Manu: repolished the introduction

Manuscript/eDFT.tex

@@ -160,15 +160,16 @@ Despite its success, the standard Kohn-Sham (KS) formulation of DFT \cite{Kohn_1
 The description of strongly multiconfigurational ground states (often
 referred to as ``strong correlation problem'') still remains a
 challenge. \cite{Gori-Giorgi_2010,Fromager_2015,Gagliardi_2017}
-Another issue, which is partly connected to the previous one, is the description of electronically-excited states. 
+Another issue, which is partly connected to the previous one, is the
+description of low-lying quasi-degenerate states. 
 
-The standard approach for modeling excited states in DFT is
+The standard approach for modeling excited states in a DFT framework is
 linear-response time-dependent DFT (TDDFT). \cite{Runge_1984,Casida,Casida_2012}
-In this case, the electronic spectrum relies on the (unperturbed) ground-state KS picture, which may break down when electron correlation is strong. 
-Moreover, in exact TDDFT, the xc energy is in fact an xc action \cite{Vignale_2008} which is a
+In this case, the electronic spectrum relies on the (unperturbed) pure-ground-state KS picture, which may break down when electron correlation is strong. 
+Moreover, in exact TDDFT, the xc energy is in fact an xc {\it action} \cite{Vignale_2008} which is a
 functional of the time-dependent density $n\equiv n(\br,t)$ and, as
-such, has memory. Standard implementations of TDDFT rely on 
-the adiabatic approximation where memory effects are neglected. In other
+such, it should incorporate memory effects. Standard implementations of TDDFT rely on 
+the adiabatic approximation where these effects are neglected. In other
 words, the xc functional is assumed to be local in time. \cite{Casida,Casida_2012}
 As a result, double electronic excitations are completely absent from the TDDFT spectrum, thus reducing further the applicability of TDDFT. \cite{Maitra_2004,Cave_2004,Mazur_2009,Romaniello_2009a,Sangalli_2011,Mazur_2011,Huix-Rotllant_2011,Elliott_2011,Maitra_2012,Sundstrom_2014,Loos_2019}
 
@@ -180,11 +181,11 @@ and excited states altogether, \ie, with the same set of orbitals.
 Interestingly, a similar approach exists in DFT. Referred to as
 Gross--Oliveira--Kohn (GOK) DFT\cite{Gross_1988a,Gross_1988b,Oliveira_1988}, it was proposed at the end of the 80's as a generalization
 of Theophilou's DFT for equiensembles. \cite{Theophilou_1979}
-In exact GOK-DFT, the ensemble xc energy not only a functional of the
+In GOK-DFT, the ensemble xc energy is a functional of the
 density but also a  
 function of the ensemble weights. Note that, unlike in conventional
-Boltzmann ensembles~\cite{Pastorczak_2013}, the weights (each state in the ensemble
-is assigned a given and fixed weight) are allowed to vary
+Boltzmann ensembles~\cite{Pastorczak_2013}, the ensemble weights [each state in the ensemble
+is assigned a given and fixed weight] are allowed to vary
 independently in a GOK ensemble.   
 The weight dependence of the xc functional plays a crucial role in the
 calculation of excitation energies.
@@ -193,12 +194,12 @@ It actually accounts for the derivative discontinuity contribution to energy gap
 %\titou{Shall we further discuss the derivative discontinuity? Why is it important and where is it coming from?}
 
 Even though GOK-DFT is in principle able to
-tackle near-degenerate situations and multiple-electron excitation
+describe near-degenerate situations and multiple-electron excitation
 processes, it has not
 been given much attention until quite recently. \cite{Franck_2014,Borgoo_2015,Kazaryan_2008,Gould_2013,Gould_2014,Filatov_2015,Filatov_2015b,Filatov_2015c,Gould_2017,Deur_2017,Gould_2018,Gould_2019,Sagredo_2018,Ayers_2018,Deur_2018,Deur_2019,Kraisler_2013, Kraisler_2014,Alam_2016,Alam_2017,Nagy_1998,Nagy_2001,Nagy_2005,Pastorczak_2013,Pastorczak_2014,Pribram-Jones_2014,Yang_2013a,Yang_2014,Yang_2017,Senjean_2015,Senjean_2016,Senjean_2018,Smith_2016}
 One of the reason is the lack, not to say the absence, of reliable
-density-functional approximations for ensembles (eDFAs) in the literature. 
-The most recent works on this topic are still fundamental and
+density-functional approximations for ensembles (eDFAs). 
+The most recent works dealing with this particular issue are still fundamental and
 exploratory, as they rely either on simple (but nontrivial) model
 systems
 \cite{Carrascal_2015,Deur_2017,Deur_2018,Deur_2019,Senjean_2015,Senjean_2016,Senjean_2018,Sagredo_2018,Senjean_2020,Fromager_2020,Gould_2019}
@@ -210,7 +211,7 @@ discontinuity problem that ocurs when crossing an integral number of
 electrons can be recast into a weight-dependent ensemble
 one. \cite{Senjean_2018,Senjean_2020}      
 
-The present work is an attempt to answer this question,  
+The present work is an attempt to address this problem,  
 with the ambition to turn, in the forthcoming future, GOK-DFT into a
 (low-cost) practical computational method for modeling excited states in molecules and extended systems.
 Starting from the ubiquitous local-density approximation (LDA), we
@@ -220,8 +221,8 @@ extracted. The present eDFA, \trashEF{is specially designed for the computation
 single and double excitations within GOK-DFT}, which can be seen as a natural
 extension of LDA, will be referred to as eLDA in the remaining of this paper.
 As a proof of concept, we apply this general strategy to
-ensemble correlation energies only [we use the orbital-dependent exact
-ensemble exchange energy for convenience] in the particular case of
+ensemble correlation energies [that we combine with 
+ensemble exact exchange energies] in the particular case of
 \emph{strict} one-dimensional (1D) and
 spin-polarized systems. \cite{Loos_2012, Loos_2013a, Loos_2014a, Loos_2014b} 
 In other words, the Coulomb interaction used in this work describes
@@ -234,9 +235,8 @@ In these extreme conditions, where magnetic effects compete with Coulombic force
 
 The paper is organized as follows. 
 Exact and approximate formulations of GOK-DFT are discussed in Section
-\ref{sec:eDFT}, with a particular emphasis on the extraction of
-individual energy levels and the calculation of individual exact exchange
-energies.
+\ref{sec:eDFT}, with a particular emphasis on the calculation of
+individual energy levels.
 In Sec.~\ref{sec:eDFA}, we detail the construction of the
 weight-dependent local correlation functional specially designed for the
 computation of single and double excitations within GOK-DFT.