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%% This BibTeX bibliography file was created using BibDesk.
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@article{Werner_2020,
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author = {Werner,Hans-Joachim and Knowles,Peter J. and Manby,Frederick R. and Black,Joshua A. and Doll,Klaus and He{\ss}elmann,Andreas and Kats,Daniel and K{\"o}hn,Andreas and Korona,Tatiana and Kreplin,David A. and Ma,Qianli and Miller,Thomas F. and Mitrushchenkov,Alexander and Peterson,Kirk A. and Polyak,Iakov and Rauhut,Guntram and Sibaev,Marat},
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date-added = {2022-03-23 22:21:43 +0100},
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date-modified = {2022-03-23 22:21:43 +0100},
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doi = {10.1063/5.0005081},
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journal = {J. Chem. Phys.},
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number = {14},
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pages = {144107},
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title = {The Molpro quantum chemistry package},
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volume = {152},
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year = {2020},
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bdsk-url-1 = {https://doi.org/10.1063/5.0005081}}
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@article{Damour_2021,
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author = {Damour,Yann and V{\'e}ril,Micka{\"e}l and Kossoski,F{\'a}bris and Caffarel,Michel and Jacquemin,Denis and Scemama,Anthony and Loos,Pierre-Fran{\c c}ois},
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date-added = {2022-03-23 22:00:31 +0100},
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date-modified = {2022-03-23 22:00:31 +0100},
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doi = {10.1063/5.0065314},
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journal = {J. Chem. Phys.},
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number = {13},
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pages = {134104},
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title = {Accurate full configuration interaction correlation energy estimates for five- and six-membered rings},
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volume = {155},
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year = {2021},
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bdsk-url-1 = {https://doi.org/10.1063/5.0065314}}
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@article{Kohn_2013,
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abstract = {Abstract The multireference problem is considered one of the great challenges in coupled-cluster (CC) theory. Most recent developments are based on state-specific approaches, which focus on a single state and avoid some of the numerical problems of more general approaches. We review various state-of-the-art methods, including Mukherjee's state-specific multireference coupled-cluster (Mk-MRCC) theory, multireference Brillouin--Wigner coupled-cluster (MR-BWCC) theory, the MRexpT method, and internally contracted multireference coupled-cluster (ic-MRCC) theory. Related methods such as extended single-reference schemes [e.g., the complete active space coupled-cluster (CASCC) theory] and canonical transformation (CT) theory are covered as well. The comparison is done on the basis of formal arguments, implementation issues, and numerical results. Although a final and generally accepted multireference CC theory is still lacking, it is emphasized that recent developments render the new MRCC schemes useful tools for solving chemical problems. {\copyright} 2012 John Wiley \& Sons, Ltd. This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods},
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author = {K{\"o}hn, Andreas and Hanauer, Matthias and M{\"u}ck, Leonie Anna and Jagau, Thomas-Christian and Gauss, J{\"u}rgen},
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date-added = {2022-03-23 21:54:39 +0100},
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date-modified = {2022-03-23 21:54:39 +0100},
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doi = {https://doi.org/10.1002/wcms.1120},
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journal = {WIREs Comput. Mol. Sci.},
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number = {2},
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pages = {176-197},
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title = {State-specific multireference coupled-cluster theory},
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volume = {3},
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year = {2013},
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bdsk-url-1 = {https://onlinelibrary.wiley.com/doi/abs/10.1002/wcms.1120},
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bdsk-url-2 = {https://doi.org/10.1002/wcms.1120}}
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@article{Lyakh_2012,
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author = {Lyakh, Dmitry I. and Musia{\l}, Monika and Lotrich, Victor F. and Bartlett, Rodney J.},
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date-added = {2022-03-23 21:54:33 +0100},
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date-modified = {2022-03-23 21:54:33 +0100},
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doi = {10.1021/cr2001417},
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file = {/home/antoinem/Zotero/storage/SPCFP9WN/Lyakh et al. - 2012 - Multireference Nature of Chemistry The Coupled-Cl.pdf;/home/antoinem/Zotero/storage/FILL3QGA/cr2001417.html},
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journal = {Chem. Rev.},
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pages = {182--243},
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publisher = {{American Chemical Society}},
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title = {Multireference {{Nature}} of {{Chemistry}}: {{The Coupled}}-{{Cluster View}}},
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volume = {112},
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year = {2012},
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bdsk-url-1 = {https://doi.org/10.1021/cr2001417}}
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@article{Mahapatra_1998,
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author = {U. S. Mahapatra and B. Datta and D. Mukherjee},
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date-added = {2022-03-23 21:54:27 +0100},
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date-modified = {2022-03-23 21:54:27 +0100},
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doi = {10.1080/002689798168448},
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journal = {Mol. Phys.},
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number = {1},
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pages = {157-171},
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title = {A state-specific multi-reference coupled cluster formalism with molecular applications},
