Manu: saving work in the theory section.

Manuscript/eDFT.tex

@@ -832,9 +832,15 @@ following ensemble local density \textit{approximation} (eLDA) will be employed:
 \beq\label{eq:eLDA_corr_fun}
 	\E{c}{\bw}[\n{}{}]\approx \int \n{}{}(\br{}) \e{c}{\bw}(\n{}{}(\br{})) d\br{},
 \eeq
-where the correlation energy per particle $\e{c}{\bw}(\n{}{})$ is \textit{weight dependent}. 
-As shown in Sec.~\ref{sec:eDFA}, the latter can be constructed, for
-example, from a finite uniform electron gas model. 
+\manu{
+where the ensemble correlation energy per particle 
+\beq\label{eq:decomp_ens_correner_per_part}
+\e{c}{\bw}(\n{}{})=\sum_{K\geq 0}w_K\be{c}{(K)}(\n{}{})
+\eeq
+}
+is \textit{weight dependent}. 
+As shown in Sec.~\ref{sec:eDFA}, the latter can be constructed
+from a finite uniform electron gas model. 
 %\titou{Manu, I think we should clearly define here what the expression of the ensemble energy with and without GOC.
 %What do you think?}
 
@@ -885,26 +891,19 @@ $\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$:
 Therefore, it can be identified as 
 an individual-density-functional correlation energy where the density-functional
 correlation energy per particle is approximated by the ensemble one for
-all the states within the ensemble. Note that the weighted sum of the
-individual KS-eLDA energy levels delivers a \manu{\it ghost-interaction-corrected (GIC)} version of
-the KS-eLDA ensemble energy: 
-\beq\label{eq:Ew-eLDA}
-\begin{split}
-\E{GIC-eLDA}{\bw}&=\sum_{I\geq0}\ew{I}\E{{eLDA}}{(I)}
-\\
-&=
-\E{eLDA}{\bw}
--\WHF[\bGam{\bw}]+\sum_{I\geq0}\ew{I}\WHF[ \bGam{(I)}].
-\end{split}
-\eeq
+all the states within the ensemble. 
 
-Let us finally stress that, to the best of our knowledge, eLDA is the first
+
+Let us stress that, to the best of our knowledge, eLDA is the first
 density-functional approximation that incorporates ensemble weight
 dependencies explicitly, thus allowing for the description of derivative
 discontinuities [see Eq.~\eqref{eq:excited_ener_level_gs_lim} and the
 comment that follows] {\it via} the last term on the right-hand side
-of Eq.~\eqref{eq:EI-eLDA}.
-\manu{
+of Eq.~\eqref{eq:EI-eLDA}. \manu{According to the decomposition of
+the ensemble
+correlation energy per particle in Eq.
+\eqref{eq:decomp_ens_correner_per_part}, the latter can be recast as
+follows:
 \beq
 \begin{split}
 &
@@ -924,12 +923,13 @@ d\br{}
 -
 \e{c}{\bw}(\n{\bGam{\bw}}{}(\br{}))
 \Big)\,\n{\bGam{\bw}}{}(\br{})
- d\br{}
+ d\br{},
 %\sum_{K>0}\delta_{IK}\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})}
 \end{split}
 \eeq
-thus leading to the following Taylor expansion
-\beq
+thus leading to the following Taylor expansion through first order in
+$\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$: 
+\beq\label{eq:Taylor_exp_DDisc_term}
 \begin{split}
 &
 \int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{})
@@ -958,6 +958,25 @@ d\br{}
 +\mathcal{O}\left([\n{\bGam{\bw}}{}-\n{\bGam{(I)}}{}]^2\right).
 \end{split}
 \eeq
+As readily seen from Eqs. \eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and \eqref{eq:Taylor_exp_DDisc_term}, the 
+role of the correlation ensemble derivative [last term on the right-hand side of Eq.
+\eqref{eq:EI-eLDA}] is, through zeroth order, to substitute the expected
+individual correlation energy per particle for the ensemble one.
+}
+
+\manu{
+Let us finally note that the weighted sum of the
+individual KS-eLDA energy levels delivers a \manu{\it ghost-interaction-corrected (GIC)} version of
+the KS-eLDA ensemble energy: 
+\beq\label{eq:Ew-eLDA}
+\begin{split}
+\E{GIC-eLDA}{\bw}&=\sum_{I\geq0}\ew{I}\E{{eLDA}}{(I)}
+\\
+&=
+\E{eLDA}{\bw}
+-\WHF[\bGam{\bw}]+\sum_{I\geq0}\ew{I}\WHF[ \bGam{(I)}].
+\end{split}
+\eeq
 }
 
 \titou{