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Response_Letter/Response_Letter.tex
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@ -30,10 +30,16 @@ We look forward to hearing from you.
\begin{itemize}
\item
{As a follow-up note to the recent benchmarking work of Eriksen et al., the authors present the correlation energy of ground state benzene calculated using CIPSI, another flavor in the Selected Configuration Interaction plus Perturbation family. In this endeavor, four combinations regarding the choice of orbitals and perturbation corrections are tested. The best result is obtained from constructing a set of localized orbitals from natural orbitals using a Boys-Foster localization procedure as well as employing a renormalized version of the PT2 correction in the perturbative stage. The final energy agrees with the theoretical estimate of the blind test and is also very close to the best post blind test estimate in the SCI+PT category.
Benchmarking is indeed an important and essential component in the development of electronic structure theories. It is good that the authors complement the benchmark dataset. Overall, the work is well-motivated and generally explained in a clear and organized manner. However, there are also a few issues to be addressed or clarified. }
{As a follow-up note to the recent benchmarking work of Eriksen et al., the authors present the correlation energy of ground state benzene calculated using CIPSI, another flavor in the Selected Configuration Interaction plus Perturbation family.
In this endeavor, four combinations regarding the choice of orbitals and perturbation corrections are tested.
The best result is obtained from constructing a set of localized orbitals from natural orbitals using a Boys-Foster localization procedure as well as employing a renormalized version of the PT2 correction in the perturbative stage.
The final energy agrees with the theoretical estimate of the blind test and is also very close to the best post blind test estimate in the SCI+PT category.
Benchmarking is indeed an important and essential component in the development of electronic structure theories.
It is good that the authors complement the benchmark dataset.
Overall, the work is well-motivated and generally explained in a clear and organized manner.
However, there are also a few issues to be addressed or clarified. }
\\
\alert{}
\alert{We thank the reviewer for his/her positive comments.}
\item
{In the manuscript, the authors reason that rPT2 should be employed because its correction behaves more linearly than its PT2 counterpart.
@ -41,7 +47,8 @@ We look forward to hearing from you.
I believe that the authors have performed numerical tests to confirm this finding in a previous paper and I think should give a citation in the sentence. }
\\
\alert{The rPT2 correction corresponds to a partial resummation of some of the higher-order diagrams from many-body perturbation theory.
As correctly pointed out by the reviewer, this correction has been thoroughly tested in Ref.~, and we have then added this reference to the sentence as requested.}
We have mentioned this in the revised version of the manuscript.
As correctly pointed out by the reviewer, this correction has been thoroughly tested in Ref.~50 for weakly and strongly correlated systems, and we have then added this reference to the corresponding sentence as requested.}
\item
{It is not very clear how the extrapolation is actually done.
@ -50,18 +57,19 @@ We look forward to hearing from you.
Then are the four points randomly selected from a bunch of them?
Whatever it is, the authors might want to clarify the confusion.}
\\
\alert{Sorry for the confusion. We have taken the four last points which correspond to the four largest variational wave functions.
This is now clearly stated in the revised manuscript.}
\alert{Sorry for the confusion. 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 (see footnote 65).}
\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.. I realize that this ia a note and the authors want to keep it short. In light of this the authors might want to include an input file in their appendix. Because their quantum package is publicly available, it might make it easy for someone to reproduce their findings. }
{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.
I realize that this is a note and the authors want to keep it short. In light of this the authors might want to include an input file in their appendix. Because their quantum package is publicly available, it might make it easy for someone to reproduce their findings. }
\\
\alert{}
\alert{Toto}
\end{itemize}
%%% REVIEWER 2 %%%
\noindent \textbf{\large Authors' answer to Reviewer \#2}
\noindent \textbf{\large Authors' answer to Reviewer \#3}
\begin{itemize}
@ -72,7 +80,8 @@ We look forward to hearing from you.
The manuscript is reasonably well written (bar the occasional grammatical error/typo, which the authors are sure to locate upon revising the manuscript), and I surely deem it suitable for future publication in JCP.
