\documentclass[aps,prb,reprint,noshowkeys,superscriptaddress]{revtex4-1}
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,wrapfig,txfonts}
\usepackage[version=4]{mhchem}

\newcommand{\ie}{\textit{i.e.}}
\newcommand{\eg}{\textit{e.g.}}
\newcommand{\alert}[1]{\textcolor{black}{#1}}
\usepackage[normalem]{ulem}
\newcommand{\titou}[1]{\textcolor{red}{#1}}
\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
\newcommand{\toto}[1]{\textcolor{green}{#1}}
\newcommand{\trashAS}[1]{\textcolor{green}{\sout{#1}}}
\newcommand{\AS}[1]{\toto{(\underline{\bf AS}: #1)}}

\newcommand{\Ec}{E_\text{c}}
\newcommand{\mEh}{$mE_h$}
\newcommand{\Eh}{$E_h$}
\newcommand{\mc}{\multicolumn}
\newcommand{\fnm}{\footnotemark}
\newcommand{\fnt}{\footnotetext}
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
\newcommand{\QP}{\textsc{quantum package}}

\usepackage[
	colorlinks=true,
    citecolor=blue,
    breaklinks=true
	]{hyperref}
\urlstyle{same}

\begin{document}

\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\CEISAM}{Universit\'e de Nantes, CNRS,  CEISAM UMR 6230, F-44000 Nantes, France}

\title{Reference correlation energies in finite Hilbert spaces: five- and six-membered rings}

\author{Micka\"el V\'eril}
\affiliation{\LCPQ}
\author{Yann Damour}
\affiliation{\LCPQ}
\author{Anthony Scemama}
\affiliation{\LCPQ}
\author{Michel Caffarel}
\affiliation{\LCPQ}
\author{Denis Jacquemin}
\affiliation{\CEISAM}
\author{Pierre-Fran\c{c}ois Loos}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}

% Abstract
\begin{abstract}
We report (frozen-core) full configuration interaction (FCI) energies in finite Hilbert spaces for various five- and six-membered rings.
In the continuity of our recent work on the benzene molecule [\href{https://doi.org/10.1063/5.0027617}{J. Chem. Phys. \textbf{153}, 176101 (2020)}], itself motivated by the blind challenge of Eriksen \textit{et al.} [\href{https://doi.org/10.1021/acs.jpclett.0c02621}{J. Phys. Chem. Lett. \textbf{11}, 8922 (2020)}] on the same system, we report reference frozen-core correlation energies for twelve cyclic molecules (cyclopentadiene, furan, imidazole, pyrrole, thiophene, benzene, pyrazine, pyridazine, pyridine, pyrimidine, tetrazine, and triazine) in the standard correlation-consistent double-$\zeta$ Dunning basis set (cc-pVDZ).
This corresponds to Hilbert spaces with sizes ranging from $10^{20}$ (for thiophene) to $10^{36}$ (for benzene).
Our estimates are based on localized-orbital-based selected configuration interaction (SCI) calculations performed with the \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI) algorithm.
The performance and convergence properties of several series of methods are investigated. 
In particular, we study the convergence properties of ii) the M{\o}ller-Plesset perturbation series up to fifth-order (MP2, MP3, MP4, and MP5), ii) the iterative approximate single-reference coupled-cluster series CC2, CC3, and CC4, and ii) the single-reference coupled-cluster series CCSD, CCSDT, and CCSDTQ.
The performance of the ground-state gold standard CCSD(T) is also investigated.
\end{abstract}

