loos/Ec
2
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

saving work in manuscript

Browse Source
This commit is contained in:
Pierre-Francois Loos 2021-06-18 11:05:01 +02:00
parent 45d6c5e9a1
commit 9656bd8466
3 changed files with 794 additions and 467 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

935
Ec.nb
View File

File diff suppressed because it is too large Load Diff

275
Manuscript/Ec.bib
View File

@ -1,13 +1,264 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2021-05-10 09:24:00 +0200
%% Created for Pierre-Francois Loos at 2021-06-18 11:02:22 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Yao_2021,
author = {Yao, Yuan and Umrigar, C. J.},
date-added = {2021-06-18 11:02:08 +0200},
date-modified = {2021-06-18 11:02:21 +0200},
doi = {10.1021/acs.jctc.1c00385},
journal = {J. Chem. Theory Comput.},
number = {0},
pages = {null},
title = {Orbital Optimization in Selected Configuration Interaction Methods},
volume = {0},
year = {0},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.1c00385}}
@article{Yao_2020,
author = {Yao,Yuan and Giner,Emmanuel and Li,Junhao and Toulouse,Julien and Umrigar,C. J.},
date-added = {2021-06-18 11:01:38 +0200},
date-modified = {2021-06-18 11:01:52 +0200},
doi = {10.1063/5.0018577},
journal = {J. Chem. Phys.},
number = {12},
pages = {124117},
title = {Almost exact energies for the Gaussian-2 set with the semistochastic heat-bath configuration interaction method},
volume = {153},
year = {2020},
Bdsk-Url-1 = {https://doi.org/10.1063/5.0018577}}
@article{Hohenberg_1964,
annote = {Hohenberg-Kohn theorem},
author = {P. Hohenberg and W. Kohn},
date-added = {2021-06-18 10:04:31 +0200},
date-modified = {2021-06-18 10:04:31 +0200},
doi = {10.1103/PhysRev.136.B864},
journal = {Phys. Rev.},
pages = {B864--B871},
title = {Inhomogeneous electron gas},
volume = {136},
year = {1964},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRev.136.B864}}
@article{Kohn_1965,
author = {W. Kohn and L. J. Sham},
date-added = {2021-06-18 10:04:20 +0200},
date-modified = {2021-06-18 10:04:20 +0200},
doi = {10.1103/PhysRev.140.A1133},
journal = {Phys. Rev.},
pages = {A1133--A1138},
title = {Self-consistent equations including exchange and correlation effects},
volume = {140},
year = {1965},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRev.140.A1133}}
@article{Grimme_2006,
author = {Grimme,Stefan},
date-added = {2021-06-18 10:02:28 +0200},
date-modified = {2021-06-18 10:02:42 +0200},
doi = {10.1063/1.2148954},
journal = {J. Chem. Phys.},
number = {3},
pages = {034108},
title = {Semiempirical hybrid density functional with perturbative second-order correlation},
volume = {124},
year = {2006},
Bdsk-Url-1 = {https://doi.org/10.1063/1.2148954}}
@article{Malrieu_2003,
author = {Jean-Paul Malrieu and Celestino Angeli},
date-added = {2021-06-18 09:15:41 +0200},
date-modified = {2021-06-18 09:15:41 +0200},
doi = {10.1080/00268976.2013.788745},
journal = {Mol. Phys.},
number = {9-11},
pages = {1092-1099},
title = {The M{\o}ller--Plesset perturbation revisited: origin of high-order divergences},
volume = {111},
year = {2013},
Bdsk-Url-1 = {https://doi.org/10.1080/00268976.2013.788745}}
@article{Lepetit_1988,
author = {Lepetit, M. B. and P{\'e}lissier, M. and Malrieu, J. P.},
date-added = {2021-06-18 09:15:35 +0200},
date-modified = {2021-06-18 09:15:35 +0200},
doi = {10.1063/1.455170},
journal = {J. Chem. Phys.},
number = {2},
pages = {998--1008},
title = {Origins of the poor convergence of many‐body perturbation theory expansions from unrestricted {Hartree-Fock} zeroth‐order descriptions},