However, I'd like to encourage the authors to take the following concerns and comments into account before submitting a revised version of the manuscript.}
\\
\alert{}
\alert{We thank the reviewer for supporting publication of the present manuscript.
The references of the blind challenge by Eriksen et al. and the subsequent work of Lee et al. (which was recently published in JCP) have been updated.}
\item
{The authors have opted for providing the reader with raw data in tabulated form, which is a real asset.
@ -80,41 +89,53 @@ 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?
In any case, the authors should comment (in more detail) on their choice of fitting function and number of data points.}
\\
\alert{}
\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.
These numbers vary by 1.9 mE$_h$.
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.
All these additional information are now provided in the revised version of the manuscript.}
\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)? Can the authors explain why?
{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?
Surely, inclusion of this point in the fitting procedure would bring about changes to the final extrapolated result?}
\\
\alert{}
\alert{Yes, the fifth point is slightly off compare to the others.
This is due to the stochastic nature of the calculation of $E_\text{PT2}$.
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$_h$].}
\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.}
\\
\alert{}
\alert{It is well known that the localized orbitals significantly speed up the convergence of SCI calculations [see ]}
\item
{Why are the rMP2-based corrections considered superior to the corresponding corrections based on MP2?
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.}
\\
\alert{}
\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
{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.}
{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.}
\\
\alert{}
\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.}
\item
{Why are FB orbitals preferred over, e.g., PM orbitals or IBOs?}
\\
\alert{Boys-Foster is the only localization criterion implemented in QUANTUM PACKAGE.}
\alert{Boys-Foster is the only localization criterion accessible 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.
This would note be possible in general but, thanks to the high symmetry of benzene, it is possible in the present case.
}
\item
{The benzene geometry was not optimized as part of the work behind 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.}
\alert{We have added the corresponding reference to the work of Schreiber et al and slightly modified the sentence accordingly.}
\end{itemize}
\end{letter}

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@ -1,13 +1,93 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-10-09 09:41:12 +0200
%% Created for Pierre-Francois Loos at 2020-10-09 13:24:13 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Deustua_2017,
Author = {Deustua, J. Emiliano and Shen, Jun and Piecuch, Piotr},
Date-Added = {2020-10-09 12:21:32 +0200},
Date-Modified = {2020-10-09 12:21:59 +0200},
Doi = {10.1103/PhysRevLett.119.223003},
Journal = {Phys. Rev. Lett.},
Pages = {223003},
Title = {Converging High-Level Coupled-Cluster Energetics by Monte Carlo Sampling and Moment Expansions},
Volume = {119},
Year = {2017},
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.119.223003},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.119.223003}}
@article{Zhang_2020,
Author = {Zhang, Ning and Liu, Wenjian and Hoffmann, Mark R.},
Date-Added = {2020-10-09 12:06:01 +0200},
Date-Modified = {2020-10-09 12:06:54 +0200},
Doi = {10.1021/acs.jctc.9b01200},
Journal = {J. Chem. Theory Comput.},
Number = {4},
Pages = {2296-2316},
Title = {Iterative Configuration Interaction with Selection},
Volume = {16},
Year = {2020},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.9b01200}}
@article{Lei_2017,
Author = {Yibo Lei and Wenjian Liu and Mark R. Hoffmann},
Date-Added = {2020-10-09 12:04:45 +0200},
Date-Modified = {2020-10-09 12:07:10 +0200},
Doi = {10.1080/00268976.2017.1308029},
Journal = {Mol. Phys.},
Number = {21-22},
Pages = {2696-2707},