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volume = {94},
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year = {1998},
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bdsk-url-1 = {https://doi.org/10.1080/002689798168448}}
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@article{Mahapatra_1999,
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author = {Mahapatra,Uttam Sinha and Datta,Barnali and Mukherjee,Debashis},
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date-added = {2022-03-23 21:54:27 +0100},
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date-modified = {2022-03-23 21:54:27 +0100},
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doi = {10.1063/1.478523},
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journal = {J. Chem. Phys.},
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number = {13},
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pages = {6171-6188},
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title = {A size-consistent state-specific multireference coupled cluster theory: Formal developments and molecular applications},
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volume = {110},
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year = {1999},
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bdsk-url-1 = {https://doi.org/10.1063/1.478523}}
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@article{Mahapatra_2008,
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author = {Mahapatra, Uttam Sinha and Chattopadhyay, Sudip and Chaudhuri, Rajat K.},
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date-added = {2022-03-23 21:54:27 +0100},
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date-modified = {2022-03-23 21:54:27 +0100},
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doi = {10.1063/1.2952666},
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issn = {0021-9606, 1089-7690},
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journal = {J. Chem. Phys.},
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month = jul,
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number = {2},
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pages = {024108},
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title = {Molecular Applications of State-Specific Multireference Perturbation Theory to {{HF}}, {{H2O}}, {{H2S}}, {{C2}}, and {{N2}} Molecules},
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volume = {129},
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year = {2008},
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bdsk-url-1 = {https://doi.org/10.1063/1.2952666}}
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@article{Jeziorski_1981,
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author = {Jeziorski, Bogumil and Monkhorst, Hendrik J.},
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date-added = {2022-03-23 21:54:18 +0100},
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date-modified = {2022-03-23 21:54:18 +0100},
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doi = {10.1103/PhysRevA.24.1668},
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journal = {Phys. Rev. A},
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pages = {1668--1681},
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title = {Coupled-cluster method for multideterminantal reference states},
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volume = {24},
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year = {1981},
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bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevA.24.1668},
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bdsk-url-2 = {https://doi.org/10.1103/PhysRevA.24.1668}}
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@book{Roos_1995a,
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abstract = {The complete active space (CAS) SCF method in conjunction with multiconfigurational second-order perturbation theory (CASPT2) has been used to study the electronic spectra of a large number of molecules. The wave functions and the transition properties are computed at the CASSCF level, while dynamic correlation contributions to the excitation energies are obtained through the perturbation treatment. The methods yield energies, which are accurate to at least 0.2 eV, except in a few cases, where the CASSCF reference function does not characterize the electronic state with sufficient accuracy. The applications comprise: the polyenes from ethene to octatetraene (cis- and trans-forms); a number of cyclic pentadienes; norbornadiene; benzene, phenol, phosphabenzene, and the azabenzenes; free base porphin; and the nucleic acid base monomers cytosine, uracil, thymine, and guanine. Finally, the photochemistry of the molecules aminobenzonitrile (ABN) and dimethylaminobenzonitrile (DMABN) has also been studied, in particular the double fluorescence that occurs in DMABN. Taken together these studies comprise large amounts of new spectroscopic data of high accuracy, which either confirm existing assignments of experimental data or lead to new predictions and qualitative as well as quantitative understanding of a large number of electronic spectra. Most studies are restricted to ground state geometries (vertical energies), but in a few cases (octatetraene, ABN, and DMABN) also excited state geometries have been determined, thus yielding 0-0 transition energies and emission spectroscopic data.},
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address = {Dordrecht},
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@ -233,8 +233,8 @@
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\begin{abstract}
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The cyclobutadiene molecule is a well-known playground for theoretical chemists and is particularly suitable to test ground- and excited-state methods.
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Indeed, due to its high spatial symmetry, especially at the $D_{4h}$ square geometry but also at the $D_{2h}$ rectangular structure, the ground and excited states of cyclobutadiene exhibit multi-configurational characters and single-reference methods, such as adiabatic time-dependent density functional theory (TD-DFT) or equation-of-motion coupled cluster (EOM-CC), are notoriously known to struggle in such situations.
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In this work, using a large panel of methods and basis sets, we provide an extensive computational study of the autoisomerization barrier (defined as the difference between the square and rectangular ground-state energies) and the vertical excitation energies of the $D_{2h}$ and $D_{4h}$ structures.
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Indeed, due to its high spatial symmetry, especially at the $D_{4h}$ square geometry but also at the $D_{2h}$ rectangular structure, the ground and excited states of cyclobutadiene exhibit multi-configurational characters and single-reference methods, such as adiabatic time-dependent density-functional theory (TD-DFT) or equation-of-motion coupled cluster (EOM-CC), are notoriously known to struggle in such situations.