volume = {89},
year = {1988},
Bdsk-Url-1 = {https://doi.org/10.1063/1.455170}}
@article{Gill_1988,
author = {Gill, Peter M. W. and Pople, John A. and Radom, Leo and Nobes, Ross H.},
date = {1988-12-15},
date-added = {2021-06-18 09:15:30 +0200},
date-modified = {2021-06-18 09:15:30 +0200},
doi = {10.1063/1.455312},
journal = {J. Chem. Phys.},
number = {12},
pages = {7307--7314},
title = {Why does unrestricted {M{\o}ller-Plesset} perturbation theory converge so slowly for spin‐contaminated wave functions?},
volume = {89},
year = {1988},
Bdsk-Url-1 = {https://doi.org/10.1063/1.455312}}
@article{Gill_1988a,
author = {P. M. W. Gill and M. W. Wong and R. H. Nobes and L. Radom},
date-added = {2021-06-18 09:15:30 +0200},
date-modified = {2021-06-18 09:15:30 +0200},
doi = {10.1016/0009-2614(88)80328-2},
journal = {Chem. Phys. Lett.},
pages = {541},
title = {How well can {RMP4} theory treat homolytic fragmentations?},
volume = {148},
year = {1988},
Bdsk-Url-1 = {https://doi.org/10.1016/0009-2614(88)80328-2}}
@article{Nobes_1987,
author = {R. H. Nobes and J. A. Pople and L. Radom and N. C. Handy and P. J. Knowles},
date-added = {2021-06-18 09:15:22 +0200},
date-modified = {2021-06-18 09:15:22 +0200},
doi = {10.1016/0009-2614(87)80545-6},
journal = {Chem. Phys. Lett.},
pages = {481},
title = {Slow convergence of the {M\oller--Plesset} perturbation series: the dissociation energy of hydrogen cyanide and the electron affinity of the cyano radical},
volume = {138},
year = {1987},
Bdsk-Url-1 = {https://doi.org/10.1016/0009-2614(87)80545-6}}
@article{Laidig_1987,
author = {William D. Laidig and Paul Saxe and Rodney J. Bartlett},
date-added = {2021-06-18 09:15:17 +0200},
date-modified = {2021-06-18 09:15:17 +0200},
doi = {10.1063/1.452291},
journal = {J. Chem. Phys.},
pages = {887},
title = {The description of \ce{N2} and \ce{F2} potential energy surfaces using multireference coupled cluster theory},
volume = {86},
year = {1987},
Bdsk-Url-1 = {https://doi.org/10.1063/1.452291}}
@article{Gill_1986,
author = {Gill, Peter M. W. and Radom, Leo},
date = {1986-11-28},
date-added = {2021-06-18 09:15:10 +0200},
date-modified = {2021-06-18 09:15:10 +0200},
doi = {10.1016/0009-2614(86)80686-8},
journal = {Chem. Phys. Lett.},
number = {1},
pages = {16--22},
title = {Deceptive convergence in {M{\o}ller-Plesset} perturbation energies},
volume = {132},
year = {1986},
Bdsk-Url-1 = {https://doi.org/10.1016/0009-2614(86)80686-8}}
@article{Handy_1985,
author = {Handy, N. C. and Knowles, P. J. and Somasundram, K.},
date-added = {2021-06-18 09:14:55 +0200},
date-modified = {2021-06-18 09:14:55 +0200},
doi = {10.1007/BF00698753},
journal = {Theoret. Chim. Acta},
number = {1},
pages = {87--100},
title = {On the convergence of the {M{\o}ller-Plesset} perturbation series},
volume = {68},
year = {1985},
Bdsk-Url-1 = {https://doi.org/10.1007/BF00698753}}
@article{Knowles_1985,
author = {P. J. Knowles and K. Somasundram and N. C. Handy and K. Hirao},
date-added = {2021-06-18 09:14:50 +0200},
date-modified = {2021-06-18 09:14:50 +0200},
doi = {10.1016/0009-2614(85)85002-8},
journal = {Chem. Phys. Lett.},
pages = {8},
title = {The Calculation of High-Order Energies in the Many-Body Perturbation Theory Series},
volume = {113},
year = {1985},
Bdsk-Url-1 = {https://doi.org/10.1016/0009-2614(85)85002-8}}
@article{Laidig_1985,
author = {William D. Laidig and George Fitzgerald and Rodney J. Bartlett},
date-added = {2021-06-18 09:14:44 +0200},
