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34
Response_Letter/Response_Letter.tex
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@ -57,8 +57,9 @@ We look forward to hearing from you.
Then are the four points randomly selected from a bunch of them? Then are the four points randomly selected from a bunch of them?
Whatever it is, the authors might want to clarify the confusion.} Whatever it is, the authors might want to clarify the confusion.}
\\ \\
\alert{Sorry for the confusion. We have taken the last four points which correspond to the four largest variational wave functions. \alert{Sorry for the confusion.
This is now clearly stated in the revised manuscript (see footnote 65).} We have taken the last four points which correspond to the four largest variational wave functions.
This is now clearly stated in the revised manuscript.}
\item \item
{Although a general description of the method is given by the authors, more technical details might be needed to better improve reproducibility, such as values of the thresholds to select the most energetically relevant determinants etc. {Although a general description of the method is given by the authors, more technical details might be needed to better improve reproducibility, such as values of the thresholds to select the most energetically relevant determinants etc.
@ -89,53 +90,56 @@ We look forward to hearing from you.
It would be valuable to the work if the authors were to indicate the variance with respect to the number of points used in the extrapolations; in the linear extrapolations, one could use, say, between 3-5 points, whereas between 4-6 points could be used in the quadratic fits? It would be valuable to the work if the authors were to indicate the variance with respect to the number of points used in the extrapolations; in the linear extrapolations, one could use, say, between 3-5 points, whereas between 4-6 points could be used in the quadratic fits?
In any case, the authors should comment (in more detail) on their choice of fitting function and number of data points.} In any case, the authors should comment (in more detail) on their choice of fitting function and number of data points.}
\\ \\
\alert{Using the 3, 4, 5, 6, and 7 last points (ie the largest wave functions), linear extrapolations yield the following correlation energy estimates: $-863.1(11)$, $-863.4(5)$, $-862.1(8)$, $-863.5(11)$, and $-864.0(9)$ mE$_h$, respectively, where the fitting error is reported in parenthesis. \alert{Using the last 3, 4, 5, and 6 points (i.e., the largest wave functions), linear extrapolations yield the following correlation energy estimates: $-863.1(11)$, $-863.4(5)$, $-862.1(8)$, and $-863.5(11)$ mE$_h$, respectively, where the fitting error is reported in parenthesis.
These numbers vary by 1.9 mE$_h$. These numbers vary by $1.4$ mE$_h$.
The four-point extrapolated estimate that we have chosen to report as our best estimate corresponds to the smallest fitting error. The four-point extrapolated estimate that we have chosen to report as our best estimate corresponds to the smallest fitting error.
Quadratic fits yield much larger variations and we never use them in practice. Quadratic fits yield much larger variations and we never use them in practice.
All these additional information are now provided in the revised version of the manuscript.} All these additional information are now provided in the revised version of the manuscript (see footnote 66).}
\item \item
{As an aside, it would seem like the fifth last point differs ever so slightly from the general trend (regardless of the choice of (r)MP2)? {As an aside, it would seem like the fifth last point differs ever so slightly from the general trend (regardless of the choice of (r)MP2)?
Can the authors explain why? Can the authors explain why?
Surely, inclusion of this point in the fitting procedure would bring about changes to the final extrapolated result?} Surely, inclusion of this point in the fitting procedure would bring about changes to the final extrapolated result?}
\\ \\
\alert{Yes, the fifth point is slightly off compare to the others. \alert{Yes, the fifth point is slightly off as compared to the others.
This is due to the stochastic nature of the calculation of $E_\text{PT2}$. This is due to the stochastic nature of the calculation of $E_\text{rPT2}$.
The associated error bars (tabulated in Table II) have a size of the order of the markers and this is now mentioned explicitly in the caption of Fig.~1. As pointed out above, taking into account this fifth point yield a slightly smaller estimate of the correlation energy [$-862.1(8)$ mE$_h$], but taking one more point seem to soften things out [$-863.5(11)$ mE$_h$].} The associated error bars (tabulated in Table II) have a size of the order of the markers and this is now mentioned explicitly in the caption of Fig.~1.
As pointed out above, taking into account this fifth point yield a slightly smaller estimate of the correlation energy [$-862.1(8)$ mE$_h$], but taking one more point seem to soften things out [$-863.5(11)$ mE$_h$].
These observations are now mentioned in the same footnote.}
\item \item
{It would be interesting if the authors could comment (even speculatively) on why results in the localized FB basis are significantly lower (and hence, in the authors' own words, more trustworthy) than the corresponding results in the NO basis.} {It would be interesting if the authors could comment (even speculatively) on why results in the localized FB basis are significantly lower (and hence, in the authors' own words, more trustworthy) than the corresponding results in the NO basis.}
\\ \\
\alert{It is well known that the localized orbitals significantly speed up the convergence of SCI calculations [see ]} \alert{It is well known that the localized orbitals significantly speed up the convergence of SCI calculations [see Refs.~???]}
\item \item
{Why are the rMP2-based corrections considered superior to the corresponding corrections based on MP2? {Why are the rMP2-based corrections considered superior to the corresponding corrections based on MP2?
