Manu: saving work

Manuscript/eDFT.tex

@@ -97,6 +97,7 @@
 \newcommand{\bmg}{\bm{\Gamma}} % orbital rotation vector
 \newcommand{\bfx}{\bf{x}}
 \newcommand{\bfr}{\bf{r}}
+\DeclareMathOperator*{\argmin}{arg\,min}
 %%%% 
 
 \begin{document}	
@@ -636,7 +637,16 @@ functional where (i) the ghost-interaction correction functional $\overline{E}^{
 Hx}[n]$ in
 Eq.~(\ref{eq:exact_GIC}) is
 neglected, for simplicity, and (ii) the weight-dependent correlation
-energy is descrive at the local density level of approximation. More
+energy is described at the local density level of approximation. 
+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
+{E}^{{\bw}}_{\rm
+c}[n]=\int d\br\;n(\br)\epsilon_{c}^{\bw}(n(\br)). 
+\eeq
+More
 details about the construction of such a functional will be given in the
 following. In order to remove ghost interactions from the variational energy
 expression used in the first step, we then employ the (in-principle-exact)
@@ -645,6 +655,36 @@ step, the response of the individual density matrices to weight
 variations (last term on the right-hand side of
 Eq.~(\ref{eq:exact_ind_ener_OEP-like})) is neglected. The complete GIC
 procedure can be summarized as follows,    
+\beq
+{\bmg}^{\bw}\approx\argmin_{{\bm\gamma}^{\bw}}
+\Big\{
+{\rm
+Tr}\left[{\bm \gamma}^{{\bw}}{\bm h}\right]+W_{\rm
+HF}\left[{\bm\gamma}^{\bw}\right]
++
+{E}^{{\bw}}_{\rm
+c}\left[n_{\bm\gamma^{\bw}}\right]
+%+E^{{\bw}}_{\rm c}\left[n_{\hat{\Gamma}^{{\bw}}}\right]
+\Big\},
+\nonumber\\
+\eeq
+and
+\beq
+E^{(I)}&&\approx{\rm
+Tr}\left[{\bmg}^{(I)}{\bm h}\right]
++\frac{1}{2} \Tr(\bmg^{(I)} \, \bG \,
+\bmg^{(I)})
+\nonumber\\
+&&+{E}^{{\bw}}_{\rm
+c}\left[n_{\bmg^{\bw}}\right]
++\int d\br\,\dfrac{\delta {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 {E}^{{\bw}}_{\rm
+c}\left[n\right]}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}}
+.
+\eeq
 
 \alert{Secs. \ref{sec:KS-eDFT}-\ref{sec:E_I} should maybe be moved to an appendix or merged
 with the theory section (?)}