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minor corrections at proof stage

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This commit is contained in:
Pierre-Francois Loos 2020-10-20 21:30:07 +02:00
parent ad8d048848
commit 6d76db7fe4
3 changed files with 174 additions and 78 deletions
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@ -10,10 +10,10 @@
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42
benzene.tex
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@ -57,20 +57,20 @@ The same comment applies to the excited-state benchmark set of Thiel and coworke
Following a similar goal, we have recently proposed a large set of highly-accurate vertical transition energies for various types of excited states thanks to the renaissance of selected configuration interaction (SCI) methods \cite{Bender_1969,Huron_1973,Buenker_1974} which can now routinely produce near full configuration interaction (FCI) quality excitation energies for small- and medium-sized organic molecules. \cite{Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c} Following a similar goal, we have recently proposed a large set of highly-accurate vertical transition energies for various types of excited states thanks to the renaissance of selected configuration interaction (SCI) methods \cite{Bender_1969,Huron_1973,Buenker_1974} which can now routinely produce near full configuration interaction (FCI) quality excitation energies for small- and medium-sized organic molecules. \cite{Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c}
% The context % The context
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 a recent article, \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: 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} (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_2014,Liu_2016,Lei_2017,Zhang_2020} and semistochastic heat-bath CI (SHCI) \cite{Holmes_2016,Holmes_2017,Sharma_2017}]}, (ii) three SCI methods including a second-order perturbative correction [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,Xu_2020} which also includes a second-order perturbative correction}, (iii) the full coupled-cluster reduction (FCCR) \cite{Xu_2018,Xu_2020} which also includes a second-order perturbative correction,
(iv) the density-matrix renornalization group approach (DMRG), \cite{White_1992,White_1993,Chan_2011} and (iv) the density-matrix renornalization group (DMRG) approach, \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} (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. 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} 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 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 target application is the non-relativistic frozen-core correlation energy of the ground state of the benzene molecule in the cc-pVDZ basis.
\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. 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. This corresponds to an active space of 30 electrons and 108 orbitals, \ie, the Hilbert space 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. 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. 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.
The outcome of this work is nicely summarized in the abstract of Ref.~\onlinecite{Eriksen_2020}: The outcome of this work is nicely summarized in the abstract of Ref.~\onlinecite{Eriksen_2020}:
@ -79,12 +79,12 @@ The outcome of this work is nicely summarized in the abstract of Ref.~\onlinecit
%%% TABLE 1 %%% %%% TABLE 1 %%%
\begin{table} \begin{table}
\caption{ \caption{
The frozen-core correlation energy (in m$E_h$) of benzene in the cc-pVDZ basis set using various methods. The frozen-core correlation energy $\Delta E$ (in m$E_h$) of benzene in the cc-pVDZ basis set using various methods.
\label{tab:energy} \label{tab:energy}
} }
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{llc} \begin{tabular}{llc}
Method & \tabc{$E_c$} & Ref. \\ Method & \tabc{$\Delta E$} & Ref. \\
\hline \hline
ASCI & $-860.0$ & \onlinecite{Eriksen_2020} \\ ASCI & $-860.0$ & \onlinecite{Eriksen_2020} \\
iCI & $-861.1$ & \onlinecite{Eriksen_2020} \\ iCI & $-861.1$ & \onlinecite{Eriksen_2020} \\
@ -114,7 +114,7 @@ The outcome of this work is nicely summarized in the abstract of Ref.~\onlinecit
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 bars associated with $E_\text{PT2}$ or $E_\text{rPT2}$ (not shown) are of the order of the size of the markers.} 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*}
@ -193,9 +193,9 @@ The statistical error on $E_\text{(r)PT2}$, corresponding to one standard deviat
\end{table} \end{table}
% CIPSI % CIPSI
For the sake of completeness and our very own curiosity, we report in this Note the frozen-core correlation energy obtained with a fourth flavor of SCI known as \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI), \cite{Huron_1973} which also includes a second-order perturbative (PT2) correction. For the sake of completeness and our very own curiosity, we report in this Note the frozen-core correlation energy obtained with a fourth flavor of SCI known as \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI), \cite{Huron_1973} which also includes a PT2 correction.
In short, the CIPSI algorithm belongs to the family of SCI+PT2 methods. In short, the CIPSI algorithm belongs to the family of SCI+PT2 methods.
The idea behind such methods is to avoid the exponential increase of the size of the CI expansion by retaining the most energetically relevant determinants only, thanks to the use of a second-order energetic criterion to select perturbatively determinants in the FCI space. The idea behind such methods is to slow down the exponential increase of the size of the CI expansion by retaining the most energetically relevant determinants only, thanks to the use of a second-order energetic criterion to select perturbatively determinants in the FCI space.
However, performing SCI calculations rapidly becomes extremely tedious when one increases the system size as one hits the exponential wall inherently linked to these methods. However, performing SCI calculations rapidly becomes extremely tedious when one increases the system size as one hits the exponential wall inherently linked to these methods.
From a historical point of view, CIPSI is probably one of the oldest SCI algorithm. From a historical point of view, CIPSI is probably one of the oldest SCI algorithm.