date-modified = {2021-06-18 09:14:44 +0200},
doi = {10.1016/0009-2614(85)80934-9},
journal = {Chem. Phys. Lett.},
pages = {151},
title = {Is Fifth-Order MBPT Enough?},
volume = {113},
year = {1985},
Bdsk-Url-1 = {https://doi.org/10.1016/0009-2614(85)80934-9}}
@article{Marie_2021,
author = {Antoine Marie and Hugh G. A. Burton and Pierre-Fran{\c c}ois Loos},
date-added = {2021-06-18 08:52:50 +0200},
date-modified = {2021-06-18 08:53:45 +0200},
doi = {10.1088/1361-648X/abe795},
journal = {J. Phys.: Condens. Matter},
pages = {283001},
title = {Perturbation Theory in the Complex Plane: Exceptional Points and Where to Find Them},
volume = {33},
year = {2021},
Bdsk-Url-1 = {https://doi.org/10.1088/1361-648X/abe795}}
@article{Aroeira_2021,
author = {Aroeira, Gustavo J. R. and Davis, Madeline M. and Turney, Justin M. and Schaefer, Henry F.},
date-added = {2021-06-18 05:49:33 +0200},
date-modified = {2021-06-18 05:49:49 +0200},
doi = {10.1021/acs.jctc.0c00888},
journal = {J. Chem. Theory Comput.},
number = {1},
pages = {182-190},
title = {Coupled Cluster Externally Corrected by Adaptive Configuration Interaction},
volume = {17},
year = {2021},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.0c00888}}
@article{Paldus_2017,
author = {J. Paldus},
date-added = {2021-06-18 05:44:17 +0200},
date-modified = {2021-06-18 05:45:05 +0200},
doi = {10.1007/s10910-016-0688-6},
journal = {J. Math. Chem},
pages = {477--502},
title = {Externally and internally corrected coupled cluster approaches: an overview},
volume = {55},
year = {2017},
Bdsk-Url-1 = {https://doi.org/10.1007/s10910-016-0688-6}}
@misc{Magoulas_2021,
archiveprefix = {arXiv},
author = {Ilias Magoulas and Karthik Gururangan and Piotr Piecuch and J. Emiliano Deustua and Jun Shen},
date-added = {2021-06-18 05:40:59 +0200},
date-modified = {2021-06-18 05:41:08 +0200},
eprint = {2102.10143},
primaryclass = {physics.chem-ph},
title = {Is Externally Corrected Coupled Cluster Always Better than the Underlying Truncated Configuration Interaction?},
year = {2021}}
@article{Lee_2021,
author = {Lee, Seunghoon and Zhai, Huanchen and Sharma, Sandeep and Umrigar, C. J. and Chan, Garnet Kin-Lic},
date-added = {2021-06-18 05:39:07 +0200},
date-modified = {2021-06-18 05:39:21 +0200},
doi = {10.1021/acs.jctc.1c00205},
journal = {J. Chem. Theory Comput.},
number = {6},
pages = {3414-3425},
title = {Externally Corrected CCSD with Renormalized Perturbative Triples (R-ecCCSD(T)) and the Density Matrix Renormalization Group and Selected Configuration Interaction External Sources},
volume = {17},
year = {2021},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.1c00205}}
@misc{g09,
author = {M. J. Frisch and G. W. Trucks and H. B. Schlegel and G. E. Scuseria and M. A. Robb and J. R. Cheeseman and G. Scalmani and V. Barone and B. Mennucci and G. A. Petersson and H. Nakatsuji and M. Caricato and X. Li and H. P. Hratchian and A. F. Izmaylov and J. Bloino and G. Zheng and J. L. Sonnenberg and M. Hada and M. Ehara and K. Toyota and R. Fukuda and J. Hasegawa and M. Ishida and T. Nakajima and Y. Honda and O. Kitao and H. Nakai and T. Vreven and Montgomery, {Jr.}, J. A. and J. E. Peralta and F. Ogliaro and M. Bearpark and J. J. Heyd and E. Brothers and K. N. Kudin and V. N. Staroverov and R. Kobayashi and J. Normand and K. Raghavachari and A. Rendell and J. C. Burant and S. S. Iyengar and J. Tomasi and M. Cossi and N. Rega and J. M. Millam and M. Klene and J. E. Knox and J. B. Cross and V. Bakken and C. Adamo and J. Jaramillo and R. Gomperts and R. E. Stratmann and O. Yazyev and A. J. Austin and R. Cammi and C. Pomelli and J. W. Ochterski and R. L. Martin and K. Morokuma and V. G. Zakrzewski and G. A. Voth and P. Salvador and J. J. Dannenberg and S. Dapprich and A. D. Daniels and {\"O}. Farkas and J. B. Foresman and J. V. Ortiz and J. Cioslowski and D. J. Fox},