Because of the improved linear proportionality in Fig.~1? Because of the improved linear proportionality in Fig.~1?
If so, the authors might want to discuss exactly why a linear relationship is to be expected.} If so, the authors might want to discuss exactly why a linear relationship is to be expected.}
\\ \\
\alert{As mentioned in the original manuscript, the rPT2 correction does indeed yield an improved linear proportionality as compared to the usual PT2 treatment. The theoretical reasons behind this has been clarified in the revised manuscript (see the response to Reviewer \#2).} \alert{As mentioned in the original manuscript, the rPT2 correction does indeed yield an improved linear proportionality as compared to the usual PT2 treatment.
The theoretical reasons behind this has been clarified in the revised manuscript (see the response to Reviewer \#2).}
\item \item
{Some of the method acronyms have not been properly introduced in the text, and some have been slightly misrepresented, e.g., MBE-FCI (many-body expanded FCI) and FCCR, which is not a selected CC model. {Some of the method acronyms have not been properly introduced in the text, and some have been slightly misrepresented, e.g., MBE-FCI (many-body expanded FCI) and FCCR, which is not a selected CC model.
Also, some references appear to be missing, e.g., for iCI and DMRG.} Also, some references appear to be missing, e.g., for iCI and DMRG.}
\\ \\
\alert{We have more explicitly defined the method acronyms, corrected the description of FCCR, and added the missing references for iCI and DMRG. \alert{We have more explicitly defined the method acronyms, corrected the description of FCCR, and added the missing references for iCI and DMRG.
An additional reference to CAD-FCIQMC has been also added for the sake of completeness.} An additional reference for CAD-FCIQMC has been also added for the sake of completeness.}
\item \item
{Why are FB orbitals preferred over, e.g., PM orbitals or IBOs?} {Why are FB orbitals preferred over, e.g., PM orbitals or IBOs?}
\\ \\
\alert{Boys-Foster is the only localization criterion accessible at the moment but we are planning on implementing other schemes in the near future. \alert{Boys-Foster is the only localization criterion available at the moment but we are planning on implementing other schemes in the near future.
That being said, because we group the MOs by symmetry classes (see footnote 63), we ensure the $\sigma$-$\pi$ separability here, like in PM. That being said, because we group the MOs by symmetry classes (see footnote 63), our localization procedure ensures the $\sigma$-$\pi$ separability, like in PM.
This would note be possible in general but, thanks to the high symmetry of benzene, it is possible in the present case. This would not be possible in general but, thanks to the high symmetry of benzene, it is feasable in the present case.
} }
\item \item
{The benzene geometry was not optimized as part of the work behind Ref.~17. {The benzene geometry was not optimized as part of the work behind Ref.~17.
However, an adequate reference may be found in Ref.~17.} However, an adequate reference may be found in Ref.~17.}
\\ \\
\alert{We have added the corresponding reference to the work of Schreiber et al and slightly modified the sentence accordingly.} \alert{We have added the corresponding reference to the work of Schreiber et al.~and slightly modified the sentence accordingly.}
\end{itemize} \end{itemize}
\end{letter} \end{letter}

14
benzene.tex
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@ -145,15 +145,21 @@ As one can see from the energies of Table \ref{tab:NOvsLO}, for a given value of
We, therefore, consider these energies more trustworthy, and we will base our best estimate of the correlation energy of benzene on these calculations. We, therefore, consider these energies more trustworthy, and we will base our best estimate of the correlation energy of benzene on these calculations.
The convergence of the CIPSI correlation energy using localized orbitals is illustrated in Fig.~\ref{fig:CIPSI}, where one can see the behavior of the correlation energy, $\Delta E_\text{var.}$ and $\Delta E_\text{var.} + E_\text{(r)PT2}$, as a function of $N_\text{det}$ (left panel). The convergence of the CIPSI correlation energy using localized orbitals is illustrated in Fig.~\ref{fig:CIPSI}, where one can see the behavior of the correlation energy, $\Delta E_\text{var.}$ and $\Delta E_\text{var.} + E_\text{(r)PT2}$, as a function of $N_\text{det}$ (left panel).
The right panel of Fig.~\ref{fig:CIPSI} is more instructive as it shows $\Delta E_\text{var.}$ as a function of $E_\text{(r)PT2}$, and their corresponding four-point linear extrapolation curves that we have used to get our final estimate of the correlation energy. The right panel of Fig.~\ref{fig:CIPSI} is more instructive as it shows $\Delta E_\text{var.}$ as a function of $E_\text{(r)PT2}$, and their corresponding four-point linear extrapolation curves that we have used to get our final estimate of the correlation energy.
\footnote{\alert{The four largest variational wave functions are considered to perform the linear extrapolation.}} \alert{In other words, the four largest variational wave functions are considered to perform the linear extrapolation.}
From this figure, one clearly sees that the rPT2-based correction behaves more linearly than its corresponding PT2 version, and is thus systematically employed in the following. From this figure, one clearly sees that the rPT2-based correction behaves more linearly than its corresponding PT2 version, and is thus systematically employed in the following.
