Manu: saving work in the discussion (Fig. 5) and the theory section.

Manuscript/eDFT.tex

@@ -965,7 +965,7 @@ individual correlation energy per particle for the ensemble one.
 }
 
 \manu{
-Let us finally note that the weighted sum of the
+Let us finally note that, while 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}
@@ -974,29 +974,35 @@ the KS-eLDA ensemble energy:
 \\
 &=
 \E{eLDA}{\bw}
--\WHF[\bGam{\bw}]+\sum_{I\geq0}\ew{I}\WHF[ \bGam{(I)}].
+-\WHF[\bGam{\bw}]+\sum_{I\geq0}\ew{I}\WHF[ \bGam{(I)}],
 \end{split}
 \eeq
-}
-
-\titou{
-The corresponding excitation energies are
+the excitation energies computed from the KS-eLDA individual energy level
+expressions in Eq. \eqref{eq:EI-eLDA} simply reads 
 \beq\label{eq:Om-eLDA}
+\begin{split}
 	\Ex{eLDA}{(I)} 
-	= 
+	=& 
 	\Ex{HF}{(I)}
-	+ \int \fdv{\E{c}{\bw}[\n{\bGam{\bw}}{}]}{\n{}{}(\br{})} 
-	\qty[ \n{\bGam{(I)}}{}(\br{}) - \n{\bGam{(0)}}{}(\br{}) ] d\br{} 
+	+ \int 
+\qty[\e{c}{{\bw}}(\n{}{})+n\pdv{\e{c}{{\bw}}(\n{}{})}{\n{}{}}]	
+_{\n{}{} =
+\n{\bGam{\bw}}{}(\br{})}
+\\
+&\times\qty[ \n{\bGam{(I)}}{}(\br{}) - \n{\bGam{(0)}}{}(\br{}) ] d\br{} 
 + \DD{c}{(I)},
+\end{split}
 \eeq
-with $\Ex{HF}{(I)} = \E{HF}{(I)} - \E{HF}{(0)}$, and where 
+where the HF-like excitation energies $\Ex{HF}{(I)} = \E{HF}{(I)} -
+\E{HF}{(0)}$ are determined from a single set of ensemble KS orbitals and 
 \beq\label{eq:DD-eLDA}
 	\DD{c}{(I)}
 	= \int \n{\bGam{\bw}}{}(\br{})
 	\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{I}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}
 \eeq
-is the ensemble correlation derivative contribution to the excitation energy.
+is the eLDA correlation ensemble derivative contribution to the $I$th excitation energy.
 }
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Density-functional approximations for ensembles}
 \label{sec:eDFA}
@@ -1325,7 +1331,7 @@ The reverse is observed for the second excitation energy.
 Figure \ref{fig:EvsL} reports the excitation energies (multiplied by $L^2$) for various methods and box sizes in the case of 5-boxium (\ie, $\nEl = 5$).
 Similar graphs are obtained for the other $\nEl$ values and they can be found in the {\SI} alongside the numerical data associated with each method.
 For small $L$, the single and double excitations can be labeled as
-``pure'', \manu{as revealed by a detailed analysis of the FCI wavefunctions}.
+``pure'', \manu{as revealed by a thorough analysis of the FCI wavefunctions}.
 In other words, each excitation is dominated by a sole, well-defined reference Slater determinant.
 However, when the box gets larger (\ie, $L$ increases), there is a strong mixing between the different excitation degrees.
 In particular, the single and double excitations strongly mix, which makes their assignment as single or double excitations more discutable. \cite{Loos_2019}
@@ -1398,7 +1404,8 @@ the same quality as the one obtained in the linear response formalism
 (such as TDLDA). On the other hand, the double
 excitation energy only deviates
 from the FCI value by a few tenth of percent.
-Moreover, we note that, in the strong correlation regime (left graph of
+Moreover, we note that, in the strong correlation regime
+(left\manu{Manu: you mean right?} graph of
 Fig.~\ref{fig:EvsN}), the single excitation
 energy obtained at the equiensemble KS-eLDA level remains in good
 agreement with FCI and is much more accurate than the TDLDA and TDA-TDLDA excitation energies which can deviate by up to $60 \%$.
@@ -1423,7 +1430,12 @@ electrons.
 \end{figure}
 %%% %%% %%%
 
-It is also interesting to investigate the influence of the ensemble correlation derivative $\DD{c}{(I)}$ [defined in Eq.~\eqref{eq:DD-eLDA}] on both the single and double excitations.
+It is also interesting to investigate the influence of the
+\manu{correlation ensemble derivative contribution} $\DD{c}{(I)}$
+\manu{to the $I$th excitation energy} [see Eq.~\eqref{eq:DD-eLDA}].
+\manu{In
+our case, both single ($I=1$) and double ($I=2$) excitations are
+considered}.
 To do so, we have reported in Fig.~\ref{fig:EvsL_DD}, in the case of 3-boxium, the error percentage (with respect to FCI) as a function of the box length $L$
 on the excitation energies obtained at the KS-eLDA with and without $\DD{c}{(I)}$ [\ie, the last term in Eq.~\eqref{eq:Om-eLDA}].
 %\manu{Manu: there is something I do not understand. If you want to