This work presents the formalism and implementation for calculations of spin–orbit couplings (SOCs) using the Breit–Pauli Hamiltonian and non-relativistic wave functions described by the restricted active space configuration interaction (RASCI) method with general excitation operators of spin-conserving spin-flipping, ionizing, and electron-attaching types. The implementation is based on the application of the Wigner–Eckart theorem within the spin space, which enables the calculation of the entire SOC matrix based on the explicit calculation of just one transition between the two spin multiplets. Numeric results for a diverse set of atoms and molecules highlight the importance of a balanced treatment of correlation and adequate basis sets and illustrate the overall robust performance of RASCI SOCs. The new implementation is a useful addition to the methodological toolkit for studying spin-forbidden processes and molecular magnetism.
REFERENCES
1.
D. R.
Yarkony
, “Spin-forbidden chemistry within the Breit-Pauli approximation
,” Int. Rev. Phys. Chem.
11
, 195
–242
(1992
).2.
C. M.
Marian
, “Spin–orbit coupling and intersystem crossing in molecules
,” Wiley Interdiscip. Rev.: Comput. Mol. Sci.
2
, 187
–203
(2012
).3.
A.
Zaitsevskii
, Relyativiskaya Teoriya Electronnogo Stroyeniya Molekul
(Moscow State University
, Moscow
, 2005
).4.
P. A. M.
Dirac
, “The quantum theory of the electron
,” Proc. R. Soc. A
117
, 610
–624
(1928
).5.
M. A.
El-Sayed
, “Triplet state. Its radiative and nonradiative properties
,” Acc. Chem. Res.
1
, 8
–16
(1968
).6.
A.
Bergantini
, M. J.
Abplanalp
, P.
Pokhilko
, A. I.
Krylov
, C. N.
Shingledecker
, E.
Herbst
, and R. I.
Kaiser
, “A combined experimental and theoretical study on the formation of interstellar propylene oxide (CH3CHCH2O)—A chiral molecule
,” Astrophys. J.
860
, 108
(2018
).7.
P.
Pokhilko
, R.
Shannon
, D.
Glowacki
, H.
Wang
, and A. I.
Krylov
, “Spin-forbidden channels in reactions of unsaturated hydrocarbons with O(3P)
,” J. Phys. Chem. A
123
, 482
–491
(2019
).8.
A. L.
Buchachenko
and V. L.
Berdinsky
, “Spin catalysis of chemical reactions
,” J. Phys. Chem.
100
, 18292
–18299
(1996
).9.
G.
Baryshnikov
, B.
Minaev
, and H.
Ågren
, “Theory and calculation of the phosphorescence phenomenon
,” Chem. Rev.
117
, 6500
–6537
(2017
).10.
R.
Poli
and J. N.
Harvey
, “Spin forbidden chemical reactions of transition metal compounds. New ideas and new computational challenges
,” Chem. Soc. Rev.
32
, 1
–8
(2003
).11.
T. J.
Penfold
, E.
Gindensperger
, C.
Daniel
, and C. M.
Marian
, “Spin-vibronic mechanism for intersystem crossing
,” Chem. Rev.
118
, 6975
–7025
(2018
).12.
Magnetic Properties of Organic Materials
, edited by P. M.
Lahti
(Marcel Dekker
, 1999
).13.
M. R.
Pederson
and T.
Baruah
, Handbook of Magnetism and Advanced Magnetic Materials
(Wiley
, 2007
).14.
J. P.
Malrieu
, R.
Caballol
, C. J.
Calzado
, C.
de Graaf
, and N.
Guihéry
, “Magnetic interactions in molecules and highly correlated materials: Physical content, analytical derivation, and rigorous extraction of magnetic Hamiltonians
,” Chem. Rev.
114
, 429
–492
(2013
).15.
H. A.
Bethe
and E. E.
Salpeter
, Quantum Mechanics of One and Two Electron Atoms
(Plenum Press
, New York
, 1977
).16.
Relativistic Electronic Structure Theory
, edited by P.
Schwerdtfeger
(Elsevier
, Amsterdam
, 2002
).17.
P. Å.
Malmqvist
, B. O.
Roos
, and B.
Schimmelpfennig
, “The restricted active space (RAS) state interaction approach with spin–orbit coupling
,” Chem. Phys. Lett.
357
, 230
–240
(2002
).18.
M.
Roemelt
, “Spin orbit coupling for molecular ab initio density matrix renormalization group calculations: Application to g-tensors
,” J. Chem. Phys.
143
, 044112
(2015
).19.
P.
Pokhilko
, E.