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In this work, using a large panel of methods and basis sets, we provide an extensive computational study of the autoisomerization barrier (defined as the difference between the square and rectangular ground-state energies) and the vertical excitation energies of the $D_{2h}$ and $D_{4h}$ arrangements.
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In particular, selected configuration interaction (SCI), multi-reference perturbation theory (CASSCF, CASPT2, and NEVPT2), and coupled cluster (CCSD, CC3, CCSDT, CC4, and CCSDTQ) calculations are performed.
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The spin-flip formalism, which is known to provide a correct description of states with multi-configurational character, is also tested within TD-DFT (where numerous exchange-correlation functionals are considered) and the algebraic diagrammatic construction [ADC(2)-s, ADC(2)-x, and ADC(3)].
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A theoretical best estimate is defined for the autoisomerization barrier and for each vertical transition energy.
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@ -247,7 +247,7 @@ A theoretical best estimate is defined for the autoisomerization barrier and for
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\label{sec:intro}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Despite the fact that excited states are involved in ubiquitous processes such as photochemistry, \cite{Bernardi_1990,Bernardi_1996,Boggio-Pasqua_2007,Klessinger_1995,Olivucci_2010,Robb_2007,VanderLugt_1969} catalysis or in solar cell technology, \cite{Delgado_2010} none of the currently existing methods has been shown to provide accurate excitation energies in all scenarios due to the complexity of the process, the size of the systems, environment effects and many other possible factors.
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Despite the fact that excited states are involved in ubiquitous processes such as photochemistry, \cite{Bernardi_1990,Bernardi_1996,Boggio-Pasqua_2007,Klessinger_1995,Olivucci_2010,Robb_2007,VanderLugt_1969} catalysis or in solar cell technology, \cite{Delgado_2010} none of the currently existing methods has shown to provide accurate excitation energies in all scenarios due to the complexity of the process, the size of the systems, environment effects and many other possible factors.
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Indeed, each computational model has its own theoretical and/or technical limitations and the number of possible chemical scenario is so vast that the design of new excited-state methodologies is still a very active field of theoretical quantum chemistry.\cite{Roos_1996,Piecuch_2002,Dreuw_2005,Krylov_2006,Sneskov_2012,Gonzales_2012,Laurent_2013,Adamo_2013,Dreuw_2015,Ghosh_2018,Blase_2020,Loos_2020a,Hait_2021,Zobel_2021}
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Speaking of difficult tasks, the cyclobutadiene (CBD) molecule has been a real challenge for experimental and theoretical chemistry for many decades. \cite{Bally_1980} Due to its antiaromaticity \cite{Minkin_1994} and large angular strain, \cite{Baeyer_1885} CBD presents a high reactivity which made its synthesis a particularly difficult exercise.
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@ -267,24 +267,26 @@ The energy of this barrier is estimated, experimentally, in the range of 1.6-10
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The lowest-energy excited states of CBD in both geometries are represented in Fig.~\ref{fig:CBD}, where
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we have reported the {\oneAg} and {\tBoneg} states for the rectangular geometry and {\sBoneg} and {\Atwog} for the square one.
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Due to the energy scale, the higher-energy states ({\sBoneg} and {\twoAg} for {\Dtwo} and {\Aoneg} and {\Btwog} for {\Dfour}) are not shown.
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Interestingly, the {\twoAg} and {\Aoneg} states have a strong contribution from doubly-excited configurations and these so-called double excitations \cite{Loos_2019} are known to be a challenge for adiabatic time-dependent density-functional theory (TD-DFT) \cite{Runge_1984,Casida_1995,Tozer_2000,Maitra_2004,Cave_2004,Levine_2006,Elliott_2011,Maitra_2012,Maitra_2017} and even for state-of-the-art methods like the approximate third-order coupled-cluster (CC3) \cite{Christiansen_1995,Koch_1997} or equation-of-motion coupled-cluster with singles, doubles and triples (EOM-CCSDT). \cite{Kucharski_1991,Kallay_2004,Hirata_2000,Hirata_2004}
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Interestingly, the {\twoAg} and {\Aoneg} states have a strong contribution from doubly-excited configurations and these so-called double excitations \cite{Loos_2019} are known to be an absolute nightmare for adiabatic time-dependent density-functional theory (TD-DFT) \cite{Runge_1984,Casida_1995,Tozer_2000,Maitra_2004,Cave_2004,Levine_2006,Elliott_2011,Maitra_2012,Maitra_2017} and even for state-of-the-art methods like the approximate third-order coupled-cluster (CC3) \cite{Christiansen_1995,Koch_1997} or equation-of-motion coupled-cluster with singles, doubles, and triples (EOM-CCSDT). \cite{Kucharski_1991,Kallay_2004,Hirata_2000,Hirata_2004}
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In order to tackle the problem of multi-configurational character and double excitations, we have explored several routes.
