correction Fig 4

Response_Letter/Response_Letter.tex

@@ -184,16 +184,20 @@ that the deviation from linearity of the ensemble energy would be zero.}
 	\item 
 	{Fig. 2: Why does the crossover point for the 1st excitation curves disappear for $L=8\pi$? }
 	\\
-	\alert{It is clear from our derivations that the individual
+	\alert{The legend of Fig.~2 was incorrect (the curves were mislabeled), but this has now been corrected.
-correlation energies should vary with both the density {\it
+	In the new Fig.~2 (which is now Fig.~4 in the revised manuscript), this crossover has disappeared and the discussion is much more fluid:
-and} the ensemble weights. There is in principle no reason to expect the
+	when the weight of a state increases, this state is stabilized while the two others increases in energy (as it should).
-same variations for different ensembles and density regimes. The fact
+	The discussion regarding this figure has been modified accordingly.}
-that, for $L=8\pi$, electron correlation is strong and therefore the
+%	\alert{It is clear from our derivations that the individual
-density is more localized, is probably the reason for the disappearance
+%correlation energies should vary with both the density {\it
-of the crossover point. We were not able to rationalize this observation
+%and} the ensemble weights. There is in principle no reason to expect the
-further but we still mention in the revised manuscript that it is an
+%same variations for different ensembles and density regimes. The fact
-illustration of the importance of both the density and the weights in
+%that, for $L=8\pi$, electron correlation is strong and therefore the
-the evaluation of individual energies within an ensemble.}
+%density is more localized, is probably the reason for the disappearance
+%of the crossover point. We were not able to rationalize this observation
+%further but we still mention in the revised manuscript that it is an
+%illustration of the importance of both the density and the weights in
+%the evaluation of individual energies within an ensemble.}
 
 	\item 
 	{Page 8: If the authors have evidence of behavior between $w=(0,0)$ and the equiensemble, instead of just these endpoints, that would be interesting to mention for the eDFT crowd. }

Revised_Manuscript/eDFT.tex

@@ -867,7 +867,7 @@ Combining these, one can build the following three-state weight-dependent correl
 
 %%% FIG 1 %%%
 \begin{figure}
-	\includegraphics[width=0.7\linewidth]{embedding}
+	\includegraphics[width=0.7\linewidth]{fig1}
 	\caption{
 	\label{fig:embedding}
 	\titou{Schematic view of the ``embedding'' scheme: the two-electron finite uniform electron gas (the impurity) is embedded in the infinite uniform electron gas (the bath).
@@ -952,7 +952,7 @@ For small $L$, the system is weakly correlated, while strong correlation effects
 
 %%% FIG 1 %%%
 \begin{figure}
-	\includegraphics[width=\linewidth]{rho}
+	\includegraphics[width=\linewidth]{fig2}
 	\caption{
 	\titou{Ground-state one-electron density $\n{}{}(x)$ of 4-boxium (\ie, $N = 4$) for $L = \pi/32$ (left) and $L = 32\pi$ (right).
 	In the weak correlation regime (small box length), the one-electron density is much more delocalized and uniform than in the strong correlation regime (large box length), where a Wigner crystal starts to appear. \cite{Rogers_2017,Rogers_2016}}
@@ -996,7 +996,7 @@ equi-triensemble (or equal-weight state-averaged) limit where $\bw = (1/3,1/3)$.
 
 %%% FIG 1 %%%
 \begin{figure*}
-	\includegraphics[width=\linewidth]{EvsW_n5}
+	\includegraphics[width=\linewidth]{fig3}
 	\caption{
 	\label{fig:EvsW}
 	Deviation from linearity of the weight-dependent KS-eLDA ensemble energy $\E{eLDA}{(\ew{1},\ew{2})}$ with (dashed lines) and without (solid lines) ghost-interaction correction (GIC) for 5-boxium (\ie, $\nEl = 5$) with a box of length $L = \pi/8$ (left), $L = \pi$ (center), and $L = 8\pi$ (right).
@@ -1044,7 +1044,7 @@ Eqs.~\eqref{eq:min_with_HF_ener_fun} and \eqref{eq:WHF}].
 
