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\documentclass[aps,prb,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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\newcommand{\Ec}{E_\text{c}}
\newcommand{\mEh}{$mE_h$}
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\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{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}
\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*}
\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 & $-193.5465$ & $-738.2$ & $-229.4100$ & $-766.7$ & & & $-209.5860$ & $-757.4$ & $-552.0485$ & $-727.5$ \\
\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 & & & $-263.6053$ & $-902.3$ & & & & & & & $-295.5712$ & $-955.4$ & & \\
\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 and AS 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}
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