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The most evident way is to rely on multi-configurational methods, which are naturally designed to tackle such scenarios.
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The most evident way is to rely on multi-configurational methods, which are naturally designed to address such scenarios.
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Among these methods, one can mention the complete active space self-consistent field (CASSCF) method, \cite{Roos_1996} its second-order perturbatively-corrected variant (CASPT2) \cite{Andersson_1990,Andersson_1992,Roos_1995a} and the second-order $n$-electron valence state perturbation theory (NEVPT2) formalism. \cite{Angeli_2001,Angeli_2001a,Angeli_2002}
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The exponential scaling of the computational cost (with respect to the size of the active space) associated with these methods is the principal limitation to their applicability to large molecules.
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Another way to deal with double excitations and multi-reference situations is to use high level truncation of the equation-of-motion (EOM) formalism \cite{Rowe_1968,Stanton_1993} of coupled-cluster (CC) theory. \cite{Kucharski_1991,Kallay_2003,Kallay_2004,Hirata_2000,Hirata_2004}
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However, to provide a correct description of these situations, one have to take into account, at the very least, contributions from the triple excitations in the CC expansion. \cite{Watson_2012,Loos_2018a,Loos_2019,Loos_2020b}
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Again, due to the scaling of CC methods with the number of basis functions, their applicability is limited to small molecules.
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Although multi-reference CC methods have been specifically designed for such purposes, \cite{Jeziorski_1981,Mahapatra_1998,Mahapatra_1999,Lyakh_2012,Kohn_2013} they are computationally demanding and still far from being black-box.
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In this context, an interesting alternative to multi-configurational and CC methods is the use of selected CI (SCI) methods \cite{Bender_1969,Whitten_1969,Huron_1973,Giner_2013,Evangelista_2014,Giner_2015,Caffarel_2016b,Holmes_2016,Tubman_2016,Liu_2016,Ohtsuka_2017,Zimmerman_2017,Coe_2018,Garniron_2018} are known to provide near full CI (FCI) energies for small molecules. \cite{Caffarel_2014,Caffarel_2016a,Scemama_2016,Holmes_2017,Li_2018,Scemama_2018,Scemama_2018b,Li_2020,Loos_2018a,Chien_2018,Loos_2019,Loos_2020b,Loos_2020c,Loos_2020e,Garniron_2019,Eriksen_2020,Yao_2020,Williams_2020,Veril_2021,Loos_2021}
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In this context, an interesting alternative to multi-configurational and CC methods is provided by selected CI (SCI) methods, \cite{Bender_1969,Whitten_1969,Huron_1973,Giner_2013,Evangelista_2014,Giner_2015,Caffarel_2016b,Holmes_2016,Tubman_2016,Liu_2016,Ohtsuka_2017,Zimmerman_2017,Coe_2018,Garniron_2018} which have proven, recently, to be able to provide near full CI (FCI) energies for small molecules for both ground- and excited-state energies. \cite{Caffarel_2014,Caffarel_2016a,Scemama_2016,Holmes_2017,Li_2018,Scemama_2018,Scemama_2018b,Li_2020,Loos_2018a,Chien_2018,Loos_2019,Loos_2020b,Loos_2020c,Loos_2020e,Garniron_2019,Eriksen_2020,Yao_2020,Williams_2020,Veril_2021,Loos_2021,Damour_2021}
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For example, the \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI) method limits the exponential increase of the size of the CI expansion by retaining the most energetically relevant determinants only, using a second-order energetic criterion to select perturbatively determinants in the FCI space. \cite{Huron_1973,Giner_2013,Giner_2015,Garniron_2017,Garniron_2018,Garniron_2019}
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Nonetheless, SCI methods remain very expensive and can be applied to a limited number of situations.
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Finally, another option to deal with these chemical scenarios is to rely on the cheaper spin-flip formalism, established by Krylov in 2001, \cite{Krylov_2001a,Krylov_2001b,Krylov_2002,Casanova_2020} where one accesses the ground and doubly-excited states via a single (spin-flip) excitation and deexcitation from the lowest triplet state, respectively.
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Obviously, spin-flip methods have their own flaws, especially spin contamination (\ie, an artificial mixing of electronic states with different spin multiplicities) due principally to spin incompleteness in the spin-flip expansion but also to the spin contamination of the reference configuration. \cite{Casanova_2020}
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One can address part of this issue by increasing the excitation order (but with its additional computational cost) or by complementing the spin-incomplete configuration set with the missing configurations. \cite{Sears_2003,Casanova_2008,Huix-Rotllant_2010,Li_2010,Li_2011a,Li_2011b,Zhang_2015,Lee_2018}
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One can address part of this issue by increasing the excitation order or by complementing the spin-incomplete configuration set with the missing configurations, \cite{Sears_2003,Casanova_2008,Huix-Rotllant_2010,Li_2010,Li_2011a,Li_2011b,Zhang_2015,Lee_2018}
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both solutions being associated with an additional computational cost.
