Manu: added some notes to explain the final energy expressions

Manuscript/eDFT.tex

@@ -426,7 +426,45 @@ Hxc}\left[n\right]}{\partial w_K}\right|_{n=n^{{\bw}}}.
 \eeq
 %+\Tr(\bmg^{(I)} \, \bG \, \bmg^{\bw})
 %-\dfrac{1}{2}\Tr(\bmg^{\bw} \, \bG \, \bmg^{\bw})+...
-
+\alert{
+Note that
+\beq
+\overline{E}^{{\bw}}_{\rm
+Hx}\left[n^{{\bw}}\right]=
+\frac{1}{2} \sum_{L\geq0}w_L \Tr(\bmg^{(L)} \, \bG \, \bmg^{(L)})
+-\frac{1}{2}\Tr(\bmg^{\bw} \, \bG \, \bmg^{\bw})
+\nonumber\\
+\eeq
+and
+\beq
+\left.\dfrac{\partial \overline{E}^{{\bw}}_{\rm
+Hx}[n]}{\partial w_K}\right|_{n=n^{\bw}}&=&
+\frac{1}{2} \Tr(\bmg^{(K)} \, \bG \, \bmg^{(K)})-\frac{1}{2}
+\Tr(\bmg^{(0)} \, \bG \, \bmg^{(0)})
+\nonumber\\
+&&-\Tr\left[\left({\bmg}^{(K)}-{\bmg}^{(0)}\right) \, \bG \, \bmg^{\bw}\right] 
++\ldots
+\eeq
+thus leading to
+\beq
+&&\overline{E}^{{\bw}}_{\rm
+Hx}\left[n^{{\bw}}\right]+\sum_{K>0}\left(\delta_{IK}-w_K\right)\left.       \dfrac{\partial \overline{E}^{{\bw}}_{\rm
+Hx}\left[n\right]}{\partial w_K}\right|_{n=n^{{\bw}}}
+\nonumber\\
+&&=
+\overline{E}^{{\bw}}_{\rm
+Hx}\left[n^{{\bw}}\right]+\frac{1}{2} \Tr(\bmg^{(I)} \, \bG \, \bmg^{(I)})
+-\frac{1}{2} \sum_{L\geq0}w_L \Tr(\bmg^{(L)} \, \bG \, \bmg^{(L)})
+\nonumber\\
+&&-\Tr\left[\left({\bmg}^{(I)}-{\bmg}^{\bw}\right) \, \bG \, \bmg^{\bw}\right]
++\ldots
+\nonumber\\
+&&=\frac{1}{2} \Tr(\bmg^{(I)} \, \bG \,
+\bmg^{(I)})-\Tr\left[\left({\bmg}^{(I)}-\dfrac{1}{2}\bmg^{\bw}\right)
+\, \bG \, \bmg^{\bw}\right]
++\ldots
+\eeq
+}
 At the eLDA level:
 
 \beq