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Pierre-Francois Loos 2021-07-26 17:08:45 +02:00
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% Abstract
\begin{abstract}
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 five- and six-membered ring 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 $10^{28}$ (for thiophene) to $10^{36}$ (for benzene).
This corresponds to Hilbert spaces with sizes ranging from $10^{29}$ to $10^{36}$.
Our estimates are based on energetically optimized-orbital 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 i) 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 iii) the single-reference coupled-cluster series CCSD, CCSDT, and CCSDTQ.
@ -142,12 +142,12 @@ Its higher-order variants [MP3, \cite{Pople_1976}
MP4, \cite{Krishnan_1980} MP5, \cite{Kucharski_1989} and MP6 \cite{He_1996a,He_1996b} which scale as $\order*{\Norb^{6}}$, $\order*{\Norb^{7}}$, $\order*{\Norb^{8}}$, and $\order*{\Norb^{9}}$ respectively] have been investigated much more scarcely.
However, it is now widely recognised that the series of MP approximations might show erratic, slowly convergent, or divergent behavior that limits its applicability and systematic improvability. \cite{Laidig_1985,Knowles_1985,Handy_1985,Gill_1986,Laidig_1987,Nobes_1987,Gill_1988,Gill_1988a,Lepetit_1988,Malrieu_2003,Marie_2021}
Again, MP perturbation theory and CC methods can be coupled.
The CCSD(T) method, \cite{Raghavachari_1989} known as the gold-standard of quantum chemistry for weakly correlated systems, is probably the most iconic example of such coupling.
The CCSD(T) method, \cite{Raghavachari_1989} where one includes iteratively the single and double excitations and perturbatively (from MP4 and partially MP5) the triple excitations, known as the ``gold-standard'' of quantum chemistry for weakly correlated systems thanks to its excellent accuracy/cost ratio, is probably the most iconic example of such coupling.
Motivated by the recent blind test of Eriksen \textit{et al.}\cite{Eriksen_2020}~reporting the performance of a large panel of emerging electronic structure methods [the many-body expansion FCI (MBE-FCI), \cite{Eriksen_2017,Eriksen_2018,Eriksen_2019a,Eriksen_2019b} adaptive sampling CI (ASCI), \cite{Tubman_2016,Tubman_2018,Tubman_2020} iterative CI (iCI), \cite{Liu_2014,Liu_2016,Lei_2017,Zhang_2020} semistochastic heat-bath CI (SHCI), \cite{Holmes_2016,Holmes_2017,Sharma_2017} the full coupled-cluster reduction (FCCR), \cite{Xu_2018,Xu_2020} density-matrix renormalization group (DMRG), \cite{White_1992,White_1993,Chan_2011} adaptive-shift FCI quantum Monte Carlo (AS-FCIQMC), \cite{Booth_2009,Cleland_2010,Ghanem_2019} and cluster-analysis-driven FCIQMC (CAD-FCIQMC) \cite{Deustua_2017,Deustua_2018}] on the non-relativistic frozen-core correlation energy of the benzene molecule in the standard correlation-consistent double-$\zeta$ Dunning basis set (cc-pVDZ), some of us have recently investigated the performance of the SCI method known as \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI). \cite{Huron_1973,Giner_2013,Giner_2015,Garniron_2018,Garniron_2019} on the very same system \cite{Loos_2020e} [see also Ref.~\onlinecite{Lee_2020} for a study of the performance of phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) \cite{Motta_2018}].
In the continuity of this recent work, we report here a significant extension by estimating the (frozen-core) FCI/cc-pVDZ correlation energy of twelve cyclic molecules (cyclopentadiene, furan, imidazole, pyrrole, thiophene, benzene, pyrazine, pyridazine, pyridine, pyrimidine, tetrazine, and triazine) with the help of CIPSI employing energetically-optimized orbitals at the same level of theory. \cite{Yao_2020,Yao_2021}
These systems are depicted in Fig.~\ref{fig:mol}.
This set of molecular systems corresponds to Hilbert spaces with sizes ranging from $10^{28}$ (for thiophene) to $10^{36}$ (for benzene).
This set of molecular systems corresponds to Hilbert spaces with sizes ranging from $10^{29}$ to $10^{36}$.
In addition to CIPSI, the performance and convergence properties of several series of methods are investigated.
In particular, we study i) the MP perturbation series up to fifth-order (MP2, MP3, MP4, and MP5), ii) the CC2, CC3, and CC4 approximate series, and ii) the ``complete'' CC series up to quadruples (\ie, CCSD, CCSDT, and CCSDTQ).
The performance of the ground-state gold standard CCSD(T) is also investigated.
@ -712,6 +712,8 @@ Our results have shown that, with a $\order*{N^7}$ scaling, MP4 provides an inte
We have evidenced that CC3 (where the triples are computed iteratively) also outperforms the perturbative-triples CCSD(T) method with the same $\order*{N^7}$ scaling but also its more expensive parent, CCSDT.
A similar trend is observed for the methods including quadruple excitations, where the $\order*{N^9}$ CC4 model has been shown to be more accurate than CCSDTQ [which scales as $\order*{N^{10}}$].
\titou{more variety in systems would be good.}
As perspectives, we are currently investigating the performance of the present approach for excited states in order to expand the QUEST database of vertical excitation energies. \cite{Veril_2021}
We hope to report on this in the near future.
The compression of the variational space brought by optimized orbitals could be also beneficial in the context of quantum Monte Carlo methods to generate compact, yet accurate multi-determinant trial wave functions. \cite{Dash_2018,Dash_2019,Scemama_2020,Dash_2021}