% Title
\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%
Electronic structure theory relies heavily on approximations.
Loosely speaking, to make any theory useful, three main approximations must be enforced.
The first fundamental approximation, known as the Born-Oppenheimer approximation, usually consists in assuming that the motion of nuclei and electrons are decoupled.
The nuclei coordinates can then be treated as parameters in the electronic Hamiltonian.
The second central approximation which makes calculations feasable by a computer is the basis set approximation where one introduces a set of pre-defined basis functions to represent the many-electron wave function of the system.
In most molecular calculations, a set of one-electron, atom-centered gaussian basis functions are introduced to expand the so-called one-electron molecular orbitals which are then used to build the many-electron Slater determinants.
The third and most relevant approximation in the present context is the ansatz (or form) of the electronic wave function $\Psi$.
For example, in configuration interaction (CI) methods, the wave function is expanded as a linear combination of Slater determinants, while in (single-reference) coupled-cluster (CC) theory, \cite{Cizek_1966,Paldus_1972,Crawford_2000,Bartlett_2007,Shavitt_2009} a reference Slater determinant $\Psi_0$ [usually taken as the Hartree-Fock (HF) wave function] is multiplied by a wave operator defined as the exponentiated excitation operator $\Hat{T} = \sum_{k=1}^n \Hat{T}_k$ (where $n$ is the number of electrons).

The truncation of $\Hat{T}$ allows to define a hierarchy of non-variational and size-extensive methods with improved accuracy:
CC with singles and doubles (CCSD), \cite{Cizek_1966,Purvis_1982} CC with singles, doubles, and triples (CCSDT), \cite{Noga_1987a,Scuseria_1988} CC with singles, doubles, triples, and quadruples (CCSDTQ), \cite{Oliphant_1991,Kucharski_1992} with corresponding computational scalings of $\order*{N^{6}}$, $\order*{N^{8}}$, and $\order*{N^{10}}$, respectively (where $N$ denotes the number of orbitals).
Parallel to the ``complete'' CC series presented above, an alternative series of approximate iterative CC models have been developed by the Aarhus group in the context of CC response theory \cite{Christiansen_1998} where one skips the most expensive terms and avoids the storage of the higher-excitation amplitudes:  CC2, \cite{Christiansen_1995a} CC3, \cite{Christiansen_1995b,Koch_1997} and CC4 \cite{Kallay_2005} 
These iterative methods scale as $\order*{N^{5}}$, $\order*{N^{7}}$, and $\order*{N^{9}}$, respectively, and can be seen as cheaper approximations of CCSD, CCSDT, and CCSDTQ.

A similar systematic truncation strategy can be applied to CI methods leading to the well-established family of methods known as CISD, CISDT, CISDTQ, \ldots~where one systematically increases the maximum excitation degree of the determinants taken into account. 
Except for full CI (FCI) where all determinants from the Hilbert space (\ie, with excitation degree up to $N$) are considered, truncated CI methods are variational but lack size-consistency.
The non-variationality of truncated CC methods being less of an issue than the size-inconsistency of the truncated CI methods, the formers have naturally overshadowed the latters in the electronic structure landscape.
However, a different strategy has recently made a come back in the context of CI methods.
Indeed, selected CI (SCI) methods where one iteratively selects the energetically relevant determinants from the FCI space has been recently shown to be highly successful to produce reference energies for ground and excited states in small- and medium-size molecules.


A rather different strategy in order to reach the holy grail FCI limit is to resort to M{\o}ller-Plesset perturbation theory. \cite{Moller_1934}
Again, at least in theory, one can obtain the exact energy of the system by ramping up the degree of the pertrubative series.
The second-order  M{\o}ller-Plesset (MP2) method is very well known but its higher-order 
MP3 \cite{Pople_1976}
MP4 \cite{Krishnan_1980}
MP5 \cite{Kucharski_1989}
MP6 \cite{He_1996a,He_1996b}
CCSD(T) \cite{Raghavachari_1989} is the gold-standard



Reviews. \cite{Crawford_2000,Piecuch_2002,Bartlett_2007,Shavitt_2009}

Coupled-cluster methods have been particularly successful for small- and medium-sized molecules  properties
\cite{Kallay_2003,Kallay_2004a,Gauss_2006,Kallay_2006,Gauss_2009}  

%%% FIG 1 %%%
\begin{figure*}
	\includegraphics[width=\linewidth]{mol}
	\caption{
	Five-membered rings (top) and six-membered rings (bottom) considered in this study.
	\label{fig:mol}}
\end{figure*}
%%% FIG 1 %%%

