From e902a0504d866e6a38bfe6f2ca2c9ec18b4a9bae Mon Sep 17 00:00:00 2001 From: Julien Toulouse Date: Wed, 24 Apr 2019 17:51:57 +0200 Subject: [PATCH] my last corrections --- Manuscript/G2-srDFT.tex | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/Manuscript/G2-srDFT.tex b/Manuscript/G2-srDFT.tex index 81b5f04..5e219ce 100644 --- a/Manuscript/G2-srDFT.tex +++ b/Manuscript/G2-srDFT.tex @@ -329,15 +329,15 @@ The local-density approximation (LDA) of the ECMD complementary functional is de \label{eq:def_lda_tot} \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} parameterized in Ref.~\onlinecite{PazMorGorBac-PRB-06} as a function of the spin densities $\qty{\n{\sigma}{}}_{\sigma=\uparrow,\downarrow}$ and the range-separation parameter $\mu$. +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 short-range correlation energy per electron of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parameterized in Ref.~\onlinecite{PazMorGorBac-PRB-06}. 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 +In order to correct such a defect, inspired by the recent functional proposed by some of the authors~\cite{FerGinTou-JCP-18}, we propose here a new Perdew-Burke-Ernzerhof (PBE)-based ECMD functional \begin{equation} \label{eq:def_pbe_tot} \titou{\bE{\PBE}{\Bas}[\n{}{},\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{where $s=|\nabla n|/n^{4/3}$ is the reduced density gradient}. \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} \begin{gather} @@ -345,10 +345,10 @@ In order to correct such a defect, we propose here a new Perdew-Burke-Ernzerhof \titou{\be{\text{c,md}}{\sr,\PBE}(\n{}{},s,\zeta,\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{1 + \beta(\n{}{},s,\zeta) \rsmu{}{3} },} \\ \label{eq:beta_cmdpbe} - \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)}.} + \titou{\beta(\n{}{},s,\zeta) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\n{2}{\UEG}(\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 $\n{2}{\UEG}(r_{12},\n{}{},\zeta) = \n{}{2} (1-\zeta^2) g(r_{12},n)$} with the parametrization of the UEG on-top pair-distribution function $g(0,n)$ given in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. +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{$\n{2}{\Bas}(\br{},\br{}) \approx \n{2}{\UEG}(\n{}{}(\br{}),\zeta(\br{}))$, where $\n{2}{\UEG}(\n{}{},\zeta) \approx \n{}{2} (1-\zeta^2) g_0(n)$} 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}.