Manu: done with my revisions

Revised_Manuscript/eDFT.tex

@@ -663,9 +663,9 @@ expressions. It is in fact difficult to readily distinguish from
 Eqs.~(\ref{eq:Xic}) and (\ref{eq:Upsic}) purely (uncoupled) individual
 contributions from mixed ones. For that purpose, we may 
 consider a density regime which has a weak deviation from the uniform
-one. In such a regime, for which eLDA is a reasonable approximation, the
+one. In such a regime, where eLDA is a reasonable approximation, the
 deviation of the individual densities from the ensemble one will be
-weak. As a result, 
+small. As a result, 
 we can} Taylor expand the density-functional
 correlation contributions
 around the $I$th KS state density
@@ -682,10 +682,11 @@ Therefore, it can be identified as
 an individual-density-functional correlation energy where the density-functional
 correlation energy per particle is approximated by the ensemble one for
 all the states within the ensemble. \manurev{This perturbation expansion
-may not hold in realistic systems, which may deviate significantly from
-the uniform density regime. Nevertheless, it
-gives more insight into the eLDA approximation and becomes useful when
-rationalizing its performance, as illustrated in Sec. \ref{sec:res}.\\}  
+is of course less relevant for the (more realistic) systems that exhibit significant
+deviations from the uniform
+density regime. Nevertheless, it
+gives more insight into the eLDA approximation and it becomes useful when
+it comes to rationalize its performance, as illustrated in Sec. \ref{sec:res}.\\}  
 Let us stress that, to the best of our knowledge, eLDA is the first
 density-functional approximation that incorporates ensemble weight
 dependencies explicitly, thus allowing for the description of derivative
@@ -987,7 +988,9 @@ Its Tamm-Dancoff approximation version  (TDA-TDLDA) is also considered. \cite{Dr
 Concerning the ensemble calculations, two sets of weight are tested: the zero-weight
 (ground-state) limit where $\bw = (0,0)$ and the
 equi-triensemble (or equal-weight state-averaged) limit where $\bw = (1/3,1/3)$.
-\titou{Note that a zero-weight calculation does correspond to a conventional ground-state KS calculation with exact exchange and LDA correlation.}
+\titou{Note that a zero-weight calculation does correspond to a
+\trashEF{conventional} ground-state KS calculation with \manu{$100\%$} exact exchange and LDA correlation.}
+\manu{Manu: OK?}
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -1014,6 +1017,12 @@ linearly-interpolated ensemble energy) is represented
 in Fig.~\ref{fig:EvsW} as a function of $\ew{1}$ or $\ew{2}$ while
 fulfilling the restrictions on the ensemble weights to ensure the GOK
 variational principle [\ie, $0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$].
+\manurev{More precisely, we follow a continuous path that connects
+ground-state [$\bw=(0,0)$] and equiensemble [$\bw=(1/3,1/3)$]
+calculations. For convenience, we use two connected paths. The first
+one, for which $\ew{2}=0$ and $0\leq \ew{1}\leq 1/3$, relies on the 
+biensemble while the second one is defined as follows:
+$\ew{1}=1/3$ and $0\leq \ew{2}\leq 1/3$.}
 To illustrate the magnitude of the ghost-interaction error, we report the KS-eLDA ensemble energy with and without GIC as explained above {[see Eqs.~\eqref{eq:Ew-GIC-eLDA} and \eqref{eq:Ew-eLDA}]}.
 As one can see in Fig.~\ref{fig:EvsW}, without GIC, the
 ensemble energy becomes less and less linear as $L$
@@ -1082,7 +1091,7 @@ systematically enhances the weight dependence, due to the lowering of the
 ground-state energy, as $\ew{2}$ increases.         
 The reverse is observed for the second excited state. 
 \manurev{Finally, we notice that the crossover point of the 
-first excited-state energies based on
+first-excited-state energies based on
 bi- and triensemble calculations, respectively, disappears in the strong correlation
 regime [see the right panel of Fig. \ref{fig:EIvsW}], thus illustrating
 the importance of (individual and ensemble) densities, in
@@ -1123,7 +1132,7 @@ Overall, one clearly sees that, with
 equal weights, KS-eLDA yields accurate excitation energies for both single and double excitations.
 This conclusion is verified for smaller and larger numbers of electrons
 (see {\SI}).
-\titou{Except for the two-electron system where we observe cases of underestimation, eLDA usually overestimates double excitations, as evidenced by the numerical data gathered in the {\SI}).}
+\titou{Except for the two-electron system where we observe cases of underestimation, eLDA usually overestimates double excitations, as evidenced by the numerical data gathered in the {\SI}.}
 
 %%% FIG 4 %%%
 \begin{figure*}