%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
%%%%%%%%%%%%%%%%%%%%%%%%%
The geometries of the twelve systems considered in the present study have been all obtained at the CC3/aug-cc-pVTZ level of geometry and have been extracted from a previous study. \cite{Loos_2020a}
The MP2, MP3, MP4, CC2, CC3, CC4, CCSD, CCSDT, and CCSDTQ calculations have been performed with Cfour, \cite{cfour} while the CCSD(T) and MP5 calculations have been performed in Gaussian 09. \cite{g09}
For all these calculations, we consider Dunning's correlation-consistent double-$\zeta$ basis (cc-pVDZ) which consists of Hilbert space sizes ranging from $10^{20}$ (for thiophene) to $10^{36}$ (for benzene).
We follow our usual procedure \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 second run is performed with localized orbitals.
This has the advantage to produce a smoother and faster convergence of the SCI energy toward the FCI limit by taking benefit of the local character of electron correlation.\cite{Angeli_2003,Angeli_2009,BenAmor_2011,Suaud_2017,Chien_2018,Eriksen_2020}
The Boys-Foster localization procedure \cite{Boys_1960} that we apply to the natural orbitals is performed in several orbital windows: i) core, ii) valence $\sigma$, iii) valence $\pi$, iv) valence $\pi^*$, v) valence $\sigma^*$, vi) the higher-lying $\sigma$ orbitals, and vii) the higher-lying $\pi$ orbitals. 
Like Pipek-Mezey, \cite{Pipek_1989} this choice of orbital windows allows to preserve a strict $\sigma$-$\pi$ separation in planar systems like benzene.

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.
We then linearly extrapolate the total SCI energy to $E_\text{(r)PT2} = 0$ (which effectively corresponds to the FCI limit). 
Note that, unlike excited-state calculations where it is important to enforce that the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, \cite{Applencourt_2018} the present wave functions do not fulfil this property as we aim for the lowest possible energy of a singlet state. 
We have found that $\expval*{\Hat{S}^2}$ is, nonetheless, very close to zero for each system.


\section{Results and discussion}

\begin{table*}
