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@ -233,7 +233,7 @@ We define the effective operator as \cite{GinPraFerAssSavTou-JCP-18}
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\label{eq:def_weebasis}
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\W{\Bas}{}(\br{1},\br{2}) =
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\begin{cases}
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\f{\Bas}{}(\br{1},\br{2})/\n{2}{}(\br{1},\br{2}), & \text{if $\n{2}{}(\br{1},\br{2}) \ne 0$,}
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\f{\Bas}{}(\br{1},\br{2})/\n{2,\Bas}{}(\br{1},\br{2}), & \text{if $\n{2,\Bas}{}(\br{1},\br{2}) \ne 0$,}
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\\
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\infty, & \text{otherwise,}
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\end{cases}
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@ -241,7 +241,7 @@ We define the effective operator as \cite{GinPraFerAssSavTou-JCP-18}
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where
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\begin{equation}
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\label{eq:n2basis}
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\n{2}{}(\br{1},\br{2})
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\n{2,\Bas}{}(\br{1},\br{2})
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= \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
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\end{equation}
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and $\Gam{pq}{rs} =\mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{p_\uparrow}\ai{q_\downarrow}}{\wf{}{\Bas}}$ are the opposite-spin pair density associated with $\wf{}{\Bas}$ and its corresponding tensor, respectively, $\SO{p}{}$ is a (real-valued) molecular orbital (MO),
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@ -254,13 +254,15 @@ and $\V{pq}{rs}=\langle pq | rs \rangle$ are the usual two-electron Coulomb inte
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With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ satisfies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
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\begin{equation}
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\label{eq:int_eq_wee}
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\mel*{\wf{}{\Bas}}{\hWee{\updw}}{\wf{}{\Bas}} = \iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{}(\br{1},\br{2}) \dbr{1} \dbr{2},
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\mel*{\wf{}{\Bas}}{\hWee{\updw}}{\wf{}{\Bas}} = \iint \W{\Bas}{}(\br{1},\br{2}) \n{2,\Bas}{}(\br{1},\br{2}) \dbr{1} \dbr{2},
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\end{equation}
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where $\hWee{\updw}$ contains only the opposite-spin component of $\hWee{}$.
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Because Eq.~\eqref{eq:int_eq_wee} can be rewritten as
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\begin{equation}
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\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},
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\end{equation}
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\begin{eqnarray}
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\iint r_{12}^{-1} \n{2,\Bas}{}(\br{1},\br{2}) \dbr{1} \dbr{2} = \phantom{xxxxxxxxx}
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\nonumber\\
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\iint \W{\Bas}{}(\br{1},\br{2}) \n{2,\Bas}{}(\br{1},\br{2}) \dbr{1} \dbr{2},
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\end{eqnarray}
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it intuitively motivates $\W{\Bas}{}(\br{1},\br{2})$ as a potential candidate for an effective interaction.
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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.
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@ -270,7 +272,7 @@ Thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{G
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\label{eq:lim_W}
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\lim_{\Bas \to \infty}\W{\Bas}{}(\br{1},\br{2}) = r_{12}^{-1}\
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\end{equation}
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for any $(\br{1},\br{2})$ such that $\n{2}{}(\br{1},\br{2}) \ne 0$.% and for any $\wf{}{\Bas}$, which guarantees a physically satisfying limit.
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for any $(\br{1},\br{2})$ such that $\n{2,\Bas}{}(\br{1},\br{2}) \ne 0$.% and for any $\wf{}{\Bas}$, which guarantees a physically satisfying limit.
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%=================================================================
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%\subsection{Range-separation function}
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@ -281,7 +283,7 @@ A key quantity is the value of the effective interaction at coalescence of oppos
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% \label{eq:wcoal}
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% \W{\Bas}{}(\br{}) = \W{\Bas}{}(\br{},{\br{}}),
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%\end{equation}
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which is necessarily \textit{finite} for an incomplete basis set as long as the on-top pair density $\n{2}{}(\br{},\br{})$ is non vanishing.
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which is necessarily \textit{finite} for an incomplete basis set as long as the on-top pair density $\n{2,\Bas}{}(\br{},\br{})$ is non vanishing.
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Because $\W{\Bas}{}(\br{1},\br{2})$ is a non-divergent two-electron interaction, it can be naturally linked to RS-DFT which employs a non-divergent long-range interaction operator.