 %%% FIG 2 %%%
 \begin{figure*}
-	\includegraphics[width=\linewidth]{EIvsW_n5}
+	\includegraphics[width=\linewidth]{fig4}
 	\caption{
 	\label{fig:EIvsW}
 	KS-eLDA individual energies, $\E{eLDA}{(0)}$ (black), $\E{eLDA}{(1)}$ (red), and $\E{eLDA}{(2)}$ (blue), as functions of the weights $\ew{1}$ (solid) and $\ew{2}$ (dashed) for 5-boxium (\ie, $\nEl = 5$) with a box of length $L = \pi/8$ (left), $L = \pi$ (center), and $L = 8\pi$ (right).}
@@ -1058,14 +1058,18 @@ energies, which is in agreement with
 the curvature of the GIC-eLDA ensemble energy discussed previously. Interestingly, the
 individual energies do not vary in the same way depending on the state
 considered and the value of the weights. 
-We see for example that, within the biensemble (\ie, $\ew{2}=0$), the energies of
+\titou{On one hand,} we see for example that, within the biensemble (\ie, $\ew{2}=0$), the energies of
-the ground and first excited-state increase with respect to the
+the ground and \titou{second} excited-state increase with respect to the
 first-excited-state weight $\ew{1}$, thus showing that, in this
 case, we
 ``deteriorate'' these states by optimizing the orbitals for the
-ensemble, rather than for each state separately. The reverse actually occurs for the ground state in the triensemble
+ensemble, rather than for each state separately. 
-as $\ew{2}$ increases. The variations in the ensemble
+\titou{The singly excited state is, on the other hand, stabilize in the biensemble, which is reasonable as the weight associated with this state increases.
-weights are essentially linear or quadratic. 
+For the triensemble, as $\ew{2}$ increases, the energy of the ground state increases, while the energy of the first excited state remains stable with a slight increase at large $L$. 
+The second excited state is obviously stabilized by the increase of its weight in the ensemble.
+These are all very sensible observations.}
+
+The variations in the ensemble weights are essentially linear or quadratic. 
 \manurev{This can be rationalized as follows. As readily seen from
 Eqs.~\eqref{eq:EI-eLDA} and \eqref{eq:ind_HF-like_ener}, the individual
 HF-like energies do not depend explicitly on the weights, which means
@@ -1081,23 +1085,25 @@ weights $\bw$ [see Eqs.~\eqref{eq:ens1RDM},
 \eqref{eq:ens_dens_from_ens_1RDM}, and
 \eqref{eq:decomp_ens_correner_per_part}], the latter contributions will contain both linear and quadratic terms in
 $\bw$, as evidenced by Eq.~\eqref{eq:Taylor_exp_DDisc_term} [see the second term on the right-hand
-side].} In the biensemble, the weight dependence of the first
+side].} 
-excitation energy is reduced as the correlation increases. On the other hand, switching from a bi- to a triensemble
+!!! In the biensemble, the weight dependence of the first
+excitation energy is \titou{increased?} as the correlation increases, \titou{while the weight dependence of the second excitation energy is reduced}. 
+On the other hand, switching from a bi- to a triensemble
 systematically enhances the weight dependence, due to the lowering of the
-ground-state energy, as $\ew{2}$ increases.         
+ground-state energy, as $\ew{2}$ increases.
-The reverse is observed for the second excited state. 
+The reverse is observed for the second excited state.  !!! \titou{THIS PARAGRAPH MIGHT NEED MODIFICATIONS.}    
-\manurev{Finally, we notice that the crossover point of the 
+\trashPFL{Finally, we notice that the crossover point of the 
 first-excited-state energies based on
 bi- and triensemble calculations, respectively, disappears in the strong correlation
 regime [see the right panel of Fig.~\ref{fig:EIvsW}], thus illustrating
 the importance of (individual and ensemble) densities, in
 addition to the  
 weights, in the evaluation of individual energies within
-an ensemble. 
+an ensemble.
 }
 %%% FIG 3 %%%
 \begin{figure}
-	\includegraphics[width=\linewidth]{EvsL_5}
+	\includegraphics[width=\linewidth]{fig5}
 	\caption{
 	\label{fig:EvsL}
 	Excitation energies (multiplied by $L^2$) associated with the single excitation $\Ex{}{(1)}$ (bottom) and double excitation $\Ex{}{(2)}$ (top) of 5-boxium for various methods and box lengths $L$.
@@ -1132,7 +1138,7 @@ This conclusion is verified for smaller and larger numbers of electrons
 
 %%% FIG 4 %%%
 \begin{figure*}
-	\includegraphics[width=\linewidth]{EvsN}
+	\includegraphics[width=\linewidth]{fig6}
 	\caption{
 	\label{fig:EvsN}
 	Error with respect to FCI in single and double excitation energies for $\nEl$-boxium for various methods and electron numbers $\nEl$ at $L=\pi/8$ (left), $L=\pi$ (center), and $L=8\pi$ (right).
@@ -1170,7 +1176,7 @@ electrons.
 
 %%% FIG 5 %%%
 \begin{figure*}
-	\includegraphics[width=\linewidth]{EvsL_DD}
+	\includegraphics[width=\linewidth]{fig7}
 	\caption{
 	\label{fig:EvsL_DD}
 	Error with respect to FCI (in \%) associated with the single excitation $\Ex{}{(1)}$ (bottom) and double excitation $\Ex{}{(2)}$ (top) as a function of the box length $L$ for 3-boxium (left), 5-boxium (center), and 7-boxium (right) at the KS-eLDA level with and without the contribution of the ensemble correlation derivative $\DD{c}{(I)}$.
@@ -1227,7 +1233,7 @@ shown).
 
 %%% FIG 6 %%%
 \begin{figure}
-	\includegraphics[width=\linewidth]{EvsN_DD}
+	\includegraphics[width=\linewidth]{fig8}
 	\caption{
 	\label{fig:EvsN_DD}
 	Error with respect to FCI in single and double excitation energies for $\nEl$-boxium (with a box length of $L=8\pi$) as a function of the number of electrons $\nEl$ at the KS-eLDA level with and without the contribution of the ensemble correlation derivative $\DD{c}{(I)}$.