@ -204,14 +204,14 @@ Recently, the determinant-driven CIPSI algorithm has been efficiently implemente
In particular, we were able to compute highly-accurate ground- and excited-state energies for small- and medium-sized molecules (including benzene). \cite{Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c} In particular, we were able to compute highly-accurate 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} 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). 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).
\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} 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.} 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 % Computational details
Being late to the party, we obviously cannot report blindly our CIPSI results. Being late to the party, we obviously cannot report blindly our CIPSI results.
However, following the philosophy of Eriksen \textit{et al.} \cite{Eriksen_2020} and Lee \textit{et al.}, \cite{Lee_2020} we will report our results with the most neutral tone, leaving the freedom to the reader to make up his/her mind. However, following the philosophy of Eriksen \textit{et al.} \cite{Eriksen_2020} and Lee \textit{et al.}, \cite{Lee_2020} we will report our results with the most neutral tone, leaving the freedom to the reader to make up his/her mind.
We then follow our usual ``protocol'' \cite{Scemama_2018,Scemama_2018b,Scemama_2019,Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c} by performing a preliminary SCI calculation using Hartree-Fock orbitals in order to generate a SCI wave function with at least $10^7$ determinants. We then follow our usual ``protocol'' \cite{Scemama_2018,Scemama_2018b,Scemama_2019,Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c} by performing a preliminary SCI calculation using Hartree-Fock orbitals in order to generate a SCI wave function with at least $10^7$ determinants.
Natural orbitals are then computed based on this wave function, and a new, larger SCI calculation is performed with this new natural set of orbitals. Natural orbitals are then computed based on this wave function, and a new SCI calculation is performed with this new natural set of orbitals.
This has the advantage to produce a smoother and faster convergence of the SCI energy toward the FCI limit. This has the advantage to produce a smoother and faster convergence of the SCI energy toward the FCI limit.
The total SCI energy is defined as the sum of the variational energy $E_\text{var.}$ (computed via diagonalization of the CI matrix in the reference space) and a second-order perturbative correction $E_\text{(r)PT2}$ which takes into account the external determinants, \ie, the determinants which do not belong to the variational space but are linked to the reference space via a nonzero matrix element. The magnitude of $E_\text{(r)PT2}$ provides a qualitative idea of the ``distance'' to the FCI limit. The total SCI energy is defined as the sum of the variational energy $E_\text{var.}$ (computed via diagonalization of the CI matrix in the reference space) and a second-order perturbative correction $E_\text{(r)PT2}$ which takes into account the external determinants, \ie, the determinants which do not belong to the variational space but are linked to the reference space via a nonzero matrix element. The magnitude of $E_\text{(r)PT2}$ provides a qualitative idea of the ``distance'' to the FCI limit.
As mentioned above, SCI+PT2 methods rely heavily on extrapolation, especially when one deals with medium-sized systems. As mentioned above, SCI+PT2 methods rely heavily on extrapolation, especially when one deals with medium-sized systems.
@ -232,24 +232,24 @@ 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].} 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. 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. 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.
\alert{Indeed, localized orbitals significantly speed up the convergence of SCI calculations by taking benefit of the local character of electron correlation.\cite{Angeli_2003,Angeli_2009,BenAmor_2011,Suaud_2017,Chien_2018,Eriksen_2020}} Indeed, localized orbitals significantly speed up the convergence of SCI calculations by taking benefit of the local character of electron correlation.\cite{Angeli_2003,Angeli_2009,BenAmor_2011,Suaud_2017,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. 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.
\alert{(In other words, the four largest variational wave functions are considered to perform the linear extrapolation.)} (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, DMRG, 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).
\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. \footnote{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.
These numbers vary by $1.4$ mE$_h$. 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. 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. 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. 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$_h$ 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}). 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}).
% Timings % Timings
@ -266,7 +266,7 @@ This work was performed using HPC resources from GENCI-TGCC (2020-gen1738) and f
PFL and AS have received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481). PFL and AS have received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).
% Data availability statement % Data availability statement
\alert{The data that support the findings of this study are openly available in Zenodo at http://doi.org/10.5281/zenodo.4075286.} The data that support the findings of this study are openly available in Zenodo at \href{http://doi.org/10.5281/zenodo.4075286}{http://doi.org/10.5281/zenodo.4075286}.
%merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked %merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked
%Control: key (0) %Control: key (0)
@ -1114,7 +1114,7 @@ PFL and AS have received funding from the European Research Council (ERC) under
(\bibinfo {year} {2018})}\BibitemShut {NoStop}% (\bibinfo {year} {2018})}\BibitemShut {NoStop}%
\bibitem [{Note2()}]{Note2}% \bibitem [{Note2()}]{Note2}%
\BibitemOpen \BibitemOpen
\bibinfo {note} {\protect \leavevmode {\protect \color {black}Using the last \bibinfo {note} {\protect \leavevmode {\protect Using the
3, 4, 5, and 6 largest wave functions to perform the linear extrapolation 3, 4, 5, and 6 largest wave functions to perform the linear extrapolation
yield the following correlation energy estimates: $-863.1(11)$, $-863.4(5)$, yield the following correlation energy estimates: $-863.1(11)$, $-863.4(5)$,
$-862.1(8)$, and $-863.5(11)$ mE$_h$, respectively. These numbers vary by $-862.1(8)$, and $-863.5(11)$ mE$_h$, respectively. These numbers vary by

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