Manu: saving work in the discussion (Fig. 2).

Manuscript/eDFT.tex

@@ -1290,7 +1290,7 @@ ensemble KS orbitals in the presence of GIE {[see Eqs.~\eqref{eq:min_with_HF_ene
 
 Figure \ref{fig:EIvsW} reports the behavior of the three KS-eLDA individual energies as functions of the weights.
 \manu{
-We first notice that, unlike in the exact theory, we do not obtain
+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
@@ -1300,13 +1300,12 @@ We see for example that, within the biensemble [$w_2=0$], the energy of
 the ground state increases with the first-excited-state weight $w_1$, thus showing that we
 ``deteriorate'' this state a little by optimizing the orbitals also for
 the first excited state. The reverse actually occurs in the triensemble
-as $w_2$ increases. In most cases, the variations in the ensemble
-weights are essentially
-linear, as expected from the Taylor expansion in
-Eq.~(\ref{eq:Taylor_exp_ind_corr_ener_eLDA}) and the expression in
-Eq.~(\ref{eq:eLDA}) of the ensemble correlation energy per particle in
-eLDA. As the correlation increases, the weight dependence of the first
-excitation energy is reduced. On the other hand, switching from a bi- to a tri-ensemble
+as $w_2$ increases. The variations in the ensemble
+weights are essentially linear or quadratic. They are induced by the
+eLDA functional, as readily seen from
+Eqs.~(\ref{eq:Taylor_exp_ind_corr_ener_eLDA}) and
+\eqref{eq:Taylor_exp_DDisc_term}. In the biensemble, the weight dependence of the first
+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 in this case, as $w_2$ increases.         
 The reverse is observed for the second excitation energy.