add E vs w figure and text

Manuscript/eDFT.tex

@@ -323,6 +323,7 @@ we finally recover from Eqs.~\eqref{eq:KS_ens_density} and
 	\right|_{\n{}{} = \n{\opGam{\bw}}{}}.
 \end{split}
 \eeq
+\titou{The last term in Eq.~\eqref{eq:exact_ener_level_dets} corresponds to the derivative discontinuity (DD).}
 
 %%%%%%%%%%%%%%%%
 \subsection{One-electron reduced density matrix formulation}
@@ -662,7 +663,7 @@ and
 Note that this approximation, where the ensemble density matrix is optimized from a non-local exchange potential [rather than a local one, as expected from Eq.~\eqref{eq:var_ener_gokdft}] is applicable to real (three-dimensional) systems. 
 As readily seen from Eq.~\eqref{eq:eHF-dens_mat_func}, \textit{ghost interactions}~\cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}
 and curvature~\cite{} will be introduced in the Hx energy:
-\beq
+\beq\label{eq:WHF}
 \begin{split}
 	\WHF[\bGam{\bw}] 
 	& = \frac{1}{2} \sum_{K\geq 0} \ew{K}^2 \Tr[\bGam{(K)} \bG \bGam{(K)}]
@@ -679,7 +680,7 @@ Turning to the density-functional ensemble correlation energy, the following eLD
 where the correlation energy per particle is \textit{weight dependent}. 
 Its construction from a finite uniform electron gas model is discussed in detail in Sec.~\ref{sec:eDFA}. 
 Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with Eq.~\eqref{eq:eLDA_corr_fun} leads to our final energy level expression within eLDA: 
-\beq
+\beq\label{eq:EI}
 \begin{split}
 	\E{}{(I)} 
 	& \approx \Tr[\bGam{(I)} \bh] + \frac{1}{2} \Tr[\bGam{(I)} \bG \bGam{(I)}]
@@ -694,7 +695,7 @@ Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with Eq.~\eqref{eq:eLDA_corr_fun} le
 	\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}.
 \end{split}
 \eeq
-\titou{T2: I think we should specify what those terms are physically...}
+\titou{T2: I think we should specify what those terms are physically... Maybe earlier in the manuscript?}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Density-functional approximations for ensembles}
@@ -850,16 +851,22 @@ In order to test the influence of correlation effects on excitation energies, we
 \label{sec:res}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-\titou{The discussion of Fig.~\ref{fig:EvsW} comes here.}
+First, we discuss the linearity of the ensemble energy.
+To do so, we consider 5-boxium with box lengths of $L = \pi/8$, $L = \pi$, and $L = 8\pi$, which correspond (qualitatively at least) to the weak, intermediate, and strong correlation regimes, respectively.
+The three-state ensemble energy $\E{}{(\ew{1},\ew{2})}$ is represented in Fig.~\ref{fig:EvsW} as a function of both $\ew{1}$ and $\ew{2}$.
+To illustrate the magnitude of the ghost interaction error (GIE), we have reported the ensemble energy with and without ghost interaction correction (GIC) as explained above [see Eqs.~\eqref{eq:WHF} and \eqref{eq:EI}].
+As one can see, the linearity of the ensemble energy deteriorates when $L$ gets larger. 
+In other words, the GIE increases as the correlation gets stronger.
+Moreover, because the GIE can easily computed via Eq.~\eqref{eq:WHF} even for real, three-dimensional systems, this provides a cheap way of quantifying strong correlation in a given electronic system.
 
 %%% FIG 1 %%%
-\begin{figure}
-	\includegraphics[width=\linewidth]{EvsW_2_pi}
+\begin{figure*}
+	\includegraphics[width=\linewidth]{EvsW_n5}
 	\caption{
 	\label{fig:EvsW}
-	Weight dependence of the ensemble energy with and without ghost interaction correction (GIC) for 2-boxium with a box length $L = \pi$.
+	Weight dependence of the ensemble energy with and without ghost interaction correction (GIC) for 5-boxium with a box length $L = \pi/8$ (left), $L = \pi$ (center), and $L = 8\pi$ (right).
 	}
-\end{figure}
+\end{figure*}
 %%% %%% %%%
 
 
@@ -887,7 +894,7 @@ This conclusion is verified for smaller and larger number of electrons (see {\SI
 \end{figure}
 %%% %%% %%%
 
-Figure \ref{fig:EvsN} reports the error (in \%) in excitation energies (compared to FCI), for the same methods, as a function of $\Nel$ and fixed $L$ (in this case $L=\pi$).
+Figure \ref{fig:EvsN} reports the error (in \%) in excitation energies (as compared to FCI), for the same methods, as a function of $\Nel$ and fixed $L$ (in this case $L=\pi$).
 The graphs associated with other $L$ values are reported as {\SI}.
 Again, the graph for $L=\pi$ is quite typical and we draw similar conclusions as in the previous paragraph: irrespectively of the number of electrons, the eLDA functional with state-averaged weights is able to accurately model single and double excitations. 
 As a rule of thumb, we see that eLDA single excitations are of the same quality as the ones obtained in the linear response formalism (such as TDHF or TDLDA), while double excitations only deviates from the FCI values by a few tenth of percent for $L=\pi$.