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\begin{document}

\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}

\author{Pierre-Fran\c{c}ois Loos}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Anthony Scemama}
\email{scemama@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}

% 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 $-8xx.xx$ 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.
\end{abstract}

% Title
\maketitle

% Intro
Although sometimes decried, one cannot denied the usefulness of benchmark sets and their corresponding reference data for the electronic structure community.
These are indeed essential for the validation of existing theoretical models and to bring to light and subsequently understand their strengths and, more importantly, their weaknesses.
In that regard, the previous benchmark datasets provided by the \textit{Simons Collaboration on the Many-Electron Problem} have been extremely valuable. \cite{Leblanc_2015,Motta_2017,Williams_2020}
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}

% 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 and it can be found in the supporting information of Ref.~\onlinecite{Eriksen_2020}.
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}.
The outcome of this work is nicely summarized in the abstract of Ref.~\onlinecite{Eriksen_2020}:
\textit{``In our assessment, the evaluated high-level methods are all found to qualitatively agree on a final correlation energy, with most methods yielding an estimate of the FCI value around $-863$ m$E_h$. However, we find the root-mean-square deviation of the energies from the studied methods to be considerable ($1.3$ m$E_h$), which in light of the acclaimed performance of each of the methods for smaller molecular systems clearly displays the challenges faced in extending reliable, near-exact correlation methods to larger systems.''}

%%% TABLE 1 %%%
\begin{table}
  \caption{
    The frozen-core correlation energy (in m$E_h$) of benzene in the cc-pVDZ basis set using various methods.
    \label{tab:energy}
  }
  \begin{ruledtabular}
    \begin{tabular}{ldc}
      Method		&	\tabc{$E_c$}			&	Ref.	\\ 
      \hline
      ASCI			&	-860.0(2)	&	\onlinecite{Eriksen_2020}	\\
      iCIPT2		&	-861.1(5)	&	\onlinecite{Eriksen_2020}	\\
      CCSDTQ		&	-862.4		&	\onlinecite{Eriksen_2020}	\\
      DMRG			&	-862.8(7)	&	\onlinecite{Eriksen_2020}	\\
      FCCR(2)		&	-863.0		&	\onlinecite{Eriksen_2020}	\\
      CAD-FCIQMC	&	-863.4		&	\onlinecite{Eriksen_2020}	\\
      AS-FCIQMC		&	-863.7(3)	&	\onlinecite{Eriksen_2020}	\\
      SHCI			&	-864.2(2)	&	\onlinecite{Eriksen_2020}	\\
      \hline
      ph-AFQMC		&	-864.3(4)	&	\onlinecite{Lee_2020}		\\
      \hline
      CIPSI			&	-8xx.x(x)	&	This work					\\
   \end{tabular}
 \end{ruledtabular}
\end{table}

% CIPSI
In this Note, we report 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.
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. 
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 an historical point of view, CIPSI is probably one of the oldest SCI algorithm. 
It was developed in 1973 by Huron, Rancurel, and Malrieu \cite{Huron_1973} (see also Ref.~\onlinecite{Evangelisti_1983}).
Recently, the determinant-driven CIPSI algorithm has been efficiently implemented \cite{Giner_2013,Giner_2015} in the open-source programming environment {\QP} by one of us (AS) 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 of small- and medium-sized molecules (including benzene). \cite{Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c}
CIPSI is also frequently use 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}
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}
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
The present calculations have been performed on the AMD partition of GENCI's Irene supercomputer. 
Each Irene's AMD node is a dual-socket \titou{Intel(R) Xeon(R) Platinum 8168 CPU@2.70 GHz with 192GiB of RAM}, with a total of 128 physical CPU cores.

% Discussion
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.
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 (NOs) 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
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 $E_\text{PT2}$ 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 $E_\text{PT2}$ provides a qualitative idea of the ``distance'' to the FCI limit.
As mentioned above, SCI+PT2 methods rely heavily on extrapolation when one deals with medium-sized systems.
We then linearly extrapolate the total SCI energy to $E_\text{PT2} = 0$ (which effectively corresponds to the FCI limit) using the two largest SCI wave functions. 
Although it is not possible to provide a theoretically sound error bar, we estimate the extrapolation error by the difference in excitation energy between the largest SCI wave function and its corresponding extrapolated value. 
We believe that it provides a very safe estimate of the extrapolation error. 