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In the present work, we investigate the accuracy of a large panel of computational methods on the autoisomerization barrier and the low-lying excited states of CBD at the {\Dtwo} and {\Dfour} ground-state geometries.
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Computational details are reported in Section \ref{sec:compmet} for SCI (Subsection \ref{sec:SCI}), EOM-CC (Subsection \ref{sec:CC}), multi-configurational (Subsection \ref{sec:Multi}) and spin-flip (Subsection \ref{sec:sf}) methods.
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@ -311,45 +313,47 @@ First, we consider the autoisomerization barrier (Subsection \ref{sec:auto}) and
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\subsection{Selected configuration interaction calculations}
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\label{sec:SCI}
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For the SCI calculations, we rely on the CIPSI algorithm which is implemented in QUANTUM PACKAGE. \cite{Garniron_2019}
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To treat electronic states on equal footing, we use a state-averaged formalism where the ground and excited states are expanded in the same number and same set of determinants but with different CI coefficients.
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To treat electronic states on equal footing, we use a state-averaged formalism where the ground and excited states are expanded with the same set of determinants but with different CI coefficients.
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Note that the determinant selection for these states are performed simultaneously via the protocol described in Refs.~\onlinecite{Scemama_2019,Garniron_2019}.
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For a given size of the variational wave function, the CIPSI energy is the sum of two terms: the variational energy obtained by diagonalization of the CI matrix in the reference space and a second-order perturbative correction which estimates the contribution of the determinants not included in the variational space (estimate error in the truncation).
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\titou{For a given size of the variational wave function, the CIPSI energy is the sum of two terms: the variational energy obtained by diagonalization of the CI matrix in the reference space and a second-order perturbative correction which estimates the contribution of the determinants not included in the variational space (estimate error in the truncation).
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It is possible to estimate the FCI limit for the total energies and compute the corresponding transition energies by extrapolating this second-order correction to zero.
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Extrapolation brings error and we can estimate this one by energy difference between excitation energies obtained with the largest SCI wave function and the FCI extrapolated value.
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These errors provide a rough idea of the quality of the FCI extrapolation and cannot be seen as true bar error, they are reported in the following tables.
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These errors provide a rough idea of the quality of the FCI extrapolation and cannot be seen as true bar error, they are reported in the following tables.}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Coupled-Cluster calculations}
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\subsection{Coupled-cluster calculations}
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\label{sec:CC}
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Different flavours of coupled-cluster (CC) calculations are performed using different codes.
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Indeed, CC theory provides a hierarchy of methods that provide increasingly accurate energies via the increase of the maximum excitation degree of the cluster operator.
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Without any truncation of the cluster operator one has the full CC (FCC) that is equivalent to the full configuration interaction (FCI) giving the exact energy and wave function of the system for a fixed atomic basis set.
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However, due to the computational exponential scaling with system size we have to use truncated CC methods.
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The CC with singles and doubles (CCSD), CC with singles, doubles and triples (CCSDT) calculations are achieved with \textcolor{red}{CFOUR}.
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The calculations in the context of CC response theory or ``approximate'' series (CC3,CC4) are performed with \textcolor{red}{DALTON}.\cite{Aidas_2014}
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The CC with singles, doubles, triples and quadruples (CCSDTQ) are done with the \textcolor{red}{CFOUR} code.
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Coupled-cluster theory provides a hierarchy of methods that yield increasingly accurate energies via the increase of the maximum excitation degree of the cluster operator: CC with singles and doubles (CCSD), CC with singles, doubles, and triples (CCSDT) as well as CC with singles, doubles, triples, and quadruples (CCSDTQ).
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%Without any truncation of the cluster operator, one has the full CC (FCC) that is equivalent to the full configuration interaction (FCI) giving the exact energy and wave function of the system for a fixed atomic basis set.
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%However, due to the computational exponential scaling with system size we have to use truncated CC methods.
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Here, we performed different types of CC calculations using different codes.
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CCSD, CCSDT, and CCSDTQ calculations are achieved with \textcolor{red}{CFOUR}.
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The calculations in the context of CC response theory or ``approximate'' series (CC3 and CC4) are performed with \textcolor{red}{DALTON}.\cite{Aidas_2014}
|
||||
The CC with singles, doubles, triples and quadruples (CCSDTQ) are done with the \titou{CFOUR} code.
|
||||
The CC2, \cite{Christiansen_1995,Hattig_2000} CC3 \cite{Christiansen_1995,Koch_1995} and CC4 \cite{Kallay_2005,Matthews_2020} methods can be seen as cheaper approximations of CCSD, \cite{Purvis_1982} CCSDT \cite{Noga_1987a} and CCSDTQ \cite{Kucharski_1991a} by skipping the most expensive terms and avoiding the storage of higher-excitations amplitudes.