date-added = {2021-05-10 08:40:20 +0200},
@ -984,18 +1235,6 @@
year = {1982},
Bdsk-Url-1 = {https://doi.org/10.1063/1.443164}}
@article{qp2,
author = {Y. Garniron and K. Gasperich and T. Applencourt and A. Benali and A. Fert{\'e} and J. Paquier and B. Pradines and R. Assaraf and P. Reinhardt and J. Toulouse and P. Barbaresco and N. Renon and G. David and J. P. Malrieu and M. V{\'e}ril and M. Caffarel and P. F. Loos and E. Giner and A. Scemama},
date-added = {2021-05-06 15:31:25 +0200},
date-modified = {2021-05-06 15:31:25 +0200},
doi = {10.1021/acs.jctc.9b00176},
journal = {J. Chem. Theory Comput.},
pages = {3591},
title = {Quantum Package 2.0: A Open-Source Determinant-Driven Suite Of Programs},
volume = {15},
year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.9b00176}}
@article{Rico_1993,
abstract = {Several single-reference excited-state methods based on single and double substitutions are considered. Quadratic configuration interaction (QCISD) and coupled-cluster theory (CCSD) are obtained in a time-dependent linear response framework, together with the CISD method. The QCISD and CCSD transition energies are size consistent, and exact for two-electron systems. The relation between the QCISD and CCSD excited-state theories and ground-state gradient expressions is developed and employed. Calculations are reported for singlet and triplet excited states of some small molecules. CCSD and QCISD are qualitatively superior to CISD. Overall, CCSD exhibits noticeably better accuracy than QCISD, and the differences are sometimes much larger than for ground-state problems. A possible explanation is suggested.},
author = {Rudolph J. Rico and Martin Head-Gordon},
@ -1210,16 +1449,6 @@
year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.121.113001}}
@misc{Yao_2020,
archiveprefix = {arXiv},
author = {Yuan Yao and Emmanuel Giner and Junhao Li and Julien Toulouse and C. J. Umrigar},
date-added = {2021-05-06 15:31:25 +0200},
date-modified = {2021-05-06 15:31:25 +0200},
eprint = {2004.10059},
primaryclass = {physics.chem-ph},
title = {Almost exact energies for the Gaussian-2 set with the semistochastic heat-bath configuration interaction method},
year = {2020}}
@article{Zimmerman_2017,
author = {Zimmerman, Paul M.},
date-added = {2021-05-06 15:31:25 +0200},

51
Manuscript/Ec.tex
View File

@ -36,16 +36,16 @@
\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}
\author{Micka\"el V\'eril}
\affiliation{\LCPQ}
\author{Michel Caffarel}
\affiliation{\LCPQ}
\author{Denis Jacquemin}
\affiliation{\CEISAM}
\author{Anthony Scemama}
\affiliation{\LCPQ}
\author{Pierre-Fran\c{c}ois Loos}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
@ -57,7 +57,7 @@ In the continuity of our recent work on the benzene molecule [\href{https://doi.
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.
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 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}
@ -74,35 +74,42 @@ The nuclei coordinates can then be treated as parameters in the electronic Hamil
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).
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,Piecuch_2002,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.