Title = {Further development of SDSPT2 for strongly correlated electrons},
Volume = {115},
Year = {2017},
Bdsk-Url-1 = {https://doi.org/10.1080/00268976.2017.1308029}}
@article{Liu_2014,
Author = {Liu, W. and Hoffmann, M.R.},
Date-Added = {2020-10-09 12:02:24 +0200},
Date-Modified = {2020-10-09 12:07:31 +0200},
Doi = {10.1007/s00214-014-1481-x},
Journal = {Theor. Chem. Acc.},
Pages = {1481},
Title = {SDS: the static--dynamic--static framework for strongly correlated electrons},
Volume = {133},
Year = {2014},
Bdsk-Url-1 = {https://doi.org/10.1007/s00214-014-1481-x}}
@article{Chan_2011,
Abstract = { The density matrix renormalization group is a method that is useful for describing molecules that have strongly correlated electrons. Here we provide a pedagogical overview of the basic challenges of strong correlation, how the density matrix renormalization group works, a survey of its existing applications to molecular problems, and some thoughts on the future of the method. },
Author = {Chan, Garnet Kin-Lic and Sharma, Sandeep},
Date-Added = {2020-10-09 11:59:58 +0200},
Date-Modified = {2020-10-09 12:00:15 +0200},
Doi = {10.1146/annurev-physchem-032210-103338},
Eprint = {https://doi.org/10.1146/annurev-physchem-032210-103338},
Journal = {Annual Review of Physical Chemistry},
Note = {PMID: 21219144},
Number = {1},
Pages = {465-481},
Title = {The Density Matrix Renormalization Group in Quantum Chemistry},
Url = {https://doi.org/10.1146/annurev-physchem-032210-103338},
Volume = {62},
Year = {2011},
Bdsk-Url-1 = {https://doi.org/10.1146/annurev-physchem-032210-103338}}
@article{White_1993,
Author = {White, S. R.},
Date-Added = {2020-10-09 11:57:24 +0200},
Date-Modified = {2020-10-09 11:58:16 +0200},
Doi = {10.1103/PhysRevB.48.10345},
Journal = {Phys. Rev. B},
Pages = {10345},
Title = {Density-matrix algorithms for quantum renormalization groups},
Volume = {48},
Year = {1993},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.69.2863}}
@article{Eriksen_2019b,
Author = {J. J. Eriksen and J. Gauss},
Date-Added = {2020-08-25 18:14:27 +0200},
@ -300,7 +380,8 @@
Pages = {126101},
Title = {The performance of phaseless auxiliary-field quantum Monte Carlo on the ground state electronic energy of benzene},
Volume = {153},
Year = {2020}}
Year = {2020},
Bdsk-Url-1 = {https://doi.org/10.1063/5.0024835}}
@article{Williams_2020,
Author = {Williams, Kiel T and Yao, Yuan and Li, Jia and Chen, Li and Shi, Hao and Motta, Mario and Niu, Chunyao and Ray, Ushnish and Guo, Sheng and Anderson, Robert J and others},
@ -487,7 +568,8 @@
Pages = {8922--8929},
Title = {The Ground State Electronic Energy of Benzene},
Volume = {11},
Year = {2020}}
Year = {2020},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpclett.0c02621}}
@article{Evangelista_2014,
Author = {Evangelista, Francesco A.},
@ -681,13 +763,9 @@
@article{Chien_2018,
Author = {Chien, Alan D. and Holmes, Adam A. and Otten, Matthew and Umrigar, C. J. and Sharma, Sandeep and Zimmerman, Paul M.},
Date-Added = {2020-08-02 17:36:33 +0200},
Date-Modified = {2020-08-02 17:36:33 +0200},
Date-Modified = {2020-10-09 13:24:02 +0200},
Doi = {10.1021/acs.jpca.8b01554},
File = {/Users/loos/Zotero/storage/J96RZ7JP/Chien et al. - 2018 - Excited States of Methylene, Polyenes, and Ozone f.pdf},
Issn = {1089-5639, 1520-5215},
Journal = {J. Phys. Chem. A},
Month = mar,
Number = {10},
Pages = {2714--2722},
Title = {Excited {{States}} of {{Methylene}}, {{Polyenes}}, and {{Ozone}} from {{Heat}}-{{Bath Configuration Interaction}}},
Volume = {122},

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benzene.tex
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@ -30,7 +30,7 @@
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\title{Note: The performance of CIPSI on the ground state electronic energy of benzene}
\title{The performance of CIPSI on the ground state electronic energy of benzene}
\author{Pierre-Fran\c{c}ois Loos}
\email{loos@irsamc.ups-tlse.fr}
@ -60,16 +60,16 @@ Following a similar goal, we have recently proposed a large set of highly-accura
In a recent preprint, \cite{Eriksen_2020} Eriksen \textit{et al.}~have proposed a blind test for a particular electronic structure problem inviting several groups around the world to contribute to this endeavour.