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\subsection{Multi-configurational calculations}
|
||||
\label{sec:Multi}
|
||||
State-averaged complete-active-space self-consistent field (SA-CASSCF) calculations are performed with \textcolor{red}{MOLPRO}.\cite{Werner_2012}
|
||||
On top of those, NEVPT2 calculations, both partially contracted (PC) and strongly contracted (SC) scheme are considered.
|
||||
The PC-NEVPT2 is theoretically more accurate to the strongly contracted version due to the larger number of perturbers and greater flexibility.
|
||||
CASPT2 is performed and extended multistate (XMS) CASPT2 for strong mixing between states with same spin and spatial symmetries is also performed.
|
||||
State-averaged CASSCF (SA-CASSCF) calculations are performed with MOLPRO. \cite{Werner_2020}
|
||||
\titou{state-average procedure?}
|
||||
On top of these, CASPT2 calculations are performed within the RS2 contraction scheme, while the NEVPT2 energies are computed within both partially contracted (PC) and strongly contracted (SC) scheme. \cite{Angeli_2001,Angeli_2001a,Angeli_2002}
|
||||
PC-NEVPT2 is theoretically more accurate than SC-NEVPT2 due to the larger number of external configurations and greater flexibility.
|
||||
%and extended multistate (XMS) CASPT2 calculations are also performed in the cas of strong mixing between states with same spin and spatial symmetries.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\subsection{Spin-flip calculations}
|
||||
\label{sec:sf}
|
||||
In both structures the CBD has a singlet ground state, for the spin-flip calculations we consider the lowest triplet state as reference.
|
||||
In both arrangements, CBD has a singlet ground state, for the spin-flip calculations we consider the lowest triplet state as reference.
|
||||
Spin-flip techniques are broadly accessible and here, among them, we explore algebraic-diagrammatic construction \cite{Schirmer_1982} (ADC) using standard ADC(2)-s \cite{Trofimov_1997,Dreuw_2015} and extended ADC(2)-x \cite{Dreuw_2015} schemes as well as the ADC(3) \cite{Dreuw_2015,Trofimov_2002,Harbach_2014} scheme.
|
||||
We also use spin-flip within the TD-DFT \cite{Casida_1995} framework.
|
||||
The standard and extended spin-flip ADC(2) (SF-ADC(2)-s and SF-ADC(2)-x respectively) and SF-ADC(3) are performed using Q-CHEM 5.2.1. \cite{Shao_2015}
|
||||
Spin-flip TD-DFT calculations are also performed using Q-CHEM 5.2.1. The B3LYP,\cite{Becke_1988b,Lee_1988a,Becke_1993b} PBE0 \cite{Adamo_1999a,Ernzerhof_1999} and BH\&HLYP hybrid GGA functionals are considered, they contain 20\%, 25\%, 50\% of exact exchange and are labeled, respectively, as SF-BLYP, SF-B3LYP, SF-PBE0, SF-BH\&HLYP.
|
||||
Spin-flip TD-DFT calculations are also performed using Q-CHEM 5.2.1.
|
||||
The B3LYP,\cite{Becke_1988b,Lee_1988a,Becke_1993b} PBE0 \cite{Adamo_1999a,Ernzerhof_1999} and BH\&HLYP hybrid GGA functionals are considered, they contain 20\%, 25\%, 50\% of exact exchange and are labeled, respectively, as SF-BLYP, SF-B3LYP, SF-PBE0, SF-BH\&HLYP.
|
||||
We also have done spin-flip TD-DFT calculations using range-separated hybrid (RSH) functionals as: CAM-B3LYP,\cite{Yanai_2004a} LC-$\omega$PBE08 \cite{Weintraub_2009a} and $\omega$B97X-V. \cite{Mardirossian_2014}
|
||||
The main difference here between these RSH functionals is the amount of exact-exchange at long-range: 75$\%$ for CAM-B3LYP and 100$\%$ for LC-$\omega$PBE08 and $\omega$B97X-V.
|
||||
To complete the use of spin-flip TD-DFT we also considered the hybrid meta-GGA functional M06-2X \cite{Zhao_2008} and the RSH meta-GGA functional M11.\cite{Peverati_2011}
|
||||
@ -364,25 +368,24 @@ For each studied quantity, i.e., the autoisomerisation barrier and the vertical
|
||||
These TBEs are provided using extrapolated CCSDTQ/aug-cc-pVTZ values when possible and using NEVPT2(12,12) otherwise.
|
||||
The extrapolation of the CCSDTQ/aug-cc-pVTZ values is done using two schemes.
