Manu: done with II.A

Manuscript/eDFT.tex

@@ -336,27 +336,62 @@ we finally recover from Eqs.~\eqref{eq:KS_ens_density} and
 	\right|_{\n{}{} = \n{\opGam{\bw}}{}}.
 \end{split}
 \eeq
-Note that, when $\bw=0$, the ensemble correlation functional becomes the
+Note that, when $\bw=0$, the ensemble correlation functional reduces to the
-conventional (ground-state) correlation functional $E_{\rm c}[n]$ and
+conventional (ground-state) correlation functional $E_{\rm c}[n]$. As a
-the ensemble density reduces to the ground-state one $n^{(0)}$. As a
+result, the regular KS-DFT expression is recovered from
-result,  
+Eq.~(\ref{eq:exact_ener_level_dets}) for the ground-state energy:
-Eq.~(\ref{eq:exact_ener_level_dets}) can be simplified as follows:
 \beq
+\E{}{(0)}=\mel*{\Det{(0)}}{\hH}{\Det{(0)}} +
+\E{c}{}[\n{\Det{(0)}}{}],
+\eeq
+or, equivalently,
+\beq\label{eq:gs_ener_level_gs_lim}
 \begin{split}
-	\E{}{(I)} 
+\E{}{(0)}&=\mel*{\Det{(0)}}{\hat{H}[\n{\Det{(0)}}{}]}{\Det{(0)}} 
-	& = \mel*{\Det{(I)}}{\hH}{\Det{(I)}} + \E{c}{}[\n{}{(0)}]
 \\
-	& + \int \fdv{\E{c}{}[\n{}{(0)}]}{\n{}{}(\br{})}
+&
-	\qty[ \n{\Det{(I)}}{}(\br{}) - \n{\opGam{\bw}}{}(\br{}) ] d\br{}
++
-	\\
+\E{c}{}[\n{\Det{(0)}}{}]
-	&+
+-\int \fdv{\E{c}{}[\n{\Det{(0)}}{}]}{\n{}{}(\br{})}\n{\Det{(0)}}{}(\br{})
-	\sum_{K>0} \qty(\delta_{IK} - \ew{K} )
+,
-	\left.
-	\pdv{\E{c}{\bw}[\n{}{}]}{\ew{K}}
-	\right|_{\n{}{} = \n{\opGam{\bw}}{}}.
 \end{split}
 \eeq
-\titou{Manu, shall we mention that the last term in Eq.~\eqref{eq:exact_ener_level_dets} corresponds to the derivative discontinuity (DD)?}
+where the ground-state-density-functional Hamiltonian reads
+\beq
+\hat{H}[\n{\Det{(0)}}{}]=\hH+
+\sum^N_{i=1}\fdv{\E{c}{}[\n{\Det{(0)}}{}]}{\n{}{}(\br{i})}.
+\eeq
+Note that, when divided by the total number $N$ of electrons, the sum of
+the last two terms on the right-hand side of
+Eq.~(\ref{eq:gs_ener_level_gs_lim}) match the correlation component of
+Levy--Zahariev's shift in the KS potential~\cite{Levy_2014}. 
+The excited-state ($I>0$) energy level expressions
+can also be simplified, thus leading to  
+\beq\label{eq:excited_ener_level_gs_lim}
+\begin{split}
+	\E{}{(I)} 
+	& = \mel*{\Det{(I)}}{\hat{H}[\n{\Det{(0)}}{}]}{\Det{(I)}} 
+      +
+      \left. 
+	\pdv{\E{c}{\bw}[\n{\Det{(0)}}{}]}{\ew{I}}
+	\right|_{\bw=0}.
+	\\
+	&
+       +
+        \E{c}{}[\n{\Det{(0)}}{}]
+        -\int \fdv{\E{c}{}[\n{\Det{(0)}}{}]}{\n{}{}(\br{})}\n{\Det{(0)}}{}(\br{})
+\end{split}
+\eeq 
+As readily seen from Eqs.~(\ref{eq:gs_ener_level_gs_lim}) and
+(\ref{eq:excited_ener_level_gs_lim}), any constant shift $\delta
+\E{c}{}[\n{\Det{(0)}}{}]/\delta n({\bf r})\rightarrow \delta
+\E{c}{}[\n{\Det{(0)}}{}]/\delta n({\bf r})+C$ in the correlation
+potential leaves the individual energy levels unchanged. As a result, in
+this context,
+the correlation derivative discontinuities that is expected to appear when the
+excitation process occurs~\cite{Levy_1995} are fully described by the ensemble
+correlation derivatives [second term on the right-hand side of
+Eq.~(\ref{eq:excited_ener_level_gs_lim})].    
 
 %%%%%%%%%%%%%%%%
 \subsection{One-electron reduced density matrix formulation}