From 81f32550d3e9e4feb9b9a733ec9e061b6f2d7256 Mon Sep 17 00:00:00 2001
From: Emmanuel Fromager <manu@Emmanuels-MacBook-Pro.local>
Date: Sat, 9 May 2020 10:48:47 +0200
Subject: [PATCH] Manu: II

---
 Manuscript/FarDFT.tex | 17 +++++++++--------
 1 file changed, 9 insertions(+), 8 deletions(-)

diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex
index a9e4bde..3689055 100644
--- a/Manuscript/FarDFT.tex
+++ b/Manuscript/FarDFT.tex
@@ -275,7 +275,7 @@ Turning to GOK-DFT, the extension of the Hohenberg--Kohn theorem to ensembles al
 \end{equation}
 where $\vne(\br{})$ is the external potential and $\F{}{\bw}[\n{}{}]$ is the universal ensemble functional
 (the weight-dependent analog of the Hohenberg--Kohn universal functional for ensembles).
-In the KS formulation, this functional is decomposed as
+In the KS formulation\cite{Gross_1988b}, this functional can be decomposed as
 \begin{equation}\label{eq:FGOK_decomp}
       \F{}{\bw}[\n{}{}] 
       = \Tr{ \hgamdens{\bw} \hT }+ \E{\Ha}{}[\n{}{}]+\E{\xc}{\bw}[\n{}{}],
@@ -285,9 +285,9 @@ $\Tr{ \hgamdens{\bw} \hT } =\Ts{\bw}[\n{}{}]$ is the noninteracting ensemble kin
 \begin{equation}
 	\hgam{\bw}[n] = \sum_{I=0}^{\nEns-1} \ew{I} \dyad{\Det{I}{\bw}[n]}
 \end{equation}
-is the KS density-functional density matrix operator, and $\lbrace
+is the density-functional KS density matrix operator, and $\lbrace
 \Det{I}{\bw}[n] \rbrace_{0 \le I \le \nEns-1}$ are single-determinant
-wave functions (or configuration state functions). 
+wave functions (or configuration state functions\cite{Gould_2017}). 
 Their dependence on the density is determined from the ensemble density constraint 
 \begin{equation}
 	\sum_{I=0}^{\nEns-1} \ew{I} \n{\Det{I}{\bw}[n]}{}(\br) = \n{}{}(\br).
@@ -324,7 +324,7 @@ GOK variational principle, \cite{Gross_1988b,Senjean_2015}
 \left(\hT+\hVne\right)]+\E{\Ha}{}[\n{\hGam{\bw}}{}]+\E{\xc}{\bw}[\n{\hGam{\bw}}{}]\right\},
 \eeq 
 where $\n{\hGam{\bw}}{}(\br)=\sum_{I=0}^{\nEns - 1}
-\ew{I}\n{\overline{\Psi}^{(I)}}{}$ is the trial ensemble density. As a
+\ew{I}\n{\overline{\Psi}^{(I)}}{}$ is a trial ensemble density. As a
 result, the orbitals
 $\lbrace \MO{p}{\bw}(\br{}) \rbrace_{1 \le p \le
 \nOrb}$ from which the KS
@@ -351,7 +351,7 @@ where $\ON{p}{(I)}$ denotes the occupation of $\MO{p}{\bw}(\br{})$ in
 the $I$th KS wave function $\Det{I}{\bw}\left[n^{\bw}\right]$. Turning
 to the excitation energies, they can be extracted from the
 density-functional ensemble as follows [see Eqs.~\eqref{eq:diff_Ew}
-and \eqref{eq:Ew-GOK} and Refs.~\onlinecite{Gross_1988b,Deur_2019}]:
+and \eqref{eq:min_KS_DM} and Refs.~\onlinecite{Gross_1988b,Deur_2019}]:
 \beq
 	\label{eq:dEdw}
 	\Omega^{(I)}= \Eps{I}{\bw} - \Eps{0}{\bw} + \left. \pdv{\E{\xc}{\bw}[\n{}{}]}{\ew{I}} \right|_{\n{}{} = \n{}{\bw}},
@@ -372,15 +372,16 @@ densities $n_{\Psi_I}(\br)$ as the non-interacting KS ensemble is expected to re
 Nevertheless, these densities can still be extracted in principle
 exactly from the KS ensemble as shown by Fromager. \cite{Fromager_2020}
 
-In the following, we will work at the (weight-dependent) LDA
+In the following, we will work at the (weight-dependent) \manu{ensemble}
+LDA \manu{(eLDA)}
 level of approximation, \ie
 \beq
 \E{\xc}{\bw}[\n{}{}] 
-&\overset{\rm LDA}{\approx}&
+&\overset{\rm \manu{e}LDA}{\approx}&
 \int \e{\xc}{\bw}(\n{}{}(\br{})) \n{}{}(\br{}) d\br{},
 \\
  \fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})} 
-&\overset{\rm LDA}{\approx}&
+&\overset{\rm \manu{e}LDA}{\approx}&
 \left. \pdv{\e{\xc}{\bw{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{}(\br{})} \n{}{}(\br{}) + \e{\xc}{\bw{}}(\n{}{}(\br{})).
 \eeq
 We will also adopt the usual decomposition, and write down the weight-dependent xc functional as