Manu: saving work

Revised_Manuscript/eDFT.tex

@@ -682,9 +682,10 @@ 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 are all but uniform. Nevertheless, it
-gives more insight into the eLDA approximation and becomes useful when 
-analyzing its performance, as shown in Sec. \ref{sec:res}.\\}  
+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}.\\}  
 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
@@ -1021,13 +1022,20 @@ drastically.
 It is important to note that, even though the GIC removes the explicit
 quadratic Hx terms from the ensemble energy, a non-negligible curvature
 remains in the GIC-eLDA ensemble energy when the electron
-correlation is strong. This is due to
-(i) the correlation eLDA
-functional, which contributes linearly (or even quadratically) to the individual
-energies [see Eqs.~\eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and
-\eqref{eq:Taylor_exp_DDisc_term}], and (ii) the optimization of the
+correlation is strong. \manurev{The latter ensemble energy is computed
+as the weighted
+sum of the individual KS-eLDA energies [see
+Eq.~(\ref{eq:Ew-eLDA})]. Therefore, its 
+curvature can only originate from the weight dependence of the
+individual energies.
+Note that such a dependence does not exist in the exact theory. Here,
+the individual density-functional eLDA correlation energies exhibit an
+explicit linear and quadratic dependence on the weights, as discussed
+further in the next paragraph. Note also that the individual KS-eLDA energies
+may gain an additional (implicit) dependence on the weights through the optimization of the
 ensemble KS orbitals in the presence of ghost-interaction errors [see
 Eqs.~\eqref{eq:min_with_HF_ener_fun} and \eqref{eq:WHF}].
+}
 
 %%% FIG 2 %%%
 \begin{figure*}
@@ -1069,11 +1077,19 @@ weights $\bw$ [see Eqs.~(\ref{eq:ens1RDM}),
 (\ref{eq:decomp_ens_correner_per_part})], the latter contributions will contain both linear and quadratic terms in
 $\bw$, as readily seen from Eq.~(\ref{eq:Taylor_exp_DDisc_term}) [see the second term on the right-hand
 side].} In the biensemble, the weight dependence of the first
-excited-state energy is reduced as the correlation increases. On the other hand, switching from a bi- to a triensemble
+excitation energy is reduced as the correlation increases. On the other hand, switching from a bi- to a triensemble
 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.
-
+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
+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
+addition to the  
+weights, in the evaluation of individual energies within
+an ensemble. 
+}
 %%% FIG 3 %%%
 \begin{figure}
 	\includegraphics[width=\linewidth]{EvsL_5}