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beta does not depend on mu

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Julien Toulouse 2019-04-19 20:38:30 +02:00
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Manuscript/G2-srDFT.tex
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@ -373,10 +373,10 @@ inspired by the recent functional proposed by some of the authors \cite{FerGinTo
\begin{subequations} \begin{subequations}
\begin{gather} \begin{gather}
\label{eq:epsilon_cmdpbe} \label{eq:epsilon_cmdpbe}
\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} }, \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{}{3} },
\\ \\
\label{eq:beta_cmdpbe} \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}(0,\{\n{\sigma}{}\})}. \beta(\{n_\sigma\},\{\nabla n_\sigma\}) = \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{gather}
\end{subequations} \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.~$\n{2,\Bas}{}(\br{},\br{}) \approx \n{2}{\UEG}(0,\{\n{\sigma}{}(\br{})\})$, with $\n{2}{\UEG}(0,\{n_\sigma\}) = 4 \; n_{\uparrow} \; n_{\downarrow} \; g(0,n)$ and the UEG on-top pair-distribution function $g(0,n)$, the $0$ standing for $r_{12}=0$, whose parametrization can be found 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.~$\n{2,\Bas}{}(\br{},\br{}) \approx \n{2}{\UEG}(0,\{\n{\sigma}{}(\br{})\})$, with $\n{2}{\UEG}(0,\{n_\sigma\}) = 4 \; n_{\uparrow} \; n_{\downarrow} \; g(0,n)$ and the UEG on-top pair-distribution function $g(0,n)$, the $0$ standing for $r_{12}=0$, whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}.