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@ 109,7 +109,7 @@ give more insight into eLDA.}


This is the usual Wigner crystal representation as mentionned in our previous works.


The asymptotic behavior is then correct in the large$R$ limit.


However, the coefficients do not exactly match the exact ones.


From a numerical and practical point of view, we have not found that enforcing the exact values does not improve the results.


From a numerical and practical point of view, we have not found that enforcing the exact values does improve the results.


Actually, it usually worsen them as enforcing the correct coefficients in the large$R$ limit usually deteriorates the results for intermediate $R$ values.


Finally, let us mention that the logarithmic behavior does not occur for the correlation energy.


It only occurs in the thermodynamic limit (where the number of electrons gets very large) in the Hartreeexchange term.


@ 123,7 +123,7 @@ give more insight into eLDA.}


I wonder if an explanatory diagram for the embedding scheme might clarify my thinking here, particularly if my thinking is wrong! }


\\


\alert{As mentioned in the original manuscript, the impurity carries the weight dependence of the functional.


Performing a simple gedanken experiment, one can imagine that, in the infinite system, the excitation will occur locally, i.e., on the impurity.


Performing a simple gedanken experiment, one can imagine that, in the infinite system, the excitation occurs locally, i.e., on the impurity.


Therefore, we can assume, as a first approximation, that the weight dependence will originate mainly from this impurity, most of the bath being unaffected by this local excitation. This is, roughly speaking, the philosophy that we have followed.


We believe this is also the reviewer's way of thinking.


We have added a figure to illustrate this in the revised version of the manuscript.}


@ 132,7 +132,7 @@ give more insight into eLDA.}


{Page 7: Another diagram suggestion: unfamiliar readers might be helped by a pedagogical diagram of your box systems, to help folks see how the different box lengths correspond to different correlationstrength regimes. }


\\


\alert{Very good suggestion. Accordingly, we have added a figure showing the electron density for a very small box and a very large box.


This illustrates how electrons localize when the density gets smaller.}


This illustrates how electrons localize when the density gets smaller, and how the density has the tendency to be more uniform for small boxes.}




\item


{Page 7: Does strong correlation always result in nonlinear ghosts uncorrected by the GICeLDA, or is it particularly difficult or changed by the embedding scheme somehow?


@ 193,8 +193,7 @@ 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.}\manu{Do


you agree?}


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. }


@ 206,7 +205,7 @@ $w=(0,0)$ up to the equiensemble case $w=(1/3,1/3)$. For convenience,


the path was just split in two parts: a first one where $w_2=0$ and


$0\leq w_1\leq 1/3$, and a second one where $w_1=1/3$ and $0\leq w_2\leq


1/3$. For clarity, this is now mentioned explicitly in the revised


manuscrit before commenting on the plots.}\manu{OK?}


manuscrit before commenting on the plots.}




\item


{Are there similar issues with combining HF exchange with LDA C as seen in the groundstate?


@ 216,7 +215,8 @@ manuscrit before commenting on the plots.}\manu{OK?}


supplemental material, similar issues appear for excited states.


Interestingly, increasing the ensemble weights (which of course cannot


be done in conventional groundstate DFT) seems to reduce


errors}\manu{We need to check the tables in the SI}


errors}


%\manu{We need to check the tables in the SI}




\item


{Figure 3 discussion: Will eLDA always overestimate double excitations?


@ 238,12 +238,8 @@ errors}\manu{We need to check the tables in the SI}


\alert{As mentioned in the original manuscript (see Results and Discussion section), we believe that it might be a consequence of how we constructed the eLDA functional, as the weight dependence of the eLDA functional is based on a twoelectron uniform electron gas.


We do not think this is due to the uniformity of its density, though.


Incorporating a $N$dependence in the functional through the curvature of the Fermi hole might be valuable in this respect.


This is left for future work.}\manu{Could we argue that the


difference in density between the ground and the excited states is


not that substantial? Or the deviations cancel out after integration? The


question does not seem to be focused on the functional itself but more


on the evaluation of this term for a system (boxium) that is not


uniform. What do you think?}


This is left for future work.


Besides, the difference in density between the ground and the excited states is not substantial in 1D systems, which makes the effect of the second term of Eqn 51 quite small.}




\item


{Figure 6: Do the groundstate and equiensemble results for doubles converge as $N$ goes to infinity?}



@ 600,7 +600,7 @@ expression}


be exact for any \textit{uniform}


system, the


densityfunctional correlation components $\be{c}{(K)}(\n{}{})$ are


weight\textit{independent}, unlike in the exact theory \cite{Fromager_2020}.


weight\textit{independent}, unlike in the exact theory. \cite{Fromager_2020}


As discussed further in Sec.~\ref{sec:eDFA}, these components can be


extracted from a


finite uniform electron gas model for which densityfunctional correlation excitation


@ 609,12 +609,11 @@ energies can be computed.


energy will be treated at the


DFT level while we rely on HF for the exchange part.


