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added the +infty condition on n2

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Emmanuel Giner 2019-04-09 00:31:29 +02:00
parent fffb4a47fd
commit bc3956a69f
1 changed files with 18 additions and 8 deletions
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26
Manuscript/G2-srDFT.tex
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@ -328,7 +328,12 @@ In the final step, we employ short-range density functionals\cite{TouGorSav-TCA-
We define the effective operator $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ as (see equation (27) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
\begin{equation}
\label{eq:def_weebasis}
\W{\wf{}{\Bas}}{}(\br{1},\br{2}) = \f{\wf{}{\Bas}}{}(\br{1},\br{2})/\n{\wf{}{\Bas}}{(2)}(\br{1},\br{2})
\W{\wf{}{\Bas}}{}(\br{1},\br{2}) = \left\{
\begin{array}{ll}
\f{\wf{}{\Bas}}{}(\br{1},\br{2})/\n{\wf{}{\Bas}}{(2)}(\br{1},\br{2}) & \mbox{if } \n{\wf{}{\Bas}}{(2)}(\br{1},\br{2}) \ne 0\\
\,\,\,\,+\infty & \mbox{if not.}
\end{array}
\right.
\end{equation}
where $\n{\wf{}{\Bas}}{(2)}(\br{1},\br{2})$ is the opposite-spin two-body density associated with $\wf{}{\Bas}$
\begin{equation}
@ -365,9 +370,9 @@ and which is necessarily \textit{finite} at for an \textit{incomplete} basis set
Of course, there exists \textit{a priori} an infinite set of functions in ${\rm I\!R}^6$ satisfying \eqref{eq:int_eq_wee}, but thanks to its very definition one can show (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that
\begin{equation}
\label{eq:lim_W}
\lim_{\Bas \to \infty}\W{\wf{}{\Bas}}{}(\br{1},\br{2}) = r_{12}^{-1}\quad \forall \,\,(\br{1},\br{2}),
\lim_{\Bas \to \infty}\W{\wf{}{\Bas}}{}(\br{1},\br{2}) = r_{12}^{-1}\
\end{equation}
and for any choice of $\wf{}{\Bas}$, which therefore guarantees a physically satisfying limit.
for all points $(\br{1},\br{2})$ such that $\n{\wf{}{\Bas}}{(2)}(\br{1},\br{2}) \ne 0$ 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 $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$, one can see 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 $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ already provides a quantitative representation of the incompleteness of the basis set $\Bas$ for weakly correlated systems.
@ -378,7 +383,7 @@ As we can map the Coulomb operator within a basis set $\Bas$ with a non divergen
To do so, we choose a range-separation \textit{function} $\rsmu{\wf{}{\Bas}}{}(\br{})$
\begin{equation}
\label{eq:mu_of_r}
\rsmu{\wf{}{\Bas}}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{}(\br{}),
\rsmu{\wf{}{\Bas}}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{}(\br{})
\end{equation}
such that the long-range interaction $\w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\br{1},\br{2})$
\begin{equation}
@ -527,10 +532,15 @@ We then naturally split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor
%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 $\wf{}{\Bas}$.
We therefore define the valence-only effective interaction
\begin{equation}
\label{eq:Wval}
\W{\wf{}{\Bas}}{\Val}(\br{1},\br{2}) = \f{\wf{}{\Bas}}{\Val}(\br{1},\br{2})/\n{\wf{}{\Bas},\Val}{(2)}(\br{1},\br{2}),
\end{equation}
\begin{equation}
% \label{eq:Wval}
\W{\wf{}{\Bas}}{\Val}(\br{1},\br{2}) = \left\{
\begin{array}{ll}
\f{\wf{}{\Bas}}{\Val}(\br{1},\br{2})/\n{\wf{}{\Bas},\Val}{(2)}(\br{1},\br{2}) & \mbox{if } \n{\wf{}{\Bas},\Val}{(2)}(\br{1},\br{2})\ne 0\\
\,\,\,\,+\infty & \mbox{if not.}
\end{array}
\right.
\end{equation}
with
\begin{subequations}
\begin{gather}