diff --git a/Manuscript/G2-srDFT.tex b/Manuscript/G2-srDFT.tex index 0225479..143b33e 100644 --- a/Manuscript/G2-srDFT.tex +++ b/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. diff --git a/Manuscript/VDZ.pdf b/Manuscript/VDZ.pdf index e6e86a0..9178e42 100644 Binary files a/Manuscript/VDZ.pdf and b/Manuscript/VDZ.pdf differ diff --git a/Manuscript/VQZ.pdf b/Manuscript/VQZ.pdf index c3a494a..c01e5f8 100644 Binary files a/Manuscript/VQZ.pdf and b/Manuscript/VQZ.pdf differ diff --git a/Manuscript/VTZ.pdf b/Manuscript/VTZ.pdf index fe13e29..7b09410 100644 Binary files a/Manuscript/VTZ.pdf and b/Manuscript/VTZ.pdf differ