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References/Karolewski_2013.pdf
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Response_Letter/Response_Letter.tex
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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. This is the usual Wigner crystal representation as mentionned in our previous works.
The asymptotic behavior is then correct in the large-$R$ limit. The asymptotic behavior is then correct in the large-$R$ limit.
However, the coefficients do not exactly match the exact ones. 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. 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. 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. 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! } 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. \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. 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 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.} 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. } {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. \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 \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? {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 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 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 illustration of the importance of both the density and the weights in
the evaluation of individual energies within an ensemble.}\manu{Do the evaluation of individual energies within an ensemble.}
you agree?}
\item \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. } {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 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 $0\leq w_1\leq 1/3$, 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 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 \item
{Are there similar issues with combining HF exchange with LDA C as seen in the ground-state? {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. supplemental material, similar issues appear for excited states.
Interestingly, increasing the ensemble weights (which of course cannot Interestingly, increasing the ensemble weights (which of course cannot
be done in conventional ground-state DFT) seems to reduce 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 \item
{Figure 3 discussion: Will eLDA always overestimate double excitations? {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. \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. 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. 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 This is left for future work.
difference in density between the ground and the excited states is 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.}
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?}
\item \item
{Figure 6: Do the ground-state and equiensemble results for doubles converge as $N$ goes to infinity?} {Figure 6: Do the ground-state and equiensemble results for doubles converge as $N$ goes to infinity?}

40
Revised_Manuscript/eDFT.tex
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@ -600,7 +600,7 @@ expression}
be exact for any \textit{uniform} be exact for any \textit{uniform}
system, the system, the
density-functional correlation components $\be{c}{(K)}(\n{}{})$ are 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 As discussed further in Sec.~\ref{sec:eDFA}, these components can be
extracted from a extracted from a
finite uniform electron gas model for which density-functional correlation excitation 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 energy will be treated at the
DFT level while we rely on HF for the exchange part. DFT level while we rely on HF for the exchange part.
This is different from the usual context where both exchange and This is different from the usual context where both exchange and
correlation are treated at the LDA level which gives compensation of correlation are treated at the LDA level which provides key error compensation features.
errors. Despite the errors Despite the errors
that might be introduced into the ensemble energy within such a scheme, that might be introduced into the ensemble energy within such a scheme,
cancellations may actually occur when computing excitation energies, cancellations may actually occur when computing excitation energies,
which are energy {\it differences}.} which are energy \textit{differences}.}
\manu{Manu: I changed a bit and complemented your sentence. Is this fine?}
The resulting KS-eLDA ensemble energy obtained via Eq.~\eqref{eq:min_with_HF_ener_fun} The resulting KS-eLDA ensemble energy obtained via Eq.~\eqref{eq:min_with_HF_ener_fun}
reads reads
@ -660,7 +659,7 @@ is the analog for ground and excited states (within an ensemble) of the HF energ
\manurev{ \manurev{
One may naturally wonder about the physical content of the above correlation energy One may naturally wonder about the physical content of the above correlation energy
expressions. It is in fact difficult to readily distinguish from 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 contributions from mixed ones. For that purpose, we may
consider a density regime which has a weak deviation from the uniform consider a density regime which has a weak deviation from the uniform
one. In such a regime, where eLDA is a reasonable approximation, the 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 an individual-density-functional correlation energy where the density-functional
correlation energy per particle is approximated by the ensemble one for correlation energy per particle is approximated by the ensemble one for
all the states within the ensemble. \manurev{This perturbation expansion 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 deviations from the uniform
density regime. Nevertheless, it density regime. Nevertheless, it
gives more insight into the eLDA approximation and it becomes useful when 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} \includegraphics[width=\linewidth]{rho}
\caption{ \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). \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} \label{fig:rho}
} }
\end{figure} \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 Concerning the ensemble calculations, two sets of weight are tested: the zero-weight
(ground-state) limit where $\bw = (0,0)$ and the (ground-state) limit where $\bw = (0,0)$ and the
equi-triensemble (or equal-weight state-averaged) limit where $\bw = (1/3,1/3)$. 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 \titou{Note that a zero-weight calculation does correspond to a ground-state KS calculation with $100\%$ exact exchange and LDA correlation.}
\trashEF{conventional} ground-state KS calculation with \manu{$100\%$} exact exchange and LDA correlation.}
\manu{Manu: OK?}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results and discussion} \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 correlation is strong. \manurev{The latter ensemble energy is computed
as the weighted as the weighted
sum of the individual KS-eLDA energies [see 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 curvature can only originate from the weight dependence of the
individual energies. individual energies.
Note that such a dependence does not exist in the exact theory. Here, 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 as $\ew{2}$ increases. The variations in the ensemble
weights are essentially linear or quadratic. weights are essentially linear or quadratic.
\manurev{This can be rationalized as follows. As readily seen from \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 HF-like energies do not depend explicitly on the weights, which means
that the above-mentioned variations originate from the eLDA correlation that the above-mentioned variations originate from the eLDA correlation
functional [second and third terms on the right-hand side of 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 Taylor expansions around the uniform density regime in
Eqs.~(\ref{eq:Taylor_exp_ind_corr_ener_eLDA}) and Eqs.~\eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and
(\ref{eq:Taylor_exp_DDisc_term}), contributions with an explicit weight \eqref{eq:Taylor_exp_DDisc_term}, contributions with an explicit weight
dependence still remain after summation. As both the ensemble density and dependence still remain after summation. As both the ensemble density and
the ensemble correlation energy per particle vary linearly with the the ensemble correlation energy per particle vary linearly with the
weights $\bw$ [see Eqs.~(\ref{eq:ens1RDM}), weights $\bw$ [see Eqs.~\eqref{eq:ens1RDM},
(\ref{eq:ens_dens_from_ens_1RDM}), and \eqref{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 \eqref{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 $\bw$, 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].} 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 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 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 \manurev{Finally, we notice that the crossover point of the
first-excited-state energies based on first-excited-state energies based on
bi- and triensemble calculations, respectively, disappears in the strong correlation 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 the importance of (individual and ensemble) densities, in
addition to the addition to the
weights, in the evaluation of individual energies within weights, in the evaluation of individual energies within