This is different from the usual context where both exchange and


correlation are treated at the LDA level which gives compensation of


errors. Despite the errors


correlation are treated at the LDA level which provides key error compensation features.


Despite the errors


that might be introduced into the ensemble energy within such a scheme,


cancellations may actually occur when computing excitation energies,


which are energy {\it differences}.}


\manu{Manu: I changed a bit and complemented your sentence. Is this fine?}


which are energy \textit{differences}.}




The resulting KSeLDA ensemble energy obtained via Eq.~\eqref{eq:min_with_HF_ener_fun}


reads


@ 660,7 +659,7 @@ is the analog for ground and excited states (within an ensemble) of the HF energ


\manurev{


One may naturally wonder about the physical content of the above correlation energy


expressions. It is in fact difficult to readily distinguish from


Eqs.~(\ref{eq:Xic}) and (\ref{eq:Upsic}) purely (uncoupled) individual


Eqs.~\eqref{eq:Xic} and \eqref{eq:Upsic} purely (uncoupled) individual


contributions from mixed ones. For that purpose, we may


consider a density regime which has a weak deviation from the uniform


one. In such a regime, where eLDA is a reasonable approximation, the


@ 682,7 +681,7 @@ Therefore, it can be identified as


an individualdensityfunctional correlation energy where the densityfunctional


correlation energy per particle is approximated by the ensemble one for


all the states within the ensemble. \manurev{This perturbation expansion


is of course less relevant for the (more realistic) systems that exhibit significant


is of course less relevant for (more realistic) systems that exhibit significant


deviations from the uniform


density regime. Nevertheless, it


gives more insight into the eLDA approximation and it becomes useful when


@ 956,7 +955,7 @@ For small $L$, the system is weakly correlated, while strong correlation effects


\includegraphics[width=\linewidth]{rho}


\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 than in the strong correlation regime (large box length).}


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}}


\label{fig:rho}


}


\end{figure}


@ 988,10 +987,7 @@ Its TammDancoff approximation version (TDATDLDA) is also considered. \cite{Dr


Concerning the ensemble calculations, two sets of weight are tested: the zeroweight


(groundstate) limit where $\bw = (0,0)$ and the


equitriensemble (or equalweight stateaveraged) limit where $\bw = (1/3,1/3)$.


\titou{Note that a zeroweight calculation does correspond to a


\trashEF{conventional} groundstate KS calculation with \manu{$100\%$} exact exchange and LDA correlation.}


\manu{Manu: OK?}




\titou{Note that a zeroweight calculation does correspond to a groundstate KS calculation with $100\%$ exact exchange and LDA correlation.}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Results and discussion}


@ 1034,7 +1030,7 @@ remains in the GICeLDA ensemble energy when the electron


correlation is strong. \manurev{The latter ensemble energy is computed


as the weighted


sum of the individual KSeLDA energies [see


Eq.~(\ref{eq:EweLDA})]. Therefore, its


Eq.~\eqref{eq:EweLDA}]. Therefore, its


curvature can only originate from the weight dependence of the


individual energies.


Note that such a dependence does not exist in the exact theory. Here,


@ 1071,20 +1067,20 @@ ensemble, rather than for each state separately. The reverse actually occurs for


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


weights are essentially linear or quadratic.


\manurev{This can be rationalized as follows. As readily seen from


Eqs.~(\ref{eq:EIeLDA}) and (\ref{eq:ind_HFlike_ener}), the individual


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


HFlike energies do not depend explicitly on the weights, which means


that the abovementioned variations originate from the eLDA correlation


functional [second and third terms on the righthand side of


Eq.~(\ref{eq:EIeLDA})]. If, for analysis purposes, we consider the


Eq.~\eqref{eq:EIeLDA}]. If, for analysis purposes, we consider the


Taylor expansions around the uniform density regime in


Eqs.~(\ref{eq:Taylor_exp_ind_corr_ener_eLDA}) and


(\ref{eq:Taylor_exp_DDisc_term}), contributions with an explicit weight


Eqs.~\eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and


\eqref{eq:Taylor_exp_DDisc_term}, contributions with an explicit weight


dependence still remain after summation. As both the ensemble density and


the ensemble correlation energy per particle vary linearly with the


weights $\bw$ [see Eqs.~(\ref{eq:ens1RDM}),


(\ref{eq:ens_dens_from_ens_1RDM}), and


(\ref{eq:decomp_ens_correner_per_part})], the latter contributions will contain both linear and quadratic terms in


$\bw$, as readily seen from Eq.~(\ref{eq:Taylor_exp_DDisc_term}) [see the second term on the righthand


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


systematically enhances the weight dependence, due to the lowering of the


@ 1093,7 +1089,7 @@ The reverse is observed for the second excited state.


\manurev{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


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



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