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\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}

\maketitle

% 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 belonging to the \textit{Simons Collaboration on the Many-Electron Problem} 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),
	(ii) the many-body expansion approach (MBE-FCI),
	(iii) three selected configuration interaction (SCI) methods including a second-order perturbative correction (ASCI, iCI, and SHCI),
	(iv) a selected coupled-cluster theory method including a second-order perturbative correction (FCCR),
	(v) the density-matrix renornalization group approach (DMRG), and
	(vi) two flavors of full configuration interaction quantum Monte Carlo (AS-FCIQMC and CAD-FCIQMC).
We refer the interested reader to Ref.~\onlinecite{Eriksen_2020} and its supporting information for additional details on each method and their corresponding references.
Soon after, Lee \textit{et al.} reported phaseless auxiliary-field quantum Monte Carlo (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 benzene molecule in the cc-pVDZ basis.
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 with conventional, deterministic FCI algorithm with current architecture.
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}.
 
%%% 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}{ccc}
      Method		&	$E_c$			&	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			&	XXX			&	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), which also includes a second-order perturbative (PT2) correction.
In short, the CIPSI algorithm belongs to the family of SCI+PT2 methods.
From an historical point of view, CIPSI is probably the oldest SCI algorithm developed in 1973 by Huron, Rancurel, and Malrieu. \cite{Huron_1973}
Recently, the determinant-driven CIPSI algorithm has been efficiently implemented in the open-source programming environment {\QP} by one of us (AS) enabling to perform massively parallel computations. \cite{}
In particular, we were able to compute highly-accurate calculations of ground- and excited-state energies of small- and medium-sized molecules. \cite{}
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{}
Moreover, a renormalized version of the PT2 correction dubbed rPT2 has been recently implemented for a more efficient extrapolation to the FCI limit. \cite{}
We refer the interested reader to Ref.~\onlinecite{} where one can find all the details regarding the implementation of the CIPSI algorithm.


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.

This work was performed using HPC resources from GENCI-TGCC (Grand Challenge 2019-gch0418) and from CALMIP (Toulouse) under allocation 2019-0510.

\bibliography{benzene}

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