	\caption{Total energy $E$ (in \Eh) and correlation energy $\Ec$ (in \mEh) for the frozen-core ground state of five-membered rings in the cc-pVDZ basis set.
	\label{tab:Tab5-VDZ}}
	\begin{ruledtabular}
	\begin{tabular}{lcccccccccc}
				&	\mc{2}{c}{Cyclopentadiene}	&	\mc{2}{c}{Furan}	&	\mc{2}{c}{Imidazole}	&	\mc{2}{c}{Pyrrole}	&	\mc{2}{c}{Thiophene}	\\
					\cline{2-3}	\cline{4-5}	\cline{6-7} \cline{8-9} \cline{10-11}
		Method	&	$E$&	$\Ec$	&	$E$	&	$\Ec$	&	$E$	&	$\Ec$	&	$E$	&	$\Ec$	&	$E$	&	$\Ec$ 	\\
		\hline
		HF		&	$-192.8083$	&				&	$-228.6433$	&				&	$-224.8354$	&				&	$-208.8286$	&		&	-551.3210	&	\\
		\hline
		MP2		&	$-193.4717$	&	$-663.4$	&	$-229.3508$	&	$-707.5$	&	$-225.5558$ &	$-720.4$	&	$-209.5243$	&	$-695.7$	&	$-551.9825$	 &	$-661.5$	\\
		MP3		&	$-193.5094$	&	$-701.0$	&	$-229.3711$	&	$-727.8$	&	$-225.5732$	&	$-737.8$	&	$-209.5492$	&	$-720.6$	&	$-552.0104$	&	$-689.4$	\\
		MP4		&	$-193.5428$	&	$-734.5$	&	$-229.4099$	&	$-766.6$	&	$-225.6126$	&	$-777.2$	&	$-209.5851$	&	$-756.5$	&	$-552.0476$	&	$-726.6$	\\
		MP5		&	$-193.5418$	&	$-733.4$	&	$-229.4032$	&	$-759.9$	&	$-225.6061$	&	$-770.8$	&	$-209.5809$	&	$-752.3$	&	$-552.0426$	&	$-721.6$\\	
		\hline
		CC2		&	$-193.4782$	&	$-669.9$	&	$-229.3605$	&	$-717.2$	&	$-225.5644$	&	$-729.0$	&	$-209.5311$	&	$-702.5$	&	$-551.9905$	&	$-669.5$	\\
		CC3		&	$-193.5449$	&	$-736.6$	&	$-229.4090$	&	$-765.7$	&	$-225.6115$	&	$-776.1$	&	$-209.5849$	&	$-756.3$	&	$-552.0473$	&	$-726.3$	\\
		CC4		&	$-193.5467$	&	$-738.4$	&	$-229.4102$	&	$-766.9$	&	$-225.6126$	&	$-777.2$	&	$-209.5862$	&	$-757.6$	&	$-552.0487$	&	$-727.7$	\\
		\hline
		CCSD	&	$-193.5156$	&	$-707.2$	&	$-229.3783$	&	$-735.0$	&	$-225.5796$	&	$-744.2$	&	$-209.5543$	&	$-725.7$	&	$-552.0155$	&	$-694.5$	\\
		CCSDT	&	$-193.5446$	&	$-736.2$	&	$-229.4076$	&	$-764.3$	&	$-225.6099$	&	$-774.6$	&	$-209.5838$	&	$-755.2$	&	$-552.0461$	&	$-725.1$	\\
		CCSDTQ	&	$-193.5465$	&	$-738.2$	&	$-229.4100$	&	$-766.7$	&	$-225.6123$	&	$-776.9$	&	$-209.5860$	&	$-757.4$	&	$-552.0485$	&	$-727.5$	\\	
		\hline
		CCSD(T)	&	$-193.5439$	&	$-735.6$	&	$-229.4073$	&	$-764.0$	&	$-225.6099$	&	$-774.5$	&	$-209.5836$	&	$-754.9$	&	$-552.0458$	&	$-724.8$
\\
		\hline
		CIPSI	&					&				&					&				&					&				&					&				&			&	\\	
	\end{tabular}
	\end{ruledtabular}
\end{table*}

