iteration T2 revision done

Response_Letter/Response_Letter.tex

@@ -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 Hartree-exchange 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 correlation-strength 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 non-linear ghosts uncorrected by the GIC-eLDA, 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 ground-state? 
@@ -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 ground-state 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 two-electron 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 ground-state and equiensemble results for doubles converge as $N$ goes to infinity?}

Revised_Manuscript/eDFT.tex

@@ -600,7 +600,7 @@ expression}
 be exact for any \textit{uniform}
 system, the 
 density-functional 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 density-functional 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 KS-eLDA 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 individual-density-functional correlation energy where the density-functional
 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{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 than in the strong correlation regime (large box length).}
+	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}}
 	\label{fig:rho}
 	}
 \end{figure}
@@ -988,10 +987,7 @@ Its Tamm-Dancoff approximation version  (TDA-TDLDA) is also considered. \cite{Dr
 Concerning the ensemble calculations, two sets of weight are tested: the zero-weight
 (ground-state) limit where $\bw = (0,0)$ and the
 equi-triensemble (or equal-weight state-averaged) limit where $\bw = (1/3,1/3)$.
-\titou{Note that a zero-weight calculation does correspond to a
-\trashEF{conventional} ground-state KS calculation with \manu{$100\%$} exact exchange and LDA correlation.}
-\manu{Manu: OK?}
-
+\titou{Note that a zero-weight calculation does correspond to a ground-state KS calculation with $100\%$ exact exchange and LDA correlation.}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Results and discussion}
@@ -1034,7 +1030,7 @@ remains in the GIC-eLDA ensemble energy when the electron
 correlation is strong. \manurev{The latter ensemble energy is computed
 as the weighted
 sum of the individual KS-eLDA energies [see
-Eq.~(\ref{eq:Ew-eLDA})]. Therefore, its 
+Eq.~\eqref{eq:Ew-eLDA}]. 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:EI-eLDA}) and (\ref{eq:ind_HF-like_ener}), the individual
+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
 that the above-mentioned variations originate from the eLDA correlation
 functional [second and third terms on the right-hand side of
-Eq.~(\ref{eq:EI-eLDA})]. If, for analysis purposes, we consider the
+Eq.~\eqref{eq:EI-eLDA}]. 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 right-hand
+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
 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 
 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
+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