minor corrections in Sec. I. This paper is great

Manuscript/eDFT.tex

@@ -148,12 +148,12 @@ Their accuracy is illustrated by computing single and double excitations in one-
 \label{sec:intro}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 Over the last two decades, density-functional theory (DFT)
-\cite{Hohenberg_1964,Kohn_1965} has become the method of choice for
+\cite{Hohenberg_1964,Kohn_1965,ParrBook} has become the method of choice for
 modeling the electronic structure of large molecular systems and
-materials. \cite{ParrBook}\manu{why this ref?}
+materials.
 The main reason is that, within DFT, the quantum contributions to the
 electronic repulsion energy --- the so-called exchange-correlation (xc)
-energy --- is rewritten as a functional of the electron density $n\equiv\n{}{}(\br{})$, the latter being a much simpler quantity than the many-electron wave function. 
+energy --- is rewritten as a functional of the electron density $\n{}{} \equiv \n{}{}(\br{})$, the latter being a much simpler quantity than the many-electron wave function. 
 The complexity of the many-body problem is then transferred to the xc
 density functional. 
 Despite its success, the standard Kohn-Sham (KS) formulation of DFT \cite{Kohn_1965} (KS-DFT) suffers, in practice, from various deficiencies. \cite{Woodcock_2002, Tozer_2003,Tozer_1999,Dreuw_2003,Sobolewski_2003,Dreuw_2004,Tozer_1998,Tozer_2000,Casida_1998,Casida_2000,Tapavicza_2008,Levine_2006}
@@ -167,11 +167,11 @@ 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) 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
+functional of the time-dependent density $\n{}{} \equiv \n{}{}(\br,t)$ and, as
 such, it should incorporate memory effects. Standard implementations of TDDFT rely on 
-the adiabatic approximation where these effects are neglected. In other
+the adiabatic approximation where these effects are neglected. \cite{Dreuw_2005} 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}
+As a result, double electronic excitations \titou{(where two electrons are simultaneously promoted by a single photon)} 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}
 
 When affordable (\ie, for relatively small molecules), time-independent
 state-averaged wave function methods
@@ -179,13 +179,13 @@ state-averaged wave function methods
 The basic idea is to describe a finite (canonical) ensemble of ground
 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
+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 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 ensemble 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.
@@ -217,31 +217,28 @@ with the ambition to turn, in the forthcoming future, GOK-DFT into a
 Starting from the ubiquitous local-density approximation (LDA), we
 design a weight-dependent ensemble correction based on a finite uniform
 electron gas from which density-functional excitation energies can be
-extracted. The present eDFA, \trashEF{is specially designed for the computation of
-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.
+extracted. The present eDFA, which can be seen as a natural
+extension of the 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 [that we combine with 
-ensemble exact exchange energies] in the particular case of
-\emph{strict} one-dimensional (1D) and
+ensemble correlation energies (that we combine with 
+ensemble exact exchange energies) in the particular case of
+\emph{strict} one-dimensional (1D) \trashPFL{and}
 spin-polarized systems. \cite{Loos_2012, Loos_2013a, Loos_2014a, Loos_2014b} 
-In other words, the Coulomb interaction used in this work describes
-particles\manu{Manu: the sentence sounds weird to me. An interaction does not
-describe particles ...} which are \emph{strictly} restricted to move within a 1D sub-space of three-dimensional space.
+In other words, the Coulomb interaction used in this work \titou{corresponds} to
+particles which are \emph{strictly} restricted to move within a 1D sub-space of three-dimensional space.
 Despite their simplicity, 1D models are scrutinized as paradigms for quasi-1D materials \cite{Schulz_1993, Fogler_2005a} such as carbon nanotubes \cite{Bockrath_1999, Ishii_2003, Deshpande_2008} or nanowires. \cite{Meyer_2009, Deshpande_2010}
 %Early models of 1D atoms using this interaction have been used to study the effects of external fields upon Rydberg atoms \cite{Burnett_1993, Mayle_2007} and the dynamics of surface-state electrons in liquid helium. \cite{Nieto_2000, Patil_2001}
 This description of 1D systems also has interesting connections with the exotic chemistry of ultra-high magnetic fields (such as those in white dwarf stars), where the electronic cloud is dramatically compressed perpendicular to the magnetic field. \cite{Schmelcher_1990, Lange_2012, Schmelcher_2012}
 In these extreme conditions, where magnetic effects compete with Coulombic forces, entirely new bonding paradigms emerge. \cite{Schmelcher_1990, Schmelcher_1997, Tellgren_2008, Tellgren_2009, Lange_2012, Schmelcher_2012, Boblest_2014, Stopkowicz_2015}
 
 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 calculation of
-individual energy levels.
+Exact and approximate formulations of GOK-DFT are discussed in Sec.~\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.
 Computational details needed to reproduce the results of the present work are reported in Sec.~\ref{sec:comp_details}.
-In Sec.~\ref{sec:res}, we illustrate the accuracy of the present eLDA by computing single and double excitations in one-dimensional many-electron systems in the weak, intermediate and strong correlation regimes.
+In Sec.~\ref{sec:res}, we illustrate the accuracy of the present eLDA functional by computing single and double excitations in 1D many-electron systems in the weak, intermediate and strong correlation regimes.
 Finally, we draw our conclusion in Sec.~\ref{sec:conclusion}.
 Atomic units are used throughout.