Manu: saving work

Manuscript/eDFT.tex

@@ -545,7 +545,7 @@ and therefore
 Combining the latter Eqs. with
 Eqs. (\ref{eq:exact_GIC}), (\ref{eq:var_princ_Gamma_ens}) leads to
 the final ensemble energy expression
-\beq
+\beq\label{eq:exact_Eens_EEXX}
 E^{{\bw}}={\rm
 Tr}\left[{\bmg}^{{\bw}}{\bm h}\right]+\frac{1}{2} \sum_{L\geq0}w_L
 \Tr(\bmg^{(L)} \, \bG \, \bmg^{(L)})
@@ -567,7 +567,7 @@ c}\left[n_{\bmg^{\bw}}\right]
 \eeq
 
 For $K>0$
-\beq
+\beq\label{eq:XE_EEXX}
 &&\dfrac{\partial E^{{\bw}}}{\partial w_K}=
 {\rm
 Tr}\left[\left({\bmg}^{(K)}-{\bmg}^{(0)}\right){\bm h}\right]
@@ -594,6 +594,40 @@ Tr}\left[\dfrac{\partial\bmg^{(L)}}{\partial w_K}{\bm h}\right]
 c}\left[n_{\bmg^{\bw}}\right]}{\delta
 n({\br})}n_{\frac{\partial \bmg^{(L)}}{\partial w_K}}(\br).
 \eeq
+If we introduce individual Fock matrices
+\beq
+{\bm F}^{(L)}={\bm h}+\bG \,\bmg^{(L)}+\overline{\bm v}^{{\bw}}_{\rm
+c}\left[n_{\bmg^{\bw}}\right],
+\eeq
+the last three terms can be simply rewritten as 
+\beq
+\sum_{L\geq0}w_L{\rm
+Tr}\left[{\bm F}^{(L)}\frac{\partial \bmg^{(L)}}{\partial w_K}\right].
+\eeq
+
+According to Eqs.~(\ref{eq:indiv_ener_from_ens}),
+(\ref{eq:exact_Eens_EEXX}), and (\ref{eq:XE_EEXX}),
+\beq
+E^{(I)}&&={\rm
+Tr}\left[{\bmg}^{(I)}{\bm h}\right]
++\frac{1}{2} \Tr(\bmg^{(I)} \, \bG \,
+\bmg^{(I)})
+\nonumber\\
+&&+\overline{E}^{{\bw}}_{\rm
+c}\left[n_{\bmg^{\bw}}\right]
++\int d\br\,\dfrac{\delta \overline{E}^{{\bw}}_{\rm
+c}\left[n_{\bmg^{\bw}}\right]}{\delta
+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 \overline{E}^{{\bw}}_{\rm
+c}\left[n\right]}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}}
+\nonumber\\
+&&
++\sum_{K>0}\left(\delta_{IK}-w_K\right)\sum_{L\geq0}w_L{\rm
+Tr}\left[{\bm F}^{(L)}\frac{\partial \bmg^{(L)}}{\partial w_K}\right]
+.
+\eeq
+
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{KS-eDFT for excited states}