In addition to coupled cluster theory with singles, doubles, triples, and quadruples (CCSDTQ), \cite{Oliphant_1991,Kucharski_1992} a large panel of highly-accurate, emerging electronic structure methods were considered:
(i) the many-body expansion FCI (MBE-FCI), \cite{Eriksen_2017,Eriksen_2018,Eriksen_2019a,Eriksen_2019b}
(ii) three SCI methods including a second-order perturbative correction \alert{[adaptive sampling CI (ASCI), \cite{Tubman_2016,Tubman_2018,Tubman_2020} iterative CI (iCI), \cite{Liu_2016} and semistochastic heat-bath CI (SHCI) \cite{Holmes_2016,Holmes_2017,Sharma_2017}]},
(iii) a selected coupled-cluster theory method which also includes a second-order perturbative correction (FCCR), \cite{Xu_2018}
(iv) the density-matrix renornalization group approach (DMRG), \cite{White_1992} and
(v) two flavors of FCI quantum Monte Carlo (FCIQMC), \cite{Booth_2009,Cleland_2010} namely AS-FCIQMC \cite{Ghanem_2019} and CAD-FCIQMC. \cite{Deustua_2018}
(ii) three SCI methods including a second-order perturbative correction \alert{[adaptive sampling CI (ASCI), \cite{Tubman_2016,Tubman_2018,Tubman_2020} iterative CI (iCI), \cite{Liu_2014,Liu_2016,Lei_2017,Zhang_2020} and semistochastic heat-bath CI (SHCI) \cite{Holmes_2016,Holmes_2017,Sharma_2017}]},
(iii) \alert{the full coupled-cluster reduction (FCCR) \cite{Xu_2018} which also includes a second-order perturbative correction},
(iv) the density-matrix renornalization group approach (DMRG), \cite{White_1992,White_1993,Chan_2011} and
(v) two flavors of FCI quantum Monte Carlo (FCIQMC), \cite{Booth_2009,Cleland_2010} namely AS-FCIQMC \cite{Ghanem_2019} and CAD-FCIQMC. \cite{Deustua_2017,Deustua_2018}
We refer the interested reader to Ref.~\onlinecite{Eriksen_2020} and its supporting information for additional details on each method and the complete list of references.
Soon after, Lee \textit{et al.}~reported phaseless auxiliary-field quantum Monte Carlo \cite{Motta_2018} (ph-AFQMC) correlation energies for the very same problem. \cite{Lee_2020}
% The system
The target application is the non-relativistic frozen-core correlation energy of the ground state of the benzene molecule in the cc-pVDZ basis.
The geometry of benzene has been computed at the MP2/6-31G* level \cite{Schreiber_2008} and it can be found in the supporting information of Ref.~\onlinecite{Eriksen_2020} alongside its nuclear repulsion and Hartree-Fock energies.
\alert{The geometry of benzene has been optimized at the MP2/6-31G* level \cite{Schreiber_2008} and its coordinates} can be found in the supporting information of Ref.~\onlinecite{Eriksen_2020} alongside its nuclear repulsion and Hartree-Fock energies.
This corresponds to an active space of 30 electrons and 108 orbitals, \ie, the Hilbert space of benzene is of the order of $10^{35}$ Slater determinants.
Needless to say that this size of Hilbert space cannot be tackled by exact diagonalization with current architectures.
