Manu: saving work

Revised_Manuscript/eDFT.tex

@@ -605,15 +605,16 @@ As discussed further in Sec.~\ref{sec:eDFA}, these components can be
 extracted from a
 finite uniform electron gas model for which density-functional correlation excitation
 energies can be computed. 
-} 
-\titou{Note also that, here, only the correlation part of the ensemble
-energy is treated at the
-DFT level while we rely on HF exchange.
-This is different from the usual context where both exchange and correlation are treated at the LDA level which gives compensation of errors.}
-\manu{Manu: I changed a bit your sentence. Is this fine? Maybe we should add
-that we are not interested in accurate ensemble energies. Error
-cancellations may occur when computing excitation
-energies, which are the quantities we are truly interested in.}
+}\titou{Note also that, here, only the correlation part of the 
+energy will be treated at the
+DFT level while we rely on HF for the exchange part.
+This is different from the usual context where both exchange and
+correlation are treated at the LDA level which gives compensation of
+errors. Despite the errors
+that might be introduced into the ensemble energy within such a scheme,
+cancellations may actually occur when computing excitation energies,
+which are energy {\it differences}.}
+\manu{Manu: I changed a bit and complemented your sentence. Is this fine?}
 
 The resulting KS-eLDA ensemble energy obtained via Eq.~\eqref{eq:min_with_HF_ener_fun}
 reads 
@@ -640,6 +641,7 @@ where
 is the analog for ground and excited states (within an ensemble) of the HF energy, and
 \begin{gather}
 \begin{split}
+\label{eq:Xic}
 	\Xi_\text{c}^{(I)} 
 	& = \int \e{c}{\bw}(\n{\bGam{\bw}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
 	\\
@@ -650,15 +652,24 @@ is the analog for ground and excited states (within an ensemble) of the HF energ
 	\\
 \end{split}
 \\
+\label{eq:Upsic}
 	\Upsilon_\text{c}^{(I)} 
 	= \int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{})
 	\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}.
 \end{gather}
-
-If, for analysis purposes, we Taylor expand the density-functional
+\manurev{
+One may naturally wonder about the physical content of the above correlation energy
+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
+deviation of the individual densities from the ensemble one will be
+weak. As a result, 
+we can} Taylor expand the density-functional
 correlation contributions
 around the $I$th KS state density
-$\n{\bGam{(I)}}{}(\br{})$, the 
+$\n{\bGam{(I)}}{}(\br{})$, \manurev{so that} the 
 second term on the right-hand side
 of Eq.~\eqref{eq:EI-eLDA} can be simplified as follows through first order in
 $\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$: 
@@ -670,9 +681,10 @@ $\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$:
 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. 
-
-
+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}.\\}  
 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