From 34607cf900b328b1757e12c6d95a202d27e7858d Mon Sep 17 00:00:00 2001
From: Emmanuel Fromager <manu@dhcp-35-235.u-strasbg.fr>
Date: Fri, 11 Oct 2019 12:43:35 +0200
Subject: [PATCH] Manu: started polishing the theory. Saving work.

---
 Manuscript/eDFT.tex | 28 +++++++++++++++++++++++-----
 1 file changed, 23 insertions(+), 5 deletions(-)

diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex
index f642624..8e905dd 100644
--- a/Manuscript/eDFT.tex
+++ b/Manuscript/eDFT.tex
@@ -381,9 +381,7 @@ Hxc}\left[n_{\bmg^{\bw}}\right]
 \Big\}
 ,
 \eeq
-where $n_{\bmg^{\bw}}$ 
-\manu{I am in favor of using $n_{{\bmg}^{{\bw}}}$ rather than $n_{\bmg^{\bw}}$,
-for clarity} is the density obtained from the density matrix
+where $n_{\bmg^{\bw}}$ is the density obtained from the density matrix
 ${\bmg}^{\bw}$ and ${\bm h}={\bm t}+{\bm v}_{\rm ext}$ is the total one-electron
 Hamiltonian matrix representation. When the minimum is reached, the
 ensemble energy and its derivatives can be used to extract individual
@@ -607,7 +605,7 @@ Tr}\left[{\bm F}^{(L)}\frac{\partial \bmg^{(L)}}{\partial w_K}\right].
 
 According to Eqs.~(\ref{eq:indiv_ener_from_ens}),
 (\ref{eq:exact_Eens_EEXX}), and (\ref{eq:XE_EEXX}),
-\beq
+\beq\label{eq:exact_ind_ener_OEP-like}
 E^{(I)}&&={\rm
 Tr}\left[{\bmg}^{(I)}{\bm h}\right]
 +\frac{1}{2} \Tr(\bmg^{(I)} \, \bG \,
@@ -628,7 +626,28 @@ Tr}\left[{\bm F}^{(L)}\frac{\partial \bmg^{(L)}}{\partial w_K}\right]
 .
 \eeq
 
+\subsection{Two-step ghost-interaction-corrected calculation}
 
+In order to compute (approximate) energy levels within generalized
+GOK-DFT we use a two-step procedure. The first step consists in
+optimizing variationally the ensemble density matrix according to
+Eq.~(\ref{eq:var_princ_Gamma_ens}) with an approximate Hxc ensemble
+functional where (i) the ghost-interaction correction functional $\overline{E}^{{\bw}}_{\rm
+Hx}[n]$ in
+Eq.~(\ref{eq:exact_GIC}) is
+neglected, for simplicity, and (ii) the weight-dependent correlation
+energy is descrive at the local density level of approximation. More
+details about the construction of such a functional will be given in the
+following. In order to remove ghost interactions from the variational energy
+expression used in the first step, we then employ the (in-principle-exact)
+expression in Eq.~(\ref{eq:exact_ind_ener_OEP-like}). In this second
+step, the response of the individual density matrices to weight
+variations (last term on the right-hand side of
+Eq.~(\ref{eq:exact_ind_ener_OEP-like})) is neglected. The complete GIC
+procedure can be summarized as follows,    
+
+\alert{Secs. \ref{sec:KS-eDFT}-\ref{sec:E_I} should maybe be moved to an appendix or merged
+with the theory section (?)}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{KS-eDFT for excited states}
 \label{sec:KS-eDFT}
@@ -719,7 +738,6 @@ The (state-independent) Levy-Zahariev shift and the so-called derivative discont
 \end{align}
 Because the Levy-Zahariev shift is state independent, it does not contribute to excitation energies [see Eq.~\eqref{eq:Ex}].
 The only remaining piece of information to define at this stage is the weight-dependent Hartree-exchange-correlation functional $\be{Hxc}{\bw}(\n{}{})$.
-
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Density-functional approximations for ensembles}
 \label{sec:eDFA}