From 892fef9e66ca623c0aa6d71b9b3c2c9a63a487b0 Mon Sep 17 00:00:00 2001 From: Emmanuel Fromager Date: Tue, 25 Feb 2020 17:29:31 +0100 Subject: [PATCH] Manu: done with III (see my comments/corrections) --- Manuscript/eDFT.tex | 37 ++++++++++++++++++++++++++++++------- 1 file changed, 30 insertions(+), 7 deletions(-) diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex index 11e4ca1..13b2f71 100644 --- a/Manuscript/eDFT.tex +++ b/Manuscript/eDFT.tex @@ -933,22 +933,23 @@ One of the main driving force behind the popularity of DFT is its ``universal'' Obviously, the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} does not have this feature as it does depend on the number of electrons constituting the FUEG. However, one can partially cure this dependency by applying a simple embedding scheme in which the two-electron FUEG (the impurity) is embedded in the IUEG (the bath). The weight-dependence of the correlation functional is then carried exclusively by the impurity [\ie, the functional defined in Eq.~\eqref{eq:ecw}], while the remaining correlation effects are provided by the bath (\ie, the usual LDA correlation functional). -Following this simple strategy, which is further theoretically justified by the generalized adiabatic connection formalism for ensembles (GACE) originally derived by Franck and Fromager, \cite{Franck_2014} we propose to \emph{shift} the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} as follows: +Following this simple strategy, which can be further theoretically justified by the generalized adiabatic connection formalism for ensembles (GACE) originally derived by Franck and Fromager, \cite{Franck_2014} we propose to \emph{shift} the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} as follows: \begin{equation} \label{eq:becw} - \be{c}{\bw}(\n{}{}) = (1-\ew{1}-\ew{2}) \be{c}{(0)}(\n{}{}) + \ew{1} \be{c}{(1)}(\n{}{}) + \ew{2} \be{c}{(2)}(\n{}{}), + \manu{\e{c}{\bw}(\n{}{})\rightarrow}\be{c}{\bw}(\n{}{}) = (1-\ew{1}-\ew{2}) \be{c}{(0)}(\n{}{}) + \ew{1} \be{c}{(1)}(\n{}{}) + \ew{2} \be{c}{(2)}(\n{}{}), \end{equation} where \begin{equation} \be{c}{(I)}(\n{}{}) = \e{c}{(I)}(\n{}{}) + \e{c}{\text{LDA}}(\n{}{}) - \e{c}{(0)}(\n{}{}). \end{equation} -The local-density approximation (LDA) correlation functional, +In the following, we will use the LDA correlation functional that has been specifically designed for 1D systems in +Ref.~\onlinecite{Loos_2013}: \begin{equation} \label{eq:LDA} \e{c}{\text{LDA}}(\n{}{}) = a_1^\text{LDA} F\qty[1,\frac{3}{2},a_3^\text{LDA}, \frac{a_1^\text{LDA}(1-a_3^\text{LDA})}{a_2^\text{LDA}} {\n{}{}}^{-1}], \end{equation} -specifically designed for 1D systems in Ref.~\onlinecite{Loos_2013} as been used, where $F(a,b,c,x)$ is the Gauss hypergeometric function, \cite{NISTbook} and +where $F(a,b,c,x)$ is the Gauss hypergeometric function, \cite{NISTbook} and \begin{subequations} \begin{align} a_1^\text{LDA} & = - \frac{\pi^2}{360}, @@ -958,7 +959,10 @@ specifically designed for 1D systems in Ref.~\onlinecite{Loos_2013} as been used a_3^\text{LDA} & = 2.408779. \end{align} \end{subequations} -Equation \eqref{eq:becw} can be recast +\manu{Note that the strategy described in Eq.~(\ref{eq:becw}) is general and +can be applied to real (higher-dimension) systems.} In order to make the +connection with the GACE formalism \cite{Franck_2014,Deur_2017} more explicit, one may +recast Eq.~\eqref{eq:becw} as \begin{equation} \label{eq:eLDA} \begin{split} @@ -968,14 +972,33 @@ Equation \eqref{eq:becw} can be recast & + \ew{1} \qty[\e{c}{(1)}(\n{}{})-\e{c}{(0)}(\n{}{})] + \ew{2} \qty[\e{c}{(2)}(\n{}{})-\e{c}{(0)}(\n{}{})], \end{split} \end{equation} -which nicely highlights the centrality of the LDA in the present eDFA. +or, equivalently, +\begin{equation} +\label{eq:eLDA_gace} +\begin{split} + \be{c}{\bw}(\n{}{}) + & = \e{c}{\text{LDA}}(\n{}{}) + \\ + & + \sum_{K>0}\int_0^{\ew{K}} +\qty[\e{c}{(K)}(\n{}{})-\e{c}{(0)}(\n{}{})]d\xi_K, +\end{split} +\end{equation} +where the $K$th correlation excitation energy (per electron) is integrated over the +ensemble weight $\xi_K$ at fixed (uniform) density $n$. +Eq.~(\ref{eq:eLDA_gace}) nicely highlights the centrality of the LDA in the present eDFA. In particular, $\be{c}{(0,0)}(\n{}{}) = \e{c}{\text{LDA}}(\n{}{})$. Consequently, in the following, we name this correlation functional ``eLDA'' as it is a natural extension of the LDA for ensembles. Finally, we note that, by construction, +\manu{ \begin{equation} \pdv{\be{c}{\bw}(\n{}{})}{\ew{J}} = \be{c}{(J)}(\n{}{}(\br{})) - \be{c}{(0)}(\n{}{}(\br{})). \end{equation} - +Manu: I guess that the "overlines" and the dependence in $\bf r$ of the +densities on the RHS should be removed. The final expression should be +\beq +\pdv{\be{c}{\bw}(\n{}{})}{\ew{J}} = \e{c}{(J)}(\n{}{}) - \e{c}{(0)}(\n{}{}). +\eeq +} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Computational details} \label{sec:comp_details}