\begin{squeezetable}
\begin{table*}
	\caption{Total energy $E$ (in \Eh) and correlation energy $\Ec$ (in \mEh) for the frozen-core ground state of six-membered rings in the cc-pVDZ basis set.
	\label{tab:Tab6-VDZ}}
	\begin{ruledtabular}
	\begin{tabular}{lcccccccccccccc}
				&	\mc{2}{c}{Benzene}	&	\mc{2}{c}{Pyrazine}	&	\mc{2}{c}{Pyridazine}	&	\mc{2}{c}{Pyridine}	&	\mc{2}{c}{Pyrimidine}	&	\mc{2}{c}{Tetrazine}	&	\mc{2}{c}{Triazine}	\\
					\cline{2-3}	\cline{4-5}	\cline{6-7} \cline{8-9} \cline{10-11} \cline{12-13} \cline{14-15}
	Method		&	$E$	&	$\Ec$	&	$E$	&	$\Ec$	&	$E$	&	$\Ec$	&	$E$	&	$\Ec$	
				&	$E$	&	$\Ec$	&	$E$	&	$\Ec$	&	$E$	&	$\Ec$	\\
	\hline
		HF		&	$-230.7222$	&			&	$-262.7030$	&			&	$-262.6699$	&			&	$-246.7152$	&			&	$-262.7137$	&			&	$-294.6157$		&		&	$-278.7173$	\\
		\hline
		MP2		&	$-231.5046$  	&	$-782.3$	&	$-263.5376$ 	&	$-834.6$ 	&	$-263.5086$ &	$-838.7$	&	$-247.5227$	&	$-807.5$	&	$-263.5437$   &	$-830.1$	&	$-295.5117$    &   $-895.9$	&			$-279.5678$   &   $-850.5$\\
		MP3		&	$-231.5386$	&	$-816.4$	&	$-263.5567$	&	$-853.7$	&	$-263.5271$	&	$-857.3$	&	$-247.5492$	&	$-834.0$	&	$-263.5633$	&	$-849.6$	&	$-295.5152$   &	$-899.5$	&	$-279.5809$   &	$-863.6$	\\
		MP4		&	$-231.5808$	&	$-858.5$	&	$-263.6059$	&	$-902.9$	&	$-263.5778$	&	$-907.9$	&	$-247.5951$	&	$-879.9$	&	$-263.6129$	&	$-899.3$	&	$-295.5743$	&  $-958.6$	&	$-279.6340$    &	$-916.7$	\\
		MP5		&	$-231.5760$   &          $-853.8$	&	$-263.5968$	&	$-893.8$	&	$-263.5681$	&	$-898.3$	&	$-247.5881$	&	$-872.9$	&	$-263.6036$ &            $-890.0$	&	$-295.5600$	&	$-944.3$	&	$-279.6228$	&	$-905.4$	\\
		\hline
		CC2		&	$-231.5117$  	&	$-789.4$	&	$-263.5475$	&	$-844.5$	&	$-263.5188$	&	$-848.9$	&	$-247.5315$   &	$-816.3$	&	$-263.5550$ 	&	$-841.3$	&	$-295.5247$    &   $-909.0$		&	$-279.5817$       &	$-864.4$	\\
		CC3		&	$-231.5814$	&	$-859.1$	&	$-263.6045$	&	$-901.5$	&	$-263.5761$	&	$-906.2$	&	$-247.5948$	&	$-879.6$	&	$-263.6120$	&	$-898.4$	&	$-295.5706$  &	$-954.9$	&	$-279.6329$    &	$-915.6$	\\
		CC4		&	$-231.5828$	&	$-860.6$	&	$-263.6056$	&	$-902.6$	&	$-263.5773$	&	$-907.5$	&	$-247.5960$		&	$-880.8$		&	$-263.6129$	&	$-899.3$		&	$-295.5716$	&	$-955.9$	&	$-279.6334$	&	$-916.1$			\\
		\hline
		CCSD	&	$-231.5440$	&	$-821.8$	&	$-263.5640$	&	$-861.0$	&	$-263.5347$	&	$-864.9$	&	$-247.5559$	&	$-840.7$	&	$-263.5716$	&	$-858.0$	&	$-295.5248$   &	$-909.1$	&	$-279.5911$   &	 $-873.8$	\\
		CCSDT	&	$-231.5802$	&	$-857.9$	&	$-263.6024$	&	$-899.4$	&	$-263.5739$	&	$-904.0$	&	$-247.5931$	&	$-877.9$	&	$-263.6097$	&	$-896.1$	&	$-295.5673$  &  $-951.6$	&	$-279.6300$    &	$-912.7$	\\
		CCSDTQ	&	$-231.5826$	&	$-860.4$		&	$-263.6053$	&	$-902.3$		&	$-263.5770$	&	$-907.1$	&	$-247.5960$	&	$-880.8$	&	$-263.6126$  & $-899.0$		&			$-295.5712$		&	$-955.4$		&	$-279.6331$		&	$-915.8$		\\
		\hline
		CCSD(T)	&	$-231.5798$	&	$-857.5$	&	$-263.6024$		&	$-899.4$	&	$-263.5740$	&	$-904.1$	&	$-247.5929$	&	$-877.7$	&	$-263.6099$	&	$-896.2$	&	$-295.5680$		&	$-952.2$	&	$-279.6305$		&	$-913.1$	\\
		\hline
		CIPSI	&				&			&				&			&				&			&				&			&				&			\\
	\end{tabular}
	\end{ruledtabular}
\end{table*}
\end{squeezetable}    


\section{Conclusion}

\begin{acknowledgements}
This work was performed using HPC resources from GENCI-TGCC (2020-gen1738) and from CALMIP (Toulouse) under allocation 2020-18005.
PFL, AS, and MC have received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).
\end{acknowledgements}

\section*{Data availability statement}
The data that support the findings of this study are openly available in Zenodo at \href{http://doi.org/XX.XXXX/zenodo.XXXXXXX}{http://doi.org/XX.XXXX/zenodo.XXXXXXX}.

\bibliography{Ec}

\end{document}
