From c23ca2aacd40f5d502865a33937dbe0937d77e45 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Wed, 24 Apr 2019 08:42:44 +0200 Subject: [PATCH] minor --- Manuscript/G2-srDFT.tex | 29 ++++++++++++++--------------- 1 file changed, 14 insertions(+), 15 deletions(-) diff --git a/Manuscript/G2-srDFT.tex b/Manuscript/G2-srDFT.tex index cbadd33..c9e4e36 100644 --- a/Manuscript/G2-srDFT.tex +++ b/Manuscript/G2-srDFT.tex @@ -255,18 +255,18 @@ and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{ = \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2}, \end{equation} and $\V{pq}{rs}=\langle pq | rs \rangle$ are the usual two-electron Coulomb integrals. -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}} = \frac{1}{2}\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 recast as +\titou{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}} = \frac{1}{2}\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 recast as \begin{equation} \iint \frac{ \n{2}{\Bas}(\br{1},\br{2})}{r_{12}} \dbr{1} \dbr{2} = \iint \W{}{\Bas}(\br{1},\br{2}) \n{2}{\Bas}(\br{1},\br{2}) \dbr{1} \dbr{2}, \end{equation} -it intuitively motivates $\W{}{\Bas}(\br{1},\br{2})$ as a potential candidate for an effective interaction. +\titou{which} 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. 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}) @@ -323,17 +323,16 @@ Therefore, we approximate $\bE{}{\Bas}[\n{}{}]$ by ECMD functionals evaluated wi The local-density approximation (LDA) of the ECMD complementary functional is defined as \begin{equation} \label{eq:def_lda_tot} - \titou{\bE{\LDA}{\Bas}[\n{}{},\zeta,\rsmu{}{\Bas}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\n{}{}(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},} + \titou{\bE{\LDA}{\Bas}[\n{}{},\rsmu{}{\Bas}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\n{}{}(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},} \end{equation} where \titou{$\zeta = (\n{\uparrow}{} - \n{\downarrow}{})/\n{}{}$ is the spin-polarization and $\be{\text{c,md}}{\sr,\LDA}(\n{}{},\zeta,\rsmu{}{})$} is the ECMD correlation energy per electron of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parametrized in Ref.~\onlinecite{PazMorGorBac-PRB-06} as a function of the spin densities $\qty{\n{\sigma}{}}_{\sigma=\uparrow,\downarrow}$ and the range-separation parameter $\mu$. The short-range LDA correlation functional relies on the transferability of the physics of the UEG which is certainly valid for large $\mu$ but is known to over correlate for small $\mu$. In order to correct such a defect, we propose here a new Perdew-Burke-Ernzerhof (PBE)-based ECMD functional -\begin{multline} +\begin{equation} \label{eq:def_pbe_tot} - \titou{\bE{\PBE}{\Bas}[\n{}{},s,\zeta,\rsmu{}{\Bas}] =} - \\ - \titou{\int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},} -\end{multline} + \titou{\bE{\PBE}{\Bas}[\n{}{},s,\rsmu{}{\Bas}] = + \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},} +\end{equation} \titou{(where $s$ is the reduced gradient)} inspired by the recent functional proposed by some of the authors. \cite{FerGinTou-JCP-18} \titou{$\be{\text{c,md}}{\sr,\PBE}\qty(\n{}{},s,\zeta,\rsmu{}{})$} interpolates between the usual PBE correlation functional, \cite{PerBurErn-PRL-96} \titou{$\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$}, at $\rsmu{}{}=0$ and the exact large-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGorBac-PRB-06} yielding \begin{subequations} @@ -345,7 +344,7 @@ In order to correct such a defect, we propose here a new Perdew-Burke-Ernzerhof \titou{\beta(\n{}{},s,\zeta) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\n{2}{\UEG}(0,\n{}{},\zeta)}.} \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 on-top pair density by its UEG version, i.e.~\titou{$\n{2}{\Bas}(\br{},\br{}) \approx \n{2}{\UEG}(0,\n{}{}(\br{}),\zeta(\br{}))$, where $\n{2}{\UEG}(r_{12},\n{}{},\zeta) \approx 4 \n{\uparrow}{} \n{\downarrow}{} g(r_{12},n)$} with the parametrization of the UEG on-top pair-distribution function $g(0,n)$ given in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. +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.~\titou{$\n{2}{\Bas}(\br{},\br{}) \approx \n{2}{\UEG}(0,\n{}{}(\br{}),\zeta(\br{}))$, where $\n{2}{\UEG}(r_{12},\n{}{},\zeta) \approx \n{}{2} (1-\zeta^2) g(r_{12},n)$} with the parametrization of the UEG on-top pair-distribution function $g(0,n)$ given 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 $\bE{}{\Bas}[\n{\modZ}{\Bas}]$ is approximated by $\bE{\LDA}{\Bas}[\n{\modZ}{\Bas},\rsmu{}{\Bas}]$ or $\bE{\PBE}{\Bas}[\n{\modZ}{\Bas},\rsmu{}{\Bas}]$ where $\rsmu{}{\Bas}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.