From 90a16fc541f4a5c685366e94ed568b6a3c02416d Mon Sep 17 00:00:00 2001
From: Emmanuel Fromager <manu@Emmanuels-MacBook-Pro.local>
Date: Wed, 30 Oct 2019 13:38:08 +0100
Subject: [PATCH] Manu: started polishing the entire theory section

---
 Manuscript/eDFT.tex | 34 ++++++++++++++++++++++------------
 1 file changed, 22 insertions(+), 12 deletions(-)

diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex
index e960008..7974788 100644
--- a/Manuscript/eDFT.tex
+++ b/Manuscript/eDFT.tex
@@ -165,20 +165,12 @@ In these extreme conditions, where magnetic effects compete with Coulombic force
 Atomic units are used throughout.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Density-functional theory for ensembles}
+\section{Theory}
 \label{sec:eDFT}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Generalized KS-eDFT}
-\label{sec:geKS}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Kohn--Sham formulation of GOK-DFT}
 
-Since Hartree and exchange energy contributions cannot be separated in
-the one-dimensional case, we introduce in the following an alternative
-formulation of KS-eDFT where, in complete analogy with the generalized
-KS scheme, a HF-like Hartree-exchange energy is employed. This
-formulation is in principle exact and applicable to higher dimensions.
 Let us start from the analog for ensembles of Levy's universal
 functional,   
 \beq\label{eq:ens_LL_func}
@@ -199,7 +191,19 @@ where $\hat{n}(\br)$ is the density operator, $n_{\Psi}$ denotes the
 density of wavefunction $\Psi$, and
 $\bw\equiv\left(w^{(1)},w^{(2)},\ldots\right)$ is the collection of
 (decreasing) ensemble weights assigned to the excited states. Note that
-$w_0=1-\sum_{K>0}w_K\geq 0$. When $\bw=0$, the
+$w_0=1-\sum_{K>0}w_K\geq 0$.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Hybrid GOK-DFT}
+\label{sec:geKS}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+Since Hartree and exchange energy contributions cannot be separated in
+the one-dimensional case, we introduce in the following an alternative
+formulation of KS-eDFT where, in complete analogy with the generalized
+KS scheme, a HF-like Hartree-exchange energy is employed. This
+formulation is in principle exact and applicable to higher dimensions.
+
+ When $\bw=0$, the
 conventional ground-state universal functional is recovered,
 \beq
 F^{\bw=0}[n]=F[n]=\underset{\Psi\rightarrow n}{\rm min}
@@ -528,7 +532,7 @@ Hxc}(n)}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}(\br)}.
 \eeq
 }
 
-\subsection{OEP-like approach}
+\subsection{Exact ensemble exchange in hybrid GOK-DFT}
 
 
 In the exact theory, the minimizing density matrix in
@@ -710,6 +714,12 @@ c}(n)}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}(\br)}.
 
 \alert{Secs. \ref{sec:KS-eDFT}-\ref{sec:E_I} should maybe be moved to an appendix or merged
 with the theory section (?)}
+
+%%%%%%%%%%%%%%%%
+\section{Implementation}
+
+%%%%%%%%%%%%%%%%
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{KS-eDFT for excited states}
 \label{sec:KS-eDFT}