Manu: gave the final expression for the individual energies

Manuscript/eDFT.tex

@@ -458,10 +458,34 @@ Hxc}(n^{\bw}(\br))\,n^{(I)}(\br)
 Hxc}(n)}{\partial n}\right|_{n=n^{\bw}(\br)}n^{\bw}(\br)\left(n^{(I)}(\br)-n^{\bw}(\br)\right)
 \nonumber\\
 &&
-+\sum_{K>0}\left(\delta_{IK}-w^{(K)}\right)\left.       \dfrac{\partial \overline{E}^{{\bw}}_{\rm
-Hxc}\left[n\right]}{\partial w^{(K)}}\right|_{n=n^{{\bw}}}.
++\int d\br\,\sum_{K>0}\left(\delta_{IK}-w^{(K)}\right)n^{{\bw}}(\br)\left.
+\dfrac{\partial \overline{\epsilon}^{{\bw}}_{\rm
+Hxc}(n)}{\partial w^{(K)}}\right|_{n=n^{{\bw}}(\br)}.
 \eeq
-
+\alert{
+or, equivalently,
+\beq
+&&E^{(I)}\rightarrow
+{\rm
+Tr}\left[{\bmg}^{(I)}{\bm h}\right]+
+\Tr\left[\left(\bmg^{(I)}-\dfrac{1}{2}\bmg^{\bw}\right) \, \bG \, \bmg^{\bw}\right]
+\nonumber\\
+&&
++\int d\br\,
+\dfrac{\delta \overline{E}^{{\bw}}_{\rm
+Hxc}\left[n^{\bw}\right]}{\delta
+n({\br})}\,n^{(I)}(\br)
+\nonumber\\
+&&
+-\int d\br\,\left.\dfrac{\partial \overline{\epsilon}^{{\bw}}_{\rm
+Hxc}(n)}{\partial n}\right|_{n=n^{\bw}(\br)}\Big(n^{\bw}(\br)\Big)^2
+\nonumber\\
+&&
++\int d\br\,\sum_{K>0}\left(\delta_{IK}-w^{(K)}\right)n^{{\bw}}(\br)\left.
+\dfrac{\partial \overline{\epsilon}^{{\bw}}_{\rm
+Hxc}(n)}{\partial w^{(K)}}\right|_{n=n^{{\bw}}(\br)}.
+\eeq
+}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{KS-eDFT for excited states}