Epifanovsky
, and A. I.
Krylov
, “General framework for calculating spin–orbit couplings using spinless one-particle density matrices: Theory and application to the equation-of-motion coupled-cluster wave functions
,” J. Chem. Phys.
151
, 034106
(2019
).20.
C.
Eckart
, “The application of group theory to the quantum dynamics of monatomic systems
,” Rev. Mod. Phys.
2
, 305
–380
(1930
).21.
E.
Wigner
, “Einige Folgerungen aus der Schrödingerschen Theorie für die Termstrukturen
,” Z. Phys.
43
, 624
–652
(1927
).22.
P.
Pokhilko
, “Development and application of robust many-body methods for strongly correlated systems: From spin-forbidden chemistry to single-molecule magnets
,” Ph.D. thesis, University of Southern California
, 2020
.23.
A. I.
Krylov
, “Equation-of-motion coupled-cluster methods for open-shell and electronically excited species: The Hitchhiker’s guide to Fock space
,” Annu. Rev. Phys. Chem.
59
, 433
–462
(2008
).24.
Y.
Shao
, Z.
Gan
, E.
Epifanovsky
, A. T. B.
Gilbert
, M.
Wormit
, J.
Kussmann
, A. W.
Lange
, A.
Behn
, J.
Deng
, X.
Feng
et al., “Advances in molecular quantum chemistry contained in the Q-Chem 4 program package
,” Mol. Phys.
113
, 184
–215
(2015
).25.
M. L.
Vidal
, P.
Pokhilko
, A. I.
Krylov
, and S.
Coriani
, “Equation-of-motion coupled-cluster theory to model L-edge X-ray absorption and photoelectron spectra
,” J. Phys. Chem. Lett.
11
, 8314
–8321
(2020
).26.
M. L.
Vidal
, X.
Feng
, E.
Epifanovsky
, A. I.
Krylov
, and S.
Coriani
, “A new and efficient equation-of-motion coupled-cluster framework for core-excited and core-ionized states
,” J. Chem. Theory Comput.
15
, 3117
–3133
(2019
).27.
O. R.
Meitei
, S. E.
Houck
, and N. J.
Mayhall
, “Spin–orbit matrix elements for a combined spin-flip and IP/EA approach
,” J. Chem. Theory Comput.
16
, 3597
–3606
(2020
).28.
J. M.
Turney
et al., “PSI4: An open-source ab initio electronic structure package
,” Wiley Interdiscip. Rev.: Comput. Mol. Sci.
2
, 556
–565
(2012
).29.
D.
Casanova
, “Efficient implementation of restricted active space configuration interaction with the hole and particle approximation
,” J. Comput. Chem.
34
, 720
–730
(2013
).30.
P.
Pokhilko
and A. I.
Krylov
, “Quantitative El-Sayed rules for many-body wavefunctions from spinless transition density matrices
,” J. Phys. Chem. Lett.
10
, 4857
–4862
(2019
).31.
A. I.
Krylov
, “From orbitals to observables and back
,” J. Chem. Phys.
153
, 080901
(2020
).32.
A. I.
Krylov
, “Size-consistent wave functions for bond-breaking: The equation-of-motion spin-flip model
,” Chem. Phys. Lett.
338
, 375
–384
(2001
).33.
A. I.
Krylov
, “Spin-flip configuration interaction: An electronic structure model that is both variational and size-consistent
,” Chem. Phys. Lett.
350
, 522
–530
(2001
).34.
A. I.
Krylov
, “The spin-flip equation-of-motion coupled-cluster electronic structure method for a description of excited states, bond-breaking, diradicals, and triradicals
,” Acc. Chem. Res.
39
, 83
–91
(2006
).35.
A. V.
Luzanov
, “Spin-flip models in the spin coupling method of many-particle amplitudes
,” J. Struct. Chem.
45
, 729
–739
(2004
).36.
A. V.
Luzanov
, “Matrix-covariant representation of high-order configuration interaction and coupled cluster theories
,” Int. J. Quantum Chem.
108
, 671
–695
(2008
).37.
Y.
Shao
, M.
Head-Gordon
, and A. I.
Krylov
, “The spin-flip approach within time-dependent density functional theory: Theory and applications to diradicals
,” J. Chem. Phys.
118
, 4807
–4818
(2003
).38.
H. R.
Zhekova
, M.
Seth
, and T.
Ziegler
, “Calculation of the exchange coupling constants of copper binuclear systems based on spin-flip constricted variational density functional theory
,” J. Chem. Phys.
135
, 184105
(2011
).39.
Y. A.
Bernard
, Y.