% Results % Results
Our final number are gathered in Table \ref{tab:extrap_dist_table}, where, following the notations of Ref.~\onlinecite{Eriksen_2020}, we report, in addition to the final variational energies $\Delta E_{\text{var.}}$, the Our final number are gathered in Table \ref{tab:extrap_dist_table}, where, following the notations of Ref.~\onlinecite{Eriksen_2020}, we report, in addition to the final variational energies $\Delta E_{\text{var.}}$, the
extrapolation distances, $\Delta E_{\text{dist}}$, defined as the difference between the final computed energy, $\Delta E_{\text{final}}$, and the extrapolated energy, $\Delta E_{\text{extrap.}}$ associated with ASCI, iCI, SHCI, DMRS, and CIPSI. extrapolation distances, $\Delta E_{\text{dist}}$, defined as the difference between the final computed energy, $\Delta E_{\text{final}}$, and the extrapolated energy, $\Delta E_{\text{extrap.}}$ associated with ASCI, iCI, SHCI, DMRS, and CIPSI.
The three flavours of SCI fall into an interval ranging from $-860.0$ m$E_h$ (ASCI) to $-864.2$ m$E_h$ (SHCI), while the other non-SCI methods yield correlation energies ranging from $-863.7$ to $-862.8$ m$E_h$ (see Table \ref{tab:energy}). Our final CIPSI number (obtained with localized orbitals and rPT2 correction via a four-point linear extrapolation) is $-863.4(5)$ m$E_h$, where the error reported in parenthesis represents the fitting error (not the extrapolation error for which it is much harder to provide a theoretically sound estimate). The three flavours of SCI fall into an interval ranging from $-860.0$ m$E_h$ (ASCI) to $-864.2$ m$E_h$ (SHCI), while the other non-SCI methods yield correlation energies ranging from $-863.7$ to $-862.8$ m$E_h$ (see Table \ref{tab:energy}). Our final CIPSI number (obtained with localized orbitals and rPT2 correction via a four-point linear extrapolation) is $-863.4(5)$ m$E_h$, where the error reported in parenthesis represents the fitting error (not the extrapolation error for which it is much harder to provide a theoretically sound estimate).
For comparison, the best post blind test SHCI estimate is $-863.3$ m$E_h$, which agrees almost perfectly with our best CIPSI estimate, while the best post blind test ASCI and iCI correlation energies are $-861.3$ and $-864.15$ m$E_h$, respectively s(see Table \ref{tab:extrap_dist_table}). \footnote{\alert{Using the last 3, 4, 5, and 6 largest wave functions to perform the linear extrapolation yield the following correlation energy estimates: $-863.1(11)$, $-863.4(5)$, $-862.1(8)$, and $-863.5(11)$ mE$_h$, respectively.
\alert{It is of interest to report the variation of the correlation energy estimates with the number of fitting points.} These numbers vary by $1.4$ mE$_h$.
The four-point extrapolated value of $-863.4(5)$ mE$_h$ that we have chosen to report as our best estimate corresponds to the smallest fitting error.
Quadratic fits yield much larger variations and are discarded in practice.
Due to the stochastic nature of $E_\text{rPT2}$, the fifth point is slightly off as compared to the others.
Taking into account this fifth point yield a slightly smaller estimate of the correlation energy [$-862.1(8)$ mE$_h$], while adding a sixth point settles down the correlation energy estimate at $-863.5(11)$ mE$_h$
}}
For comparison, the best post blind test SHCI estimate is $-863.3$ m$E_h$, which agrees almost perfectly with our best CIPSI estimate, while the best post blind test ASCI and iCI correlation energies are $-861.3$ and $-864.15$ m$E_h$, respectively (see Table \ref{tab:extrap_dist_table}).
%%$ FIG. 1 %%% %%$ FIG. 1 %%%
\begin{figure*} \begin{figure*}
@ -166,7 +172,7 @@ For comparison, the best post blind test SHCI estimate is $-863.3$ m$E_h$, which
Right: $\Delta E_\text{var.}$ (in m$E_h$) as a function of $E_\text{PT2}$ or $E_\text{rPT2}$. Right: $\Delta E_\text{var.}$ (in m$E_h$) as a function of $E_\text{PT2}$ or $E_\text{rPT2}$.
The four-point linear extrapolation curves (dashed lines) are also reported. The four-point linear extrapolation curves (dashed lines) are also reported.
The theoretical estimate of $-863$ m$E_h$ from Ref.~\onlinecite{Eriksen_2020} is marked by a black line for comparison purposes. The theoretical estimate of $-863$ m$E_h$ from Ref.~\onlinecite{Eriksen_2020} is marked by a black line for comparison purposes.
\alert{The statistical error bar associated with $E_\text{PT2}$ or $E_\text{rPT2}$ are of the order of the size of the markers.} \alert{The statistical error bars associated with $E_\text{PT2}$ or $E_\text{rPT2}$ (not shown) are of the order of the size of the markers.}
\label{fig:CIPSI} \label{fig:CIPSI}
} }
\end{figure*} \end{figure*}