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Although this choice is not unique, we choose here the range-separation function
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\begin{equation}
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@ -373,11 +375,11 @@ inspired by the recent functional proposed by some of the authors \cite{FerGinTo
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\be{\text{c,md}}{\sr,\PBE}(\{\n{\sigma}{}\},\{\nabla \n{\sigma}{}\},\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\{\n{\sigma}{}\},\{\nabla \n{\sigma}{}\})}{1 + \beta(\{n_\sigma\},\{\nabla n_\sigma\}, \rsmu{}{})\rsmu{}{3} },
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\\
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\label{eq:beta_cmdpbe}
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\beta(\{n_\sigma\},\{\nabla n_\sigma\},\rsmu{}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\{\n{\sigma}{}\},\{\nabla \n{\sigma}{}\})}{\n{2}{\UEG}(\{\n{\sigma}{}\})}.
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\beta(\{n_\sigma\},\{\nabla n_\sigma\},\rsmu{}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\{\n{\sigma}{}\},\{\nabla \n{\sigma}{}\})}{\n{2}{\UEG}(0,\{\n{\sigma}{}\})}.
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\end{gather}
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\end{subequations}
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The difference between the ECMD functional defined in Ref.~\onlinecite{FerGinTou-JCP-18} and the present expression \eqref{eq:epsilon_cmdpbe}-\eqref{eq:beta_cmdpbe} is that we approximate here the \textit{exact} on-top pair density by its UEG version, i.e.~$\n{2}{}(\br{},\br{}) \approx \n{2}{\UEG}(0,\n{}{}(\br{}))$, with $\n{2}{\UEG}(0,n) = 4 \; n_{\uparrow} \; n_{\downarrow} \; g(0,n)$ and the UEG on-top pair-distribution function $g(0,n)$ whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}.
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This represents a major computational saving without loss of accuracy for weakly correlated systems as we eschew the computation of $\n{2}{}(\br{})$.
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The difference between the ECMD functional defined in Ref.~\onlinecite{FerGinTou-JCP-18} and the present expression \eqref{eq:epsilon_cmdpbe}-\eqref{eq:beta_cmdpbe} is that we approximate here the on-top pair density by its UEG version, i.e.~$\n{2,\Bas}{}(\br{},\br{}) \approx \n{2}{\UEG}(0,\{\n{\sigma}{}(\br{})\})$, with $\n{2}{\UEG}(0,\{n_\sigma\}) = 4 \; n_{\uparrow} \; n_{\downarrow} \; g(0,n)$ and the UEG on-top pair-distribution function $g(0,n)$ whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}.
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This represents a major computational saving without loss of accuracy for weakly correlated systems as we eschew the computation of $\n{2,\Bas}{}(\br{},\br{})$.
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Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modZ}{}]$ is then equal to $\bE{\LDA}{\Bas}[\n{\modZ}{},\rsmu{\Bas}{}]$ or $\bE{\PBE}{\Bas}[\n{\modZ}{},\rsmu{\Bas}{}]$ where $\rsmu{\Bas}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
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@ -390,7 +392,7 @@ We then naturally split the basis set as $\Bas = \Cor \bigcup \BasFC$ (where $\C
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\begin{equation}
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\W{\Bas}{\FC}(\br{1},\br{2}) =
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\begin{cases}
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\f{\Bas}{\FC}(\br{1},\br{2})/\n{2}{\FC}(\br{1},\br{2}), & \text{if $\n{2}{\FC}(\br{1},\br{2})\ne 0$},
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\f{\Bas}{\FC}(\br{1},\br{2})/\n{2,\Bas}{\FC}(\br{1},\br{2}), & \text{if $\n{2,\Bas}{\FC}(\br{1},\br{2})\ne 0$},
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\\
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\infty, & \text{otherwise,}
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\end{cases}
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@ -402,7 +404,7 @@ with
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\f{\Bas}{\FC}(\br{1},\br{2})
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= \sum_{pq \in \Bas} \sum_{rstu \in \BasFC} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
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\\
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\n{2}{\FC}(\br{1},\br{2})
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\n{2,\Bas}{\FC}(\br{1},\br{2})
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= \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
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\end{gather}
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\end{subequations}
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@ -548,8 +550,8 @@ See {\SI} for raw data associated with the atomization energies of the four diat
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{acknowledgements}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The authors would like to thank the \emph{Centre National de la Recherche Scientifique} (CNRS) for funding.
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This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738), and CALMIP (Toulouse) under allocation 2019-18005.
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The authors would like to thank the \emph{Centre National de la Recherche Scientifique} (CNRS) and the \emph{Institut des Sciences du Calcul et des Donn\'ees} for funding.
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This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738) and CALMIP (Toulouse) under allocation 2019-18005.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{acknowledgements}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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