From d680734f2fe6782f3c5f7fa4ce6198df5cb9a4f5 Mon Sep 17 00:00:00 2001 From: Emmanuel Fromager Date: Tue, 5 May 2020 17:18:32 +0200 Subject: [PATCH] Manu: saving work --- Revised_Manuscript/eDFT.tex | 42 ++++++++++++++++++++++++------------- 1 file changed, 27 insertions(+), 15 deletions(-) diff --git a/Revised_Manuscript/eDFT.tex b/Revised_Manuscript/eDFT.tex index d60c719..6e92049 100644 --- a/Revised_Manuscript/eDFT.tex +++ b/Revised_Manuscript/eDFT.tex @@ -605,15 +605,16 @@ As discussed further in Sec.~\ref{sec:eDFA}, these components can be extracted from a finite uniform electron gas model for which density-functional correlation excitation energies can be computed. -} -\titou{Note also that, here, only the correlation part of the ensemble -energy is treated at the -DFT level while we rely on HF exchange. -This is different from the usual context where both exchange and correlation are treated at the LDA level which gives compensation of errors.} -\manu{Manu: I changed a bit your sentence. Is this fine? Maybe we should add -that we are not interested in accurate ensemble energies. Error -cancellations may occur when computing excitation -energies, which are the quantities we are truly interested in.} +}\titou{Note also that, here, only the correlation part of the +energy will be treated at the +DFT level while we rely on HF for the exchange part. +This is different from the usual context where both exchange and +correlation are treated at the LDA level which gives compensation of +errors. Despite the errors +that might be introduced into the ensemble energy within such a scheme, +cancellations may actually occur when computing excitation energies, +which are energy {\it differences}.} +\manu{Manu: I changed a bit and complemented your sentence. Is this fine?} The resulting KS-eLDA ensemble energy obtained via Eq.~\eqref{eq:min_with_HF_ener_fun} reads @@ -640,6 +641,7 @@ where is the analog for ground and excited states (within an ensemble) of the HF energy, and \begin{gather} \begin{split} +\label{eq:Xic} \Xi_\text{c}^{(I)} & = \int \e{c}{\bw}(\n{\bGam{\bw}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{} \\ @@ -650,15 +652,24 @@ is the analog for ground and excited states (within an ensemble) of the HF energ \\ \end{split} \\ +\label{eq:Upsic} \Upsilon_\text{c}^{(I)} = \int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{}) \left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}. \end{gather} - -If, for analysis purposes, we Taylor expand the density-functional +\manurev{ +One may naturally wonder about the physical content of the above correlation energy +expressions. It is in fact difficult to readily distinguish from +Eqs.~(\ref{eq:Xic}) and (\ref{eq:Upsic}) purely (uncoupled) individual +contributions from mixed ones. For that purpose, we may +consider a density regime which has a weak deviation from the uniform +one. In such a regime, for which eLDA is a reasonable approximation, the +deviation of the individual densities from the ensemble one will be +weak. As a result, +we can} Taylor expand the density-functional correlation contributions around the $I$th KS state density -$\n{\bGam{(I)}}{}(\br{})$, the +$\n{\bGam{(I)}}{}(\br{})$, \manurev{so that} the second term on the right-hand side of Eq.~\eqref{eq:EI-eLDA} can be simplified as follows through first order in $\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$: @@ -670,9 +681,10 @@ $\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$: Therefore, it can be identified as an individual-density-functional correlation energy where the density-functional correlation energy per particle is approximated by the ensemble one for -all the states within the ensemble. - - +all the states within the ensemble. \manurev{This perturbation expansion +may not hold in realistic systems, which are all but uniform. Nevertheless, it +gives more insight into the eLDA approximation and becomes useful when +analyzing its performance, as shown in Sec. \ref{sec:res}.\\} Let us stress that, to the best of our knowledge, eLDA is the first density-functional approximation that incorporates ensemble weight dependencies explicitly, thus allowing for the description of derivative