Coupled-cluster methods have been particularly successful at computing accurately various properties for small- and medium-sized molecules.
\cite{Kallay_2003,Kallay_2004a,Gauss_2006,Kallay_2006,Gauss_2009}
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.
However, a different strategy has recently made a come back in the context of CI methods. \cite{Bender_1969,Whitten_1969,Huron_1973}
Indeed, selected CI (SCI) methods, \cite{Booth_2009,Giner_2013,Evangelista_2014,Giner_2015,Holmes_2016,Tubman_2016,Liu_2016,Ohtsuka_2017,Zimmerman_2017,Coe_2018,Garniron_2018} where one iteratively selects the energetically relevant determinants from the FCI space (usually) based on a perturbative criterion, has been recently shown to be highly successful to produce reference energies for ground and excited states in small- and medium-size molecules \cite{Holmes_2017,Li_2018,Li_2020,Loos_2018a,Chien_2018,Loos_2019,Loos_2020b,Loos_2020c,Loos_2020e,Garniron_2019,Eriksen_2020,Yao_2020,Veril_2021} thanks to efficient deterministic, stochastic or hybrid algorithms well suited for massive parallelization.
We refer the interested reader to Refs.~\onlinecite{Loos_2020a,Eriksen_2021} for recent reviews.
SCI methods are based on a simple and natural observation: amongst the very large number of determinants belonging to the FCI space, only a relative small fraction of them significantly contributes to the energy.
Accordingly to this well-known fact, the SCI+PT2 family of methods performs a sparse exploration of the FCI space by selecting iteratively only the most energetically relevant determinants of the variational space and supplementing it with a second-order perturbative correction (PT2). \cite{Huron_1973,Garniron_2017,Sharma_2017,Garniron_2018,Garniron_2019}
Although the formal scaling of such algorithms remain exponential, the prefactor is greatly reduced which explains their current attractiveness in the electronic structure community and much wider applicability than their standard FCI parent.
Note that, very recently, several groups \cite{Aroeira_2021,Lee_2021,Magoulas_2021} have coupled CC and SCI methods via the externally-corrected CC methodology, \cite{Paldus_2017} showing promising performances for weakly and strongly correlated systems.
A rather different strategy in order to reach the holy grail FCI limit is to resort to M{\o}ller-Plesset (MP) perturbation theory, \cite{Moller_1934}
which popularity originates from its black-box nature, size-extensivity, and relatively low computational scaling, making it easily applied in a broad range of molecular systems.
Again, at least in theory, one can obtain the exact energy of the system by ramping up the degree of the perturbative series. \cite{Marie_2021}
The second-order M{\o}ller-Plesset (MP2) method \cite{Moller_1934} [which scales as $\order*{N^{5}}$] has been broadly adopted in quantum chemistry for several decades, and is now included in double-hybrid functionals \cite{Grimme_2006} alongside exact Hartree-Fock exchange within density-functional theory. \cite{Hohenberg_1964,Kohn_1965}
Its higher-order variants [MP3, \cite{Pople_1976}
MP4, \cite{Krishnan_1980} MP5, \cite{Kucharski_1989} and MP6 \cite{He_1996a,He_1996b} which scales as $\order*{N^{6}}$, $\order*{N^{7}}$, and $\order*{N^{8}}$, respectively] have been investigated much more scarcely.
However, it is now widely recognised that the series of MP approximations might show erratic, slow, or divergent behavior that limit 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}
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 successful example of such coupling.
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}
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} the density-matrix renormalization group (DMRG) approach, \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 \textit{configuration interaction using a perturbative selection made iteratively} (CIPSI) method \cite{Huron_1973,Giner_2013,Giner_2015,Garniron_2018,Garniron_2019} on the very same system [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 the CIPSI method employing energetically-optimized orbitals at the same level of theory. \cite{Yao_2020,Yao_2021}
This set of molecular systems corresponds to Hilbert spaces with sizes ranging from $10^{20}$ (for thiophene) to $10^{36}$ (for benzene).
In addition to CIPSI, the performance and convergence properties of several series of methods are investigated.
In particular, we study the convergence properties of 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.
%%% FIG 1 %%%
\begin{figure*}