Manu: done with the final energy expressions

Manuscript/eDFT.tex

@@ -627,7 +627,9 @@ Tr}\left[{\bm F}^{(L)}\frac{\partial \bmg^{(L)}}{\partial w_K}\right]
 .
 \eeq
 
-\subsection{Two-step ghost-interaction-corrected calculation}
+\subsection{Ensemble
+correlation LDA and ghost interaction correction 
+}
 
 In order to compute (approximate) energy levels within generalized
 GOK-DFT we use a two-step procedure. The first step consists in
@@ -642,7 +644,7 @@ At this
 level of approximation, the two correlation functionals $\overline{E}^{{\bw}}_{\rm
 c}[n]$ and ${E}^{{\bw}}_{\rm
 c}[n]$ are actually identical and can be expressed as
-\beq
+\beq\label{eq:eLDA_corr_fun}
 {E}^{{\bw}}_{\rm
 c}[n]=\int d\br\;n(\br)\epsilon_{c}^{\bw}(n(\br)). 
 \eeq
@@ -683,7 +685,27 @@ n({\br})}\left(n_{\bmg^{(I)}}(\br)-n_{\bmg^{\bw}}(\br)\right)
 \nonumber\\
 &&+\sum_{K>0}\left(\delta_{IK}-w_K\right)\left.       \dfrac{\partial {E}^{{\bw}}_{\rm
 c}\left[n\right]}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}}
-.
+,
+\eeq
+thus leading to the final implementable expression [see Eq.~(\ref{eq:eLDA_corr_fun})]
+\beq
+E^{(I)}&&\approx{\rm
+Tr}\left[{\bmg}^{(I)}{\bm h}\right]
++\frac{1}{2} \Tr(\bmg^{(I)} \, \bG \,
+\bmg^{(I)})
+\nonumber\\
+&&+\int d\br\,
+{\epsilon}^{{\bw}}_{\rm
+c}(n_{\bmg^{\bw}}(\br))\,n_{\bmg^{(I)}}(\br)
+\nonumber\\
+&&
++\int d\br\,\left.\dfrac{\partial {\epsilon}^{{\bw}}_{\rm
+c}(n)}{\partial n}\right|_{n=n_{\bmg^{\bw}}(\br)}n_{\bmg^{\bw}}(\br)\left(n_{\bmg^{(I)}}(\br)-n_{\bmg^{\bw}}(\br)\right)
+\nonumber\\
+&&
++\int d\br\,\sum_{K>0}\left(\delta_{IK}-w_K\right)n_{\bmg^{\bw}}(\br)\left.
+\dfrac{\partial {\epsilon}^{{\bw}}_{\rm
+c}(n)}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}(\br)}.
 \eeq
 
 \alert{Secs. \ref{sec:KS-eDFT}-\ref{sec:E_I} should maybe be moved to an appendix or merged