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Emmanuel Giner 2019-04-17 16:09:57 +02:00
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Manuscript/G2-srDFT.tex
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@ -298,7 +298,7 @@ coincides with the effective interaction at coalescence, i.e.~$\w{}{\lr,\rsmu{\B
%\subsection{Short-range correlation functionals} %\subsection{Short-range correlation functionals}
%================================================================= %=================================================================
Once $\rsmu{\Bas}{}(\br{})$ is defined, it can be used within RS-DFT functionals to approximate $\bE{}{\Bas}[\n{}{}]$. Once $\rsmu{\Bas}{}(\br{})$ is defined, it can be used within RS-DFT functionals to approximate $\bE{}{\Bas}[\n{}{}]$.
As in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we consider here a specific class of short-range correlation functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05} As in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we consider here a specific class of short-range correlation functionals known as \manu{correlation energy with multi determinantal reference (ECMD)} whose general definition reads \cite{TouGorSav-TCA-05}
\begin{multline} \begin{multline}
\label{eq:ec_md_mu} \label{eq:ec_md_mu}
\bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}]
@ -414,7 +414,7 @@ Defining $\n{\modZ}{\FC}$ as the FC (i.e.~valence-only) one-electron density obt
The most computationally intensive task of the present approach is the evaluation of $\W{\Bas}{}(\br{})$ [see Eq.~\eqref{eq:wcoal}] at each quadrature grid point. The most computationally intensive task of the present approach is the evaluation of $\W{\Bas}{}(\br{})$ [see Eq.~\eqref{eq:wcoal}] at each quadrature grid point.
Yet embarrassingly parallel, this step scales, in the general case (i.e.~$\wf{}{\Bas}$ is a determinant expansion), as $\Ng \Nb^4$ (where $\Nb$ is the number of basis functions in $\Bas$) but is reduced to $\order*{ \Ng \Ne^2 \Nb^2}$ when $\wf{}{\Bas}$ is a single Slater determinant. Yet embarrassingly parallel, this step scales, in the general case (i.e.~$\wf{}{\Bas}$ is a determinant expansion), as $\Ng \Nb^4$ (where $\Nb$ is the number of basis functions in $\Bas$) but is reduced to $\order*{ \Ng \Ne^2 \Nb^2}$ when $\wf{}{\Bas}$ is a single Slater determinant.
As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, this choice for $\wf{}{\Bas}$ already provides, for weakly correlated systems, a quantitative representation of the incompleteness of $\Bas$. As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, this choice for $\wf{}{\Bas}$ already provides, for weakly correlated systems, a quantitative representation of the incompleteness of $\Bas$.
Hence, unless otherwise stated, we will stick to this choice throughout the current study. Hence\trashMG{, unless otherwise stated,} we will stick to this choice throughout the current study.
In our current implementation, the computational bottleneck is the four-index transformation to get the two-electron integrals in the MO basis which appear in Eqs.~\eqref{eq:n2basis} and \eqref{eq:fbasis}. In our current implementation, the computational bottleneck is the four-index transformation to get the two-electron integrals in the MO basis which appear in Eqs.~\eqref{eq:n2basis} and \eqref{eq:fbasis}.
Nevertheless, this step usually has to be performed for most correlated WFT calculations. Nevertheless, this step usually has to be performed for most correlated WFT calculations.
Modern integral decomposition techniques (such as density fitting \cite{Whi-JCP-73}) or atomic-orbital-based algorithms could be employed to significantly speed up this step. Modern integral decomposition techniques (such as density fitting \cite{Whi-JCP-73}) or atomic-orbital-based algorithms could be employed to significantly speed up this step.