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two-electron integrals

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Julien Toulouse 2019-04-19 15:57:16 +02:00
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Manuscript/G2-srDFT.tex
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@ -193,7 +193,7 @@ According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming t
\approx \E{\modY}{\Bas} \approx \E{\modY}{\Bas}
+ \bE{}{\Bas}[\n{\modZ}{\Bas}], + \bE{}{\Bas}[\n{\modZ}{\Bas}],
\end{equation} \end{equation}
where where
\begin{equation} \begin{equation}
\label{eq:E_funcbasis} \label{eq:E_funcbasis}
\bE{}{\Bas}[\n{}{}] \bE{}{\Bas}[\n{}{}]
@ -244,13 +244,13 @@ where
\n{2}{}(\br{1},\br{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}, = \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
\end{equation} \end{equation}
and $\Gam{pq}{rs} =\mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{p_\uparrow}\ai{q_\downarrow}}{\wf{}{\Bas}}$ are the opposite-spin pair density associated with $\wf{}{\Bas}$ and its corresponding tensor (respectively), $\SO{p}{}$ is a molecular orbital (MO), and $\Gam{pq}{rs} =\mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{p_\uparrow}\ai{q_\downarrow}}{\wf{}{\Bas}}$ are the opposite-spin pair density associated with $\wf{}{\Bas}$ and its corresponding tensor, respectively, $\SO{p}{}$ is a (real-valued) molecular orbital (MO),
\begin{equation} \begin{equation}
\label{eq:fbasis} \label{eq:fbasis}
\f{\Bas}{}(\br{1},\br{2}) \f{\Bas}{}(\br{1},\br{2})
= \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2}, = \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
\end{equation} \end{equation}
and $\V{pq}{rs}$ are the usual two-electron Coulomb integrals. and $\V{pq}{rs}=\langle pq | rs \rangle$ are the usual two-electron Coulomb integrals.
With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ satisfies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ satisfies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
\begin{equation} \begin{equation}
\label{eq:int_eq_wee} \label{eq:int_eq_wee}