Manu: saving work

Manuscript/eDFT.tex

@@ -121,7 +121,9 @@
 
 \begin{document}	
 
-\title{Weight-dependent local density-functional approximations for ensembles}
+\title{Weight-dependent local density-functional \manu{approximation to
+ensemble correlation energies}}
+%\title{Weight-dependent local density-functional approximations for ensembles}
 
 \author{Pierre-Fran\c{c}ois Loos}
 \email{loos@irsamc.ups-tlse.fr}
@@ -1272,10 +1274,11 @@ drastically.
 %correlation is strong. It is not clear to me which integral ($W_{01}?$)
 %drives the all thing.\\} 
 It is important to note that, even though the GIC removes the explicit
-quadratic terms from the ensemble energy, a non-negligible curvature
-remains in the GIC-eLDA ensemble energy. \manu{This might be due to
+quadratic \manu{Hx} terms from the ensemble energy, a non-negligible curvature
+remains in the GIC-eLDA ensemble energy \manu{when the electron
+correlation is strong}. \manu{This is due to
 \textit{(i)} the correlation eLDA
-functional, which induces linear or quadratic weight dependencies of the individual
+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 \textit{(ii)} the optimization of the
 ensemble KS orbitals in the presence of ghost-interaction errors {[see
@@ -1317,10 +1320,10 @@ the ground and first 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 individually. The reverse actually occurs for the ground state in the triensemble
+ensemble, rather than for each state separately. The reverse actually occurs for the ground state in the triensemble
 as $\ew{2}$ increases. The variations in the ensemble
 weights are essentially linear or quadratic. They are induced by the
-eLDA functional, as readily seen from
+eLDA correlation functional, as readily seen from
 Eqs.~\eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and
 \eqref{eq:Taylor_exp_DDisc_term}. 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
@@ -1445,7 +1448,7 @@ correlation ensemble derivative contribution $\DD{c}{(I)}$
 to the $I$th excitation energy [see Eq.~\eqref{eq:DD-eLDA}].
 In our case, both single ($I=1$) and double ($I=2$) excitations are considered.
 To do so, we have reported in Fig.~\ref{fig:EvsL_DD}, for $\nEl = 3$, $5$, and $7$, the error percentage (with respect to FCI) as a function of the box length $L$
-on the excitation energies obtained at the KS-eLDA with and without $\DD{c}{(I)}$ [\ie, the last term in Eq.~\eqref{eq:Om-eLDA}].
+on the excitation energies obtained at the KS-eLDA level with and without $\DD{c}{(I)}$ [\ie, the last term in Eq.~\eqref{eq:Om-eLDA}].
 %\manu{Manu: there is something I do not understand. If you want to
 %evaluate the importance of the ensemble correlation derivatives you
 %should only remove the following contribution from the $K$th KS-eLDA
@@ -1457,13 +1460,12 @@ on the excitation energies obtained at the KS-eLDA with and without $\DD{c}{(I)}
 %%rather than $E^{(I)}_{\rm HF}$
 %}
 We first stress that although for $\nEl=3$ both single and double excitation energies are 
-systematically improved, as the strength of electron correlation
-increases, when
+systematically improved (as the strength of electron correlation
+increases) when
 taking into account  
 the correlation ensemble derivative, this is not
 always the case for larger numbers of electrons.
-The influence of the correlation ensemble derivative becomes substantial in the strong correlation regime.
-For 3-boxium, in the zero-weight limit, its contribution is
+For 3-boxium, in the zero-weight limit, the ensemble derivative is
 significantly larger for the single
 excitation as compared to the double excitation; the reverse is observed in the equal-weight triensemble
 case.