The correlation energies reported in Ref.~\onlinecite{Eriksen_2020} are gathered in Table \ref{tab:energy} alongside the best ph-AFQMC estimate from Ref.~\onlinecite{Lee_2020} based on a CAS(6,6) trial wave function.
@ -115,8 +115,8 @@ Recently, the determinant-driven CIPSI algorithm has been efficiently implemente
In particular, we were able to compute highly-accurate calculations of ground- and excited-state energies for small- and medium-sized molecules (including benzene). \cite{Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c}
CIPSI is also frequently used to provide accurate trial wave function for QMC calculations. \cite{Caffarel_2014,Caffarel_2016a,Caffarel_2016b,Giner_2013,Giner_2015,Scemama_2015,Scemama_2016,Scemama_2018,Scemama_2018b,Scemama_2019,Dash_2018,Dash_2019}
The particularity of the current implementation is that the selection step and the PT2 correction are computed \textit{simultaneously} via a hybrid semistochastic algorithm \cite{Garniron_2017,Garniron_2019} (which explains the statistical error associated with the PT2 correction in the following).
Moreover, a renormalized version of the PT2 correction (dubbed rPT2 below) has been recently implemented for a more efficient extrapolation to the FCI limit. \cite{Garniron_2019}
We refer the interested reader to Ref.~\onlinecite{Garniron_2019} where one can find all the details regarding the implementation of the CIPSI algorithm.
\alert{Moreover, a renormalized version of the PT2 correction (dubbed rPT2 below) has been recently implemented and tested for a more efficient extrapolation to the FCI limit thanks to a partial resummation of the higher-order of perturbation. \cite{Garniron_2019}
We refer the interested reader to Ref.~\onlinecite{Garniron_2019} where one can find all the details regarding the implementation of the rPT2 correction and the CIPSI algorithm.}
% Computational details
Being late to the party, we obviously cannot report blindly our CIPSI results.
@ -141,9 +141,11 @@ $\sigma^*$ [25--36];
higher-lying $\pi$ [39,41--43,46,49,50,53--57,71--74,82--85,87,92,93,98];
higher-lying $\sigma$ [37,38,40,44,45,47,48,51,52,58--70,75--81,86,88--91,94--97,99--114].}
Like Pipek-Mezey, \cite{Pipek_1989} this choice of orbital windows allows to preserve a strict $\sigma$-$\pi$ separation in planar systems like benzene.
As one can see from the energies of Table \ref{tab:NOvsLO}, for a given value of $N_\text{det}$, the variational energy as well as the PT2-corrected energies are much lower with localized orbitals than with natural orbitals. We, therefore, consider these energies more trustworthy, and we will base our best estimate of the correlation energy of benzene on these calculations.
As one can see from the energies of Table \ref{tab:NOvsLO}, for a given value of $N_\text{det}$, the variational energy as well as the PT2-corrected energies are much lower with localized orbitals than with natural orbitals. \cite{Chien_2018,Eriksen_2020}
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 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.}}
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
@ -151,6 +153,7 @@ Our final number are gathered in Table \ref{tab:extrap_dist_table}, where, follo
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).
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}).
\alert{It is of interest to report the variation of the correlation energy estimates with the number of fitting points.}
%%$ FIG. 1 %%%
\begin{figure*}
@ -163,6 +166,7 @@ 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 $E_\text{rPT2}$.
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.
\alert{The statistical error bar associated with $E_\text{PT2}$ or $E_\text{rPT2}$ are of the order of the size of the markers.}
\label{fig:CIPSI}
}
\end{figure*}
@ -253,7 +257,7 @@ We thank Janus Eriksen and Cyrus Umrigar for useful comments.
This work was performed using HPC resources from GENCI-TGCC (2020-gen1738) and from CALMIP (Toulouse) under allocation 2020-18005.
% Data availability statement
The data that supports the findings of this study are available within the article.
\alert{The data that support the findings of this study are openly available in Zenodo at http://doi.org/[DOI].}
\bibliography{benzene}

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