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benzene.tex
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% Abstract % Abstract
\begin{abstract} \begin{abstract}
Following the recent work of Eriksen \textit{et al.}~[\href{https://arxiv.org/abs/2008.02678}{arXiv:2008.02678 [physics.chem-ph]}], we report the performance of the \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI) method on the non-relativistic frozen-core correlation energy of the benzene molecule in the cc-pVDZ basis. Following our usual protocol, we obtain a correlation energy of $-863.4(5)$ m$E_h$ which agrees with the theoretical estimate of $-863$ m$E_h$ proposed by Eriksen \textit{et al.}~using an extensive array of highly-accurate new electronic structure methods. Following the recent work of Eriksen \textit{et al.}~[\href{https://arxiv.org/abs/2008.02678}{arXiv:2008.02678 [physics.chem-ph]}], we report the performance of the \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI) method on the non-relativistic frozen-core correlation energy of the benzene molecule in the cc-pVDZ basis. Following our usual protocol, we obtain a correlation energy of $-863.4$ m$E_h$ which agrees with the theoretical estimate of $-863$ m$E_h$ proposed by Eriksen \textit{et al.}~using an extensive array of highly-accurate new electronic structure methods.
\end{abstract} \end{abstract}
% Title % Title
@ -86,19 +86,19 @@ The outcome of this work is nicely summarized in the abstract of Ref.~\onlinecit
\begin{tabular}{llc} \begin{tabular}{llc}
Method & \tabc{$E_c$} & Ref. \\ Method & \tabc{$E_c$} & Ref. \\
\hline \hline
ASCI & $-860.0(2)$ & \onlinecite{Eriksen_2020} \\ ASCI & $-860.0$ & \onlinecite{Eriksen_2020} \\
iCIPT2 & $-861.1(5)$ & \onlinecite{Eriksen_2020} \\ iCI & $-861.1$ & \onlinecite{Eriksen_2020} \\
CCSDTQ & $-862.4$ & \onlinecite{Eriksen_2020} \\ CCSDTQ & $-862.4$ & \onlinecite{Eriksen_2020} \\
DMRG & $-862.8(7)$ & \onlinecite{Eriksen_2020} \\ DMRG & $-862.8$ & \onlinecite{Eriksen_2020} \\
FCCR(2) & $-863.0$ & \onlinecite{Eriksen_2020} \\ FCCR & $-863.0$ & \onlinecite{Eriksen_2020} \\
MBE-FCI & $-863.0$ & \onlinecite{Eriksen_2020} \\ MBE-FCI & $-863.0$ & \onlinecite{Eriksen_2020} \\
CAD-FCIQMC & $-863.4$ & \onlinecite{Eriksen_2020} \\ CAD-FCIQMC & $-863.4$ & \onlinecite{Eriksen_2020} \\
AS-FCIQMC & $-863.7(3)$ & \onlinecite{Eriksen_2020} \\ AS-FCIQMC & $-863.7$ & \onlinecite{Eriksen_2020} \\
SHCI & $-864.2(2)$ & \onlinecite{Eriksen_2020} \\ SHCI & $-864.2$ & \onlinecite{Eriksen_2020} \\
\hline \hline
ph-AFQMC & $-864.3(4)$ & \onlinecite{Lee_2020} \\ ph-AFQMC & $-864.3(4)$ & \onlinecite{Lee_2020} \\
\hline \hline
CIPSI & $-863.4(5)$ & This work \\ CIPSI & $-863.4$ & This work \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\end{table} \end{table}
@ -150,7 +150,7 @@ From this figure, one clearly sees that the rPT2-based correction behaves more l
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. 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}).
%%$ FIG. 1 %%% %%$ FIG. 1 %%%
\begin{figure*} \begin{figure*}
@ -214,7 +214,7 @@ The statistical error on $E_\text{(r)PT2}$, corresponding to one standard deviat
%%% TABLE II %%% %%% TABLE II %%%
\begin{table} \begin{table}
\caption{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. \caption{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 for the best blind-test and post-blind-test estimates of the correlation energy of benzene in the cc-pVDZ basis.
The final variational energies $\Delta E_{\text{var.}}$ are also reported. The final variational energies $\Delta E_{\text{var.}}$ are also reported.
See Ref.~\onlinecite{Eriksen_2020} for more details. See Ref.~\onlinecite{Eriksen_2020} for more details.
All correlation energies are given in m$E_h$. All correlation energies are given in m$E_h$.
@ -224,11 +224,17 @@ The statistical error on $E_\text{(r)PT2}$, corresponding to one standard deviat
\begin{tabular}{lcccc} \begin{tabular}{lcccc}
Method & $\Delta E_{\text{var.}}$ & $\Delta E_{\text{final}}$ & $\Delta E_{\text{extrap.}}$ & $\Delta E_{\text{dist}}$ \\ Method & $\Delta E_{\text{var.}}$ & $\Delta E_{\text{final}}$ & $\Delta E_{\text{extrap.}}$ & $\Delta E_{\text{dist}}$ \\
\hline \hline
\mc{4}{l}{Best blind-test estimates} \\
ASCI & $-737.1$ & $-835.4$ & $-860.0$ & $-24.6$ \\ ASCI & $-737.1$ & $-835.4$ & $-860.0$ & $-24.6$ \\
iCI & $-730.0$ & $-833.7$ & $-861.1$ & $-27.4$ \\ iCI & $-730.0$ & $-833.7$ & $-861.1$ & $-27.4$ \\
SHCI & $-827.2$ & $-852.8$ & $-864.2$ & $-11.4$ \\ SHCI & $-827.2$ & $-852.8$ & $-864.2$ & $-11.4$ \\
DMRG & $-859.2$ & $-859.2$ & $-862.8$ & $-3.6$ \\ DMRG & $-859.2$ & $-859.2$ & $-862.8$ & $-3.6$ \\
\hline \hline
\mc{4}{l}{Best post-blind-test estimates} \\
ASCI & $-772.4$ & $-835.2$ & $-861.3$ & $-26.1$ \\
iCI & $-770.7$ & $-842.8$ & $-864.2$ & $-21.3$ \\
SHCI & $-835.2$ & $-854.9$ & $-863.3$ & $-8.4$ \\
\hline
CIPSI & $-814.8$ & $-850.2$ & $-863.4$ & $-13.2$ \\ CIPSI & $-814.8$ & $-850.2$ & $-863.4$ & $-13.2$ \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}