Manu: saving work

Revised_Manuscript/eDFT.tex

@@ -1055,21 +1055,7 @@ Figure \ref{fig:EIvsW} reports the behavior of the three KS-eLDA individual ener
 Unlike in the exact theory, we do not obtain
 straight horizontal lines when plotting these
 energies, which is in agreement with  
-the curvature of the GIC-eLDA ensemble energy discussed previously. Interestingly, the
+the curvature of the GIC-eLDA ensemble energy discussed previously. The variations in the ensemble weights are essentially linear or quadratic. 
-individual energies do not vary in the same way depending on the state
-considered and the value of the weights. 
-\titou{On one hand,} we see for example that, within the biensemble (\ie, $\ew{2}=0$), the energies of
-the ground and \titou{second} excited-state increase with respect to the
-first-excited-state weight $\ew{1}$, thus showing that, in this
-case, we
-``deteriorate'' these states by optimizing the orbitals for the
-ensemble, rather than for each state separately. 
-\titou{The singly excited state is, on the other hand, stabilize in the biensemble, which is reasonable as the weight associated with this state increases.
-For the triensemble, as $\ew{2}$ increases, the energy of the ground state increases, while the energy of the first excited state remains stable with a slight increase at large $L$. 
-The second excited state is obviously stabilized by the increase of its weight in the ensemble.
-These are all very sensible observations.}
-
-The variations in the ensemble weights are essentially linear or quadratic. 
 \manurev{This can be rationalized as follows. As readily seen from
 Eqs.~\eqref{eq:EI-eLDA} and \eqref{eq:ind_HF-like_ener}, the individual
 HF-like energies do not depend explicitly on the weights, which means
@@ -1085,22 +1071,39 @@ weights $\bw$ [see Eqs.~\eqref{eq:ens1RDM},
 \eqref{eq:ens_dens_from_ens_1RDM}, and
 \eqref{eq:decomp_ens_correner_per_part}], the latter contributions will contain both linear and quadratic terms in
 $\bw$, as evidenced by Eq.~\eqref{eq:Taylor_exp_DDisc_term} [see the second term on the right-hand
-side].} 
+side].}\\ 
-!!! In the biensemble, the weight dependence of the first
+Interestingly, the
-excitation energy is \titou{increased?} as the correlation increases, \titou{while the weight dependence of the second excitation energy is reduced}. 
+individual energies do not vary in the same way depending on the state
-On the other hand, switching from a bi- to a triensemble
+considered and the value of the weights. 
-systematically enhances the weight dependence, due to the lowering of the
+\titou{On one hand,} we see for example that, within the biensemble (\ie, $\ew{2}=0$), the energies of
-ground-state energy, as $\ew{2}$ increases.
+the ground and \titou{second} excited-state increase with respect to the
-The reverse is observed for the second excited state.  !!! \titou{THIS PARAGRAPH MIGHT NEED MODIFICATIONS.}    
+first-excited-state weight $\ew{1}$, thus showing that, in this
-\trashPFL{Finally, we notice that the crossover point of the 
+case, we
-first-excited-state energies based on
+``deteriorate'' these states by optimizing the orbitals for the
-bi- and triensemble calculations, respectively, disappears in the strong correlation
+ensemble, rather than for each state separately. 
-regime [see the right panel of Fig.~\ref{fig:EIvsW}], thus illustrating
+\titou{The singly excited state is, on the other hand, stabilized in the biensemble, which is reasonable as the weight associated with this state increases.
-the importance of (individual and ensemble) densities, in
+For the triensemble, as $\ew{2}$ increases, the energy of the ground state increases, while the energy of the first excited state remains stable with a slight increase at large $L$. 
-addition to the  
+The second excited state is obviously stabilized by the increase of its weight in the ensemble.
-weights, in the evaluation of individual energies within
+\manurev{
-an ensemble.
+These are all very sensible observations.\\
+Let us finally stress that the (well-known) poor performance of the
+combined full HF-exchange/LDA correlation scheme in 
+ground-state DFT [$\bw=(0,0)$] is substantially improved for the
+ground state within the equiensemble [$\bw=(1/3,1/3)$]} (see the \SI).
+This is a
+remarkable and promising result. A similar improvement is observed for
+the first excited state, at least in the weak correlation regime,
+without deteriorating too much the second excited-state energy. 
 }
+%\manu{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]{fig5}