diff --git a/Manuscript/G2-srDFT.tex b/Manuscript/G2-srDFT.tex index 873d17f..c46ab51 100644 --- a/Manuscript/G2-srDFT.tex +++ b/Manuscript/G2-srDFT.tex @@ -233,7 +233,7 @@ We define the effective operator as \cite{GinPraFerAssSavTou-JCP-18} \label{eq:def_weebasis} \W{\Bas}{}(\br{1},\br{2}) = \begin{cases} - \f{\Bas}{}(\br{1},\br{2})/\n{2}{}(\br{1},\br{2}), & \text{if $\n{2}{}(\br{1},\br{2}) \ne 0$,} + \f{\Bas}{}(\br{1},\br{2})/\n{2,\Bas}{}(\br{1},\br{2}), & \text{if $\n{2,\Bas}{}(\br{1},\br{2}) \ne 0$,} \\ \infty, & \text{otherwise,} \end{cases} @@ -241,7 +241,7 @@ We define the effective operator as \cite{GinPraFerAssSavTou-JCP-18} where \begin{equation} \label{eq:n2basis} - \n{2}{}(\br{1},\br{2}) + \n{2,\Bas}{}(\br{1},\br{2}) = \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}, \end{equation} 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), @@ -254,13 +254,15 @@ and $\V{pq}{rs}=\langle pq | rs \rangle$ are the usual two-electron Coulomb inte With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ satisfies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) \begin{equation} \label{eq:int_eq_wee} - \mel*{\wf{}{\Bas}}{\hWee{\updw}}{\wf{}{\Bas}} = \iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{}(\br{1},\br{2}) \dbr{1} \dbr{2}, + \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}, \end{equation} where $\hWee{\updw}$ contains only the opposite-spin component of $\hWee{}$. Because Eq.~\eqref{eq:int_eq_wee} can be rewritten as -\begin{equation} - \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}, -\end{equation} +\begin{eqnarray} + \iint r_{12}^{-1} \n{2,\Bas}{}(\br{1},\br{2}) \dbr{1} \dbr{2} = \phantom{xxxxxxxxx} +\nonumber\\ +\iint \W{\Bas}{}(\br{1},\br{2}) \n{2,\Bas}{}(\br{1},\br{2}) \dbr{1} \dbr{2}, +\end{eqnarray} 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. @@ -270,7 +272,7 @@ Thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{G \label{eq:lim_W} \lim_{\Bas \to \infty}\W{\Bas}{}(\br{1},\br{2}) = r_{12}^{-1}\ \end{equation} -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. +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. %================================================================= %\subsection{Range-separation function} @@ -281,7 +283,7 @@ A key quantity is the value of the effective interaction at coalescence of oppos % \label{eq:wcoal} % \W{\Bas}{}(\br{}) = \W{\Bas}{}(\br{},{\br{}}), %\end{equation} -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. +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. 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. Although this choice is not unique, we choose here the range-separation function \begin{equation} @@ -373,11 +375,11 @@ inspired by the recent functional proposed by some of the authors \cite{FerGinTo \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} }, \\ \label{eq:beta_cmdpbe} - \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}{}\})}. + \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}{}\})}. \end{gather} \end{subequations} -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}. -This represents a major computational saving without loss of accuracy for weakly correlated systems as we eschew the computation of $\n{2}{}(\br{})$. +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}. +This represents a major computational saving without loss of accuracy for weakly correlated systems as we eschew the computation of $\n{2,\Bas}{}(\br{},\br{})$. 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}. @@ -390,7 +392,7 @@ We then naturally split the basis set as $\Bas = \Cor \bigcup \BasFC$ (where $\C \begin{equation} \W{\Bas}{\FC}(\br{1},\br{2}) = \begin{cases} - \f{\Bas}{\FC}(\br{1},\br{2})/\n{2}{\FC}(\br{1},\br{2}), & \text{if $\n{2}{\FC}(\br{1},\br{2})\ne 0$}, + \f{\Bas}{\FC}(\br{1},\br{2})/\n{2,\Bas}{\FC}(\br{1},\br{2}), & \text{if $\n{2,\Bas}{\FC}(\br{1},\br{2})\ne 0$}, \\ \infty, & \text{otherwise,} \end{cases} @@ -402,7 +404,7 @@ with \f{\Bas}{\FC}(\br{1},\br{2}) = \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}, \\ - \n{2}{\FC}(\br{1},\br{2}) + \n{2,\Bas}{\FC}(\br{1},\br{2}) = \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}, \end{gather} \end{subequations} @@ -548,8 +550,8 @@ See {\SI} for raw data associated with the atomization energies of the four diat %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{acknowledgements} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -The authors would like to thank the \emph{Centre National de la Recherche Scientifique} (CNRS) for funding. -This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738), and CALMIP (Toulouse) under allocation 2019-18005. +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. +This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738) and CALMIP (Toulouse) under allocation 2019-18005. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \end{acknowledgements} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%