|
||||
The first one uses CC4 values for the extrapolation and proceed as follows
|
||||
|
||||
\begin{equation}
|
||||
\label{eq:aug-cc-pVTZ}
|
||||
\Delta E^{\text{CCSDTQ}}_{\text{aug-cc-pVTZ}} = \Delta E^{\text{CCSDTQ}}_{\text{aug-cc-pVDZ}} + \left[ \Delta E^{\text{CC4}}_{\text{aug-cc-pVTZ}} - \Delta E^{\text{CC4}}_{\text{aug-cc-pVDZ}} \right]
|
||||
\end{equation}
|
||||
and we evaluate the CCSDTQ/aug-cc-pVDZ values as
|
||||
|
||||
\begin{equation}
|
||||
\label{eq:aug-cc-pVDZ}
|
||||
\Delta E^{\text{CCSDTQ}}_{\text{aug-cc-pVDZ}} = \Delta E^{\text{CCSDTQ}}_{6-31\text{G}+\text{(d)}} + \left[ \Delta E^{\text{CC4}}_{\text{aug-cc-pVDZ}} - \Delta E^{\text{CC4}}_{6-31\text{G}+\text{(d)}} \right]
|
||||
\end{equation}
|
||||
when CC4/aug-cc-pVTZ values have been obtained. If it is not the case we extrapolate CC4/aug-cc-pVTZ values using the CCSDT ones as follows
|
||||
|
||||
when CC4/aug-cc-pVTZ values have been obtained.
|
||||
If it is not the case we extrapolate CC4/aug-cc-pVTZ values using the CCSDT ones as follows
|
||||
\begin{equation}
|
||||
\label{eq:CC4_aug-cc-pVTZ}
|
||||
\Delta E^{\text{CC4}}_{\text{aug-cc-pVTZ}} = \Delta E^{\text{CC4}}_{\text{aug-cc-pVDZ}} + \left[ \Delta E^{\text{CCSDT}}_{\text{aug-cc-pVTZ}} - \Delta E^{\text{CCSDT}}_{\text{aug-cc-pVDZ}} \right]
|
||||
\end{equation}
|
||||
Then if the CC4 values have not been obtained then we use the second scheme which is the same as the first one but instead of the CC4 values we use CCSDT to extrapolate CCSDTQ. If none of the two schemes is possible then we use the NEVPT2(12,12) values. Note that a NEVPT2(12,12) value is used only once for one vertical excitation of the \Dfour structure.
|
||||
|
||||
Then if the CC4 values have not been obtained then we use the second scheme which is the same as the first one but instead of the CC4 values we use CCSDT to extrapolate CCSDTQ.
|
||||
If none of the two schemes is possible then we use the NEVPT2(12,12) values.
|
||||
Note that a NEVPT2(12,12) value is used only once for one vertical excitation of the \Dfour structure.
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
@ -434,8 +437,7 @@ Then we compare results for multi-reference methods, we can see a difference of
|
||||
%%% TABLE I %%%
|
||||
\begin{squeezetable}
|
||||
\begin{table}
|
||||
\caption{Autoisomerization barrier (in \kcalmol) of CBD computed at various levels of theory.}
|
||||
|
||||
\caption{Autoisomerization barrier (in \kcalmol) of CBD computed with various computational methods and basis sets.}
|
||||
\label{tab:auto_standard}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{llrrrr}
|
||||
@ -479,7 +481,8 @@ CIPSI & $7.91\pm 0.21$ & $8.58\pm 0.14$ & & \\
|
||||
|
||||
\begin{figure*}
|
||||
\includegraphics[scale=0.5]{AB_AVTZ.pdf}
|
||||
\caption{Autoisomerization barrier energies for CBD using the aug-cc-pVTZ basis. Purple histograms are for the SF-TD-DFT functionals, orange histograms are for the SF-ADC schemes, green histograms are for the multi-reference methods, blue histograms are for the CC methods and the black one is for the TBE.}
|
||||
\caption{Autoisomerization barrier (in \kcalmol) of CBD at various levels of theory using the aug-cc-pVTZ basis.}
