From bc3956a69faab552e0a7af4af7541f6ff2ea6225 Mon Sep 17 00:00:00 2001 From: Emmanuel Giner Date: Tue, 9 Apr 2019 00:31:29 +0200 Subject: [PATCH] added the +infty condition on n2 --- Manuscript/G2-srDFT.tex | 26 ++++++++++++++++++-------- 1 file changed, 18 insertions(+), 8 deletions(-) diff --git a/Manuscript/G2-srDFT.tex b/Manuscript/G2-srDFT.tex index d9c9e5a..6b4a744 100644 --- a/Manuscript/G2-srDFT.tex +++ b/Manuscript/G2-srDFT.tex @@ -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}