Shao
, and A. I.
Krylov
, “General formulation of spin-flip time-dependent density functional theory using non-collinear kernels: Theory, implementation, and benchmarks
,” J. Chem. Phys.
136
, 204103
(2012
).40.
D.
Casanova
and A. I.
Krylov
, “Spin-flip methods in quantum chemistry
,” Phys. Chem. Chem. Phys.
22
, 4326
–4342
(2020
).41.
D.
Casanova
and M.
Head-Gordon
, “Restricted active space spin-flip configuration interaction approach: Theory, implementation and examples
,” Phys. Chem. Chem. Phys.
11
, 9779
–9790
(2009
).42.
M. E.
Sandoval-Salinas
, A.
Carreras
, and D.
Casanova
, “Triangular graphene nanofragments: Open-shell character and doping
,” Phys. Chem. Chem. Phys.
21
, 9069
–9076
(2019
).43.
M.
Desroches
, P.
Mayorga Burrezo
, J.
Boismenu-Lavoie
, M.
Peña Álvarez
, C. J.
Gómez-García
, J. M.
Matxain
, D.
Casanova
, J.-F.
Morin
, and J.
Casado
, “Breaking bonds and forming nanographene diradicals with pressure
,” Angew. Chem., Int. Ed.
56
, 16212
–16217
(2017
).44.
A.
Pérez-Guardiola
, M. E.
Sandoval-Salinas
, D.
Casanova
, E.
San-Fabián
, A. J.
Pérez-Jiménez
, and J. C.
Sancho-García
, “The role of topology in organic molecules: Origin and comparison of the radical character in linear and cyclic oligoacenes and related oligomers
,” Phys. Chem. Chem. Phys.
20
, 7112
–7124
(2018
).45.
Z.
Li
, T. Y.
Gopalakrishna
, Y.
Han
, Y.
Gu
, L.
Yuan
, W.
Zeng
, D.
Casanova
, and J.
Wu
, “[6]Cyclo-para-phenylmethine: An analog of benzene showing global aromaticity and open-shell diradical character
,” J. Am. Chem. Soc.
141
, 16266
–16270
(2019
).46.
L.
Chunchen
, M. E.
Sandoval-Salinas
, Y.
Hong
, T. Y.
Gopalakrishna
, H.
Phan
, N.
Aratani
, T. S.
Herng
, J.
Ding
, H.
Yamada
, D.
Kim
, D.
Casanova
, and J.
Wu
, “Macrocyclic polyradicaloids with unusual super-ring structure and global aromaticity
,” Chem
4
, 1586
–1595
(2018
).47.
A. V.
Luzanov
, D.
Casanova
, X.
Feng
, and A. I.
Krylov
, “Quantifying charge resonance and multiexciton character in coupled chromophores by charge and spin cumulant analysis
,” J. Chem. Phys.
142
, 224104
(2015
).48.
D.
Casanova
and A. I.
Krylov
, “Quantifying local excitation, charge resonance, and multiexciton character in correlated wave functions of multichromophoric systems
,” J. Chem. Phys.
144
, 014102
(2016
).49.
A. E.
Rudenko
, N. E.
Clayman
, K. L.
Walker
, J. K.
Maclaren
, P. M.
Zimmerman
, and R. M.
Waymouth
, “Ligand-induced reductive elimination of ethane from azopyridine palladium dimethyl complexes
,” J. Am. Chem. Soc.
140
, 11408
–11415
(2018
).50.
P. M.
Zimmerman
, F.
Bell
, D.
Casanova
, and M.
Head-Gordon
, “Mechanism for singlet fission in pentacene and tetracene: From single exciton to two triplets
,” J. Am. Chem. Soc.
133
, 19944
–19952
(2011
).51.
D.
Casanova
, “Electronic structure study of singlet-fission in tetracene derivatives
,” J. Chem. Theory Comput.
10
, 324
–334
(2014
).52.
N. V.
Korovina
, S.
Das
, Z.
Nett
, X.
Feng
, J.
Joy
, R.
Haiges
, A. I.
Krylov
, S. E.
Bradforth
, and M. E.
Thompson
, “Singlet fission in a covalently linked cofacial alkynyltetracene dimer
,” J. Am. Chem. Soc.
138
, 617
–627
(2016
).53.
X.
Feng
, D.
Casanova
, and A. I.
Krylov
, “Intra- and inter-molecular singlet fission in covalently linked dimers
,” J. Phys. Chem. C
120
, 19070
–19077
(2016
).54.
D.
Casanova
, “Theoretical modeling of singlet fission
,” Chem. Rev.