|
||||
%Purple, orange, green, blue, and black histograms are for SF-TD-DFT, SF-ADC, multi-reference methods, CC and TBE.}
|
||||
\label{fig:AB}
|
||||
\end{figure*}
|
||||
|
||||
@ -611,10 +614,10 @@ CASPT2(4,4) &6-31+G(d)& $1.440$ & $3.162$ & $4.115$ \\
|
||||
& aug-cc-pVDZ & $1.414$ & $2.971$ & $4.068$ \\
|
||||
& aug-cc-pVTZ & $1.412$ & $2.923$ & $4.072$ \\
|
||||
& aug-cc-pVQZ & $1.417$ & $2.911$ & $4.081$ \\[0.1cm]
|
||||
XMS-CASPT2(4,4) &6-31+G(d)& & & $4.151$ \\
|
||||
& aug-cc-pVDZ & & & $4.105$ \\
|
||||
& aug-cc-pVTZ & & & $4.114$ \\
|
||||
& aug-cc-pVQZ && & $4.125$ \\[0.1cm]
|
||||
%XMS-CASPT2(4,4) &6-31+G(d)& & & $4.151$ \\
|
||||
%& aug-cc-pVDZ & & & $4.105$ \\
|
||||
%& aug-cc-pVTZ & & & $4.114$ \\
|
||||
%& aug-cc-pVQZ && & $4.125$ \\[0.1cm]
|
||||
SC-NEVPT2(4,4) &6-31+G(d)& $1.407$ & $2.707$ & $4.145$ \\
|
||||
& aug-cc-pVDZ & $1.381$ & $2.479$ & $4.109$ \\
|
||||
& aug-cc-pVTZ & $1.379$ & $2.422$ & $4.108$ \\
|
||||
@ -635,10 +638,10 @@ CASPT2(12,12) &6-31+G(d)& $1.508$ & $3.407$ & $4.099$ \\
|
||||
& aug-cc-pVDZ & $1.489$ & $3.256$ & $4.044$ \\
|
||||
& aug-cc-pVTZ & $1.480$ & $3.183$ & $4.043$ \\
|
||||
& aug-cc-pVQZ & $1.482$ & $3.163$ & $4.047$ \\[0.1cm]
|
||||
XMS-CASPT2(12,12) &6-31+G(d)& && $4.111$ \\
|
||||
& aug-cc-pVDZ & & & $4.056$ \\
|
||||
& aug-cc-pVTZ & & & $4.059$ \\
|
||||
& aug-cc-pVQZ & & & $4.065$ \\[0.1cm]
|
||||
%XMS-CASPT2(12,12) &6-31+G(d)& && $4.111$ \\
|
||||
%& aug-cc-pVDZ & & & $4.056$ \\
|
||||
%& aug-cc-pVTZ & & & $4.059$ \\
|
||||
%& aug-cc-pVQZ & & & $4.065$ \\[0.1cm]
|
||||
SC-NEVPT2(12,12) &6-31+G(d)& $1.522$ & $3.409$ & $4.130$ \\
|
||||
& aug-cc-pVDZ & $1.511$ & $3.266$ & $4.093$ \\
|
||||
& aug-cc-pVTZ & $1.501$ & $3.188$ & $4.086$ \\
|
||||
@ -938,13 +941,13 @@ CC4 & $-0.11$ & & $0.003$ & $-0.006$ & & $0.011$ & $-0.013$ \\
|
||||
CCSDTQ & $0.00$ & & $0.000$ & $0.000$ & $0.000$ & $0.000$ & $0.000$ \\[0.1cm]
|
||||
SA2-CASSCF(4,4) & & $0.208$ & $1.421$ & $0.292$ & $0.290$ & $0.734$ & $1.390$ \\
|
||||
CASPT2(4,4) & & $-0.050$ & $-0.202$ & $-0.077$ & $-0.016$ & $0.006$ & $-0.399$ \\
|
||||
XMS-CASPT2(4,4) & & & & $-0.035$ & & & \\
|
||||
%XMS-CASPT2(4,4) & & & & $-0.035$ & & & \\
|
||||
SC-NEVPT2(4,4) & & $-0.083$ & $-0.703$ & $-0.041$ & $-0.120$ & $-0.072$ & $-0.979$ \\
|
||||
PC-NEVPT2(4,4) & & $-0.080$ & $-0.757$ & $-0.066$ & $-0.118$ & $-0.097$ & $-1.031$ \\
|
||||
MRCI(4,4) & & $0.106$ & $0.553$ & $0.121$ & $0.127$ & $0.324$ & $0.381$ \\[0.1cm]
|
||||
SA2-CASSCF(12,12) & $2.66$ & $0.224$ & $0.719$ & $0.068$ & $0.226$ & $0.443$ & $0.600$ \\
|
||||
CASPT2(12,12) & & $0.018$ & $0.058$ & $-0.106$ & $0.039$ & $0.038$ & $-0.108$ \\
|
||||
XMS-CASPT2(12,12) & & & & $-0.090$ & & & \\
|
||||
%XMS-CASPT2(12,12) & & & & $-0.090$ & & & \\
|
||||
SC-NEVPT2(12,12) & $-0.65$ & $0.039$ & $0.063$ & $-0.063$ & $0.021$ & $0.046$ & $-0.142$ \\
|
||||
PC-NEVPT2(12,12) & & $0.000$ & $-0.062$ & $-0.093$ & $-0.013$ & $-0.024$ & $-0.278$ \\[0.1cm]
|
||||
%CIPSI & & $-0.001\pm 0.030$ & $0.017\pm 0.035$ & $-0.120\pm 0.090$ & $0.025\pm 0.029$ & $0.130\pm 0.050$ & \\
|
||||
|
||||
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