correction Fig 4
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@ 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


correlation energies should vary with both the density {\it


and} the ensemble weights. There is in principle no reason to expect the


same variations for different ensembles and density regimes. The fact


that, for $L=8\pi$, electron correlation is strong and therefore the


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.}


\alert{The legend of Fig.~2 was incorrect (the curves were mislabeled), but this has now been corrected.


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:


when the weight of a state increases, this state is stabilized while the two others increases in energy (as it should).


The discussion regarding this figure has been modified accordingly.}


% \alert{It is clear from our derivations that the individual


%correlation energies should vary with both the density {\it


%and} the ensemble weights. There is in principle no reason to expect the


%same variations for different ensembles and density regimes. The fact


%that, for $L=8\pi$, electron correlation is strong and therefore the


%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. }



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@ 867,7 +867,7 @@ Combining these, one can build the following threestate weightdependent 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 twoelectron 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{Groundstate oneelectron density $\n{}{}(x)$ of 4boxium (\ie, $N = 4$) for $L = \pi/32$ (left) and $L = 32\pi$ (right).


In the weak correlation regime (small box length), the oneelectron 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 @@ equitriensemble (or equalweight stateaveraged) 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 weightdependent KSeLDA ensemble energy $\E{eLDA}{(\ew{1},\ew{2})}$ with (dashed lines) and without (solid lines) ghostinteraction correction (GIC) for 5boxium (\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}


KSeLDA 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 5boxium (\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 GICeLDA 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


the ground and first excitedstate increase with respect to the


\titou{On one hand,} we see for example that, within the biensemble (\ie, $\ew{2}=0$), the energies of


the ground and \titou{second} excitedstate increase with respect to the


firstexcitedstate 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


as $\ew{2}$ increases. The variations in the ensemble


weights are essentially linear or quadratic.


ensemble, rather than for each state separately.


\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.


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:EIeLDA} and \eqref{eq:ind_HFlike_ener}, the individual


HFlike 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 righthand


side].} In the biensemble, the weight dependence of the first


excitation energy is reduced as the correlation increases. On the other hand, switching from a bi to a triensemble


side].}


!!! 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


groundstate energy, as $\ew{2}$ increases.


The reverse is observed for the second excited state.


\manurev{Finally, we notice that the crossover point of the


groundstate energy, as $\ew{2}$ increases.


The reverse is observed for the second excited state. !!! \titou{THIS PARAGRAPH MIGHT NEED MODIFICATIONS.}


\trashPFL{Finally, we notice that the crossover point of the


firstexcitedstate 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 5boxium 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 3boxium (left), 5boxium (center), and 7boxium (right) at the KSeLDA 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 KSeLDA level with and without the contribution of the ensemble correlation derivative $\DD{c}{(I)}$.



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