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@ -43,7 +43,7 @@
% 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 ground state of the benzene molecule in the cc-pVDZ basis. Following our usual protocol, we obtain a correlation energy of \titou{$-86x.x(x)$} m$E_h$ which agrees with the best 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 \titou{$-86x.x(x)$} 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}
% Title
@ -57,16 +57,15 @@ 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}
% 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.
A large panel of highly-accurate methods were considered:
(i) coupled cluster theory with singles, doubles, triples, and quadruples (CCSDTQ), \cite{Oliphant_1991,Kucharski_1992}
(ii) the many-body expansion FCI (MBE-FCI), \cite{Eriksen_2018,Eriksen_2019}
(iii) three SCI methods including a second-order perturbative correction (ASCI, \cite{Tubman_2016,Tubman_2018,Tubman_2020} iCI, \cite{Liu_2016} and SHCI \cite{Holmes_2016,Holmes_2017,Sharma_2017}),
(iv) a selected coupled-cluster theory method including a second-order perturbative correction (FCCR), \cite{Xu_2018}
(v) the density-matrix renornalization group approach (DMRG), \cite{White_1992} and
(vi) 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}
We refer the interested reader to Ref.~\onlinecite{Eriksen_2020} and its supporting information for additional details on each method and a complete list of their corresponding 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}
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_2018,Eriksen_2019}
(ii) three SCI methods including a second-order perturbative correction (ASCI, \cite{Tubman_2016,Tubman_2018,Tubman_2020} iCI, \cite{Liu_2016} and 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}
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.
@ -114,13 +113,13 @@ It was developed in 1973 by Huron, Rancurel, and Malrieu \cite{Huron_1973} (see
Recently, the determinant-driven CIPSI algorithm has been efficiently implemented \cite{Giner_2013,Giner_2015} in the open-source programming environment {\QP} by our group enabling to perform massively parallel computations. \cite{Garniron_2017,Garniron_2018,Garniron_2019}
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 in the following) has been recently implemented for a more efficient extrapolation to the FCI limit (see below). \cite{Garniron_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.
% Computational details
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} 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.
Natural orbitals are then computed based on this wave function, and a new, larger SCI calculation is performed with this new set of orbitals.
This has the advantage to produce a smoother and faster convergence of the SCI energy toward the FCI limit.
@ -142,14 +141,14 @@ 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.
The convergence of the CIPSI correlation energy using localized orbitals is illustrated in Fig.~\ref{fig:CIPSI}, where one can see the behavior of $\Delta E_\text{var.}$ 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 $\Delta E_\text{var.}$ (in m$E_h$) as a function of $E_\text{(r)PT2}$, and their corresponding \titou{five}-point linear extrapolation curves that we have used to get our final estimate of the correlation energy.
From this figure, one clearly sees that the rPT2-based correction behaves more linearly than its corresponding PT2 version, and thus systematically employed in the following.
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 \titou{five}-point linear extrapolation curves that we have used to get our final estimate of the correlation energy.
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
Our final numbers 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 the ASCI, iCI, SHCI, CIPSI, and DMRG results.
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 methods yield correlation energies ranging from $-863.7$ to $-862.8$ m$E_h$. Our final CIPSI number (obtained with localized orbitals and rPT2 correction via a \titou{five}-point linear extrapolation) is \titou{$-86x.x(x)$} m$E_h$.
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.
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 \titou{five}-point linear extrapolation) is \titou{$-86x.x(x)$} m$E_h$.
For comparison, the best post blind test SHCI estimate is $-863.3$ m$E_h$, which agrees almost perfectly with our best CIPSI estimate.
\titou{Although it is not possible to provide a theoretically sound error bar, we estimate the extrapolation error by ...
We believe that it provides a very safe estimate of the extrapolation error.}
@ -169,11 +168,11 @@ In total, the present calculation has required \titou{340k} core hours, almost h
\hspace{0.08\linewidth}
\includegraphics[width=0.4\linewidth]{fig1b}
\caption{
Convergence of the CIPSI correlation energy using localized orbitals.
Left: $\Delta E_\text{var.}$, $\Delta E_\text{var.} + E_\text{PT2}$, and $\Delta E_\text{var.} + E_\text{rPT2}$ (in m$E_h$) as functions of the number of determinants in the variational space.
Right: $\Delta E_\text{var.} + E_\text{PT2}$ and $\Delta E_\text{var.} + E_\text{rPT2}$ (in m$E_h$) as functions of $E_\text{PT2}$ or $E_\text{rPT2}$.
Convergence of the CIPSI correlation energy of benzene using localized orbitals.
Left: $\Delta E_\text{var.}$, $\Delta E_\text{var.} + E_\text{PT2}$, and $\Delta E_\text{var.} + E_\text{rPT2}$ (in m$E_h$) as functions of the number of determinants in the variational space $N_\text{det}$.
Right: $\Delta E_\text{var.}$ (in m$E_h$) as a function of $E_\text{PT2}$ or $E_\text{rPT2}$.
The \titou{five}-point linear extrapolation curves (dashed lines) are also reported.
The theoretical best 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.
\label{fig:CIPSI}
}
\end{figure*}
@ -182,7 +181,7 @@ In total, the present calculation has required \titou{340k} core hours, almost h
%\begin{squeezetable}
\begin{table*}
\caption{Variational energy $E_\text{var.}$, second-order perturbative correction $E_\text{PT2}$ and its renormalized version $E_\text{rPT2}$ (in $E_h$) as a function of the number of determinants $N_\text{det}$ for the ground-state of the benzene molecule computed in the cc-pVDZ basis set.
The statistical error on $E_\text{PT2}$, corresponding to one standard deviation, are reported in parenthesis.}
The statistical error on $E_\text{(r)PT2}$, corresponding to one standard deviation, are reported in parenthesis.}
\label{tab:NOvsLO}
\begin{ruledtabular}
\begin{tabular}{rcccccc}
@ -226,10 +225,10 @@ The statistical error on $E_\text{PT2}$, corresponding to one standard deviation
%%% TABLE II %%%
\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 the ASCI, iCI, SHCI, CIPSI, and DMRG results.
\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.
The final variational energies $\Delta E_{\text{var.}}$ are also reported.
See Ref.~\onlinecite{Eriksen_2020} for more details.
All the energies are given in m$E_h$.
All correlation energies are given in m$E_h$.
\label{tab:extrap_dist_table}
}
\begin{ruledtabular}