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Pierre-Francois Loos 2019-04-23 15:11:48 +02:00
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Manuscript/G2-srDFT.tex
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@ -209,7 +209,6 @@ where
is the basis-dependent complementary density functional, $\hT$ is the kinetic operator and $\hWee{} = \sum_{i<j} r_{ij}^{-1}$ is the interelectronic repulsion operator. is the basis-dependent complementary density functional, $\hT$ is the kinetic operator and $\hWee{} = \sum_{i<j} r_{ij}^{-1}$ is the interelectronic repulsion operator.
In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron normalized wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis set (CBS), respectively. In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron normalized wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis set (CBS), respectively.
Both wave functions yield the same target density $\n{}{}$ (assumed to be representable in \trashPFL{the basis set} $\Bas$). Both wave functions yield the same target density $\n{}{}$ (assumed to be representable in \trashPFL{the basis set} $\Bas$).
Importantly, in the CBS limit (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$. Importantly, in the CBS limit (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$.
This implies that This implies that
\begin{equation} \begin{equation}
@ -270,7 +269,6 @@ Because Eq.~\eqref{eq:int_eq_wee} can be \titou{recast} as
\end{equation} \end{equation}
it intuitively motivates $\W{}{\Bas}(\br{1},\br{2})$ as a potential candidate for an effective interaction. it intuitively motivates $\W{}{\Bas}(\br{1},\br{2})$ as a potential candidate for an effective interaction.
Note that the divergence condition of $\W{}{\Bas}(\br{1},\br{2})$ in Eq.~\eqref{eq:def_weebasis} ensures that one-electron systems are free of correction as the present approach must only correct the basis set incompleteness error originating from the e-e cusp. Note that the divergence condition of $\W{}{\Bas}(\br{1},\br{2})$ in Eq.~\eqref{eq:def_weebasis} ensures that one-electron systems are free of correction as the present approach must only correct the basis set incompleteness error originating from the e-e cusp.
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. 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.
Thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) Thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
\begin{equation} \begin{equation}