The three flavours of SCI fall into an interval ranging from $-863.7$ to $-862.8$ m$E_h$.
The CIPSI number is ?
\titou{Note that, even though benzene is big, we have already reported excitation energies of benzene with the 6-31+G(d) basis in Ref.~\onlinecite{Loos_2019}.}

%%% 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. 
	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$E_h$.
	\label{extrap_dist_table}
	}
	\begin{ruledtabular}
		\begin{tabular}{lcccc}
			Method & $\Delta E_{\text{var.}}$ & $\Delta E_{\text{final}}$ & $\Delta E_{\text{extrap.}}$ & $\Delta E_{\text{dist}}$ \\
			\hline
			ASCI	&	$-737.1$	&	$-835.4$	&	$-860.0$	&	$-24.6$	\\
			iCI		&	$-730.0$	&	$-833.7$	&	$-861.1$	&	$-27.4$	\\
			SHCI	&	$-827.2$	&	$-852.8$	&	$-864.2$	&	$-11.4$	\\
			CIPSI	&	$-8xx.x$	&	$-8xx.x$	&	$-8xx.x$	&	$-xx.x$	\\
			DMRG	&	$-859.2$	&	$-859.2$	&	$-862.8$	&	$-3.6$	\\
		\end{tabular}
	\end{ruledtabular}
\end{table}

%%$ FIG. 1 %%%
\begin{figure*}
	\includegraphics[width=0.4\linewidth]{fig1a}
	\hspace{0.08\linewidth}
	\includegraphics[width=0.4\linewidth]{fig1b}
	\caption{
	Convergence of the CIPSI correlation energy for benzene.
	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 $E_\text{rPT2}$.
	The two-point linear extrapolation curves (dashed lines) are also reported.
	The theoretical best estimate of $-863$ m$E_h$ from Ref.~\onlinecite{Eriksen_2020} is reported for comparison purposes.
	\label{fig:CIPSI}
	}
\end{figure*}

%%% TABLE 2 %%%
%\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 errors ($\delta E_\text{PT2}$ and $\delta E_\text{rPT2}$), corresponding to one standard deviation, are also reported.}
\label{tab:Cr2}
\begin{ruledtabular}
\begin{tabular}{rddddd}
\tabc{$N_\text{det}$}	&	\tabc{$E_\text{var.}$}	&	\tabc{$E_\text{PT2}$}	&	\tabc{$\delta E_\text{PT2}$}	&	\tabc{$E_\text{rPT2}$}	&	\tabc{$\delta E_\text{rPT2}$}	\\	
\hline
       12418  &	-231.17094496  &	-0.36188641     &	0.00040193  &	-0.32968372    &	 0.00036616 \\
       24841  &	-231.24480922  &	-0.28851015     &	0.00046895  &	-0.26939524    &	 0.00043788 \\
       49683  &	-231.31129020  &	-0.22584693     &	0.00043741  &	-0.21493067    &	 0.00041627 \\
       99367  &	-231.35989232  &	-0.18072875     &	0.00030762  &	-0.17412788    &	 0.00029638 \\
      198754  &	-231.38819148  &	-0.15638840     &	0.00023911  &	-0.15163900    &	 0.00023185 \\
      397561  &	-231.40076198  &	-0.14599792     &	0.00024280  &	-0.14195331    &	 0.00023608 \\
      795126  &	-231.40804005  &	-0.14060686     &	0.00023393  &	-0.13690453    &	 0.00022777 \\
     1590261  &	-231.41441889  &	-0.13567145     &	0.00024941  &	-0.13227230    &	 0.00024316 \\
     3180532  &	-231.41959399  &	-0.13129538     &	0.00023846  &	-0.12814025    &	 0.00023273 \\
     6361098  &	-231.42421895  &	-0.12727083     &	0.00024438  &	-0.12432787    &	 0.00023873 \\
    12722217  &	-231.43050041  &	-0.12234780     &	0.00023326  &	-0.11965826    &	 0.00022813 \\
    25444447  &	-231.43883900  &	-0.11574659     &	0.00022627  &	-0.11336968    &	 0.00022163	 \\
    Extrap.	  &         	   &    				&      		     &           		\\
  \end{tabular}
\end{ruledtabular}
\end{table*}
%\end{squeezetable}
%%% %%% %%% %%%


% Acknowledgements
This work was performed using HPC resources from GENCI-TGCC (Grand Challenge 2019-gch0418) and from CALMIP (Toulouse) under allocation 2020-18005.

\bibliography{benzene}

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