Manu: minor changes in the introduction

Manuscript/eDFT.tex

@@ -188,7 +188,7 @@ 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 GOK-DFT, the ensemble xc energy is a functional of the
-density but also a  
+density {\it and} 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
@@ -213,12 +213,13 @@ or atoms. \cite{Yang_2014,Yang_2017,Gould_2019_insights}
 Despite all these efforts, it is still unclear how weight dependencies
 can be incorporated into density-functional approximations. This problem is actually central not
 only in GOK-DFT but also in conventional (ground-state) DFT as the infamous derivative
-discontinuity problem that ocurs when crossing an integral number of
+discontinuity problem that occurs 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 address this problem,  
-with the ambition to turn, in the forthcoming future, GOK-DFT into a
+The present work is an attempt to address the ensemble weight dependence problem
+in GOK-DFT,  
+with the ambition to turn the theory, in the forthcoming future, 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
 design a weight-dependent ensemble correction based on a finite uniform
@@ -239,13 +240,13 @@ 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 Sec.~\ref{sec:eDFT}, 
-with a particular emphasis on the calculation of individual energy levels.
+with a particular emphasis on the extraction 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 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}.
+Finally, we draw our conclusions in Sec.~\ref{sec:conclusion}.
 Atomic units are used throughout.
 
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