D and Appendices), one can obtain an effective non-divergent electron-electron interaction, here referred to as $\wbasis$, which reproduces the expectation value of the Coulomb electron-electron interaction operator over a given wave function $\wf{}{\Bas}$. As we are interested in the behaviour at the coalescence point, we focus on the opposite-spin part of the electron-electron interaction. More specifically, the effective electron-electron interaction associated to a given wave function $\wf{}{\Bas}$ is defined as
+As originally derived in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} (see Sec.~II D~and Appendices), one can obtain an effective non-divergent electron-electron interaction, here referred to as $\wbasis$, which reproduces the expectation value of the Coulomb electron-electron interaction operator over a given wave function $\wf{}{\Bas}$. As we are interested in the behaviour at the coalescence point, we focus on the opposite-spin part of the electron-electron interaction. More specifically, the effective electron-electron interaction associated to a given wave function $\wf{}{\Bas}$ is defined as
 \begin{equation}
  \label{eq:wbasis}
  \wbasis =
@@ -402,23 +402,23 @@ where $\twodmrdiagpsi$ is the opposite-spin pair density associated with $\wf{}{
 \begin{equation}
  \twodmrdiagpsi = \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}, 
 \end{equation}
-and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$ its associated tensor in a basis of spatial orthonormal orbitals $\{\SO{p}{}\}$, and $\fbasis$ is
+and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$ its associated tensor in a basis of spatial orthonormal orbitals $\{\SO{p}{}\}$, and 
 \begin{equation}
         \label{eq:fbasis}
         \fbasis
         = \sum_{pqrstu \in \Bas}  \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
 \end{equation}
-with the usual two-electron Coulomb integrals $\V{pq}{rs}=\langle pq | rs \rangle$.
+with the usual two-electron Coulomb integrals $\V{pq}{rs}= \braket{pq}{rs}$.
 With such a definition, one can show that $\wbasis$ satisfies 
-\begin{eqnarray}
- \frac{1}{2}\iint \dr{1} \dr{2} \wbasis \twodmrdiagpsi =
-\nonumber\\
-\frac{1}{2} \iint \dr{1} \dr{2} \frac{\twodmrdiagpsi}{|\br{1}-\br{2}|}. 
-\end{eqnarray}
+\begin{multline}
+	\frac{1}{2}\iint \dr{1} \dr{2} \wbasis \twodmrdiagpsi =
+	\\
+	\frac{1}{2} \iint \dr{1} \dr{2} \frac{\twodmrdiagpsi}{\abs{\br{1}-\br{2}}}. 
+\end{multline}
 As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, the effective interaction $\wbasis$ is necessarily finite at coalescence for an incomplete basis set, and tends to the usual Coulomb interaction in the CBS limit for any choice of wave function $\psibasis$, \ie,
 \begin{equation}
  \label{eq:cbs_wbasis}
- \lim_{\Bas \to \text{CBS}} \wbasis = \frac{1}{|\br{1}-\br{2}|},\quad \forall\,\psibasis. 
+ \lim_{\Bas \to \text{CBS}} \wbasis = \frac{1}{\abs{\br{1}-\br{2}}},\quad \forall\,\psibasis. 
 \end{equation}
 The condition in Eq.~\eqref{eq:cbs_wbasis} is fundamental as it guarantees the correct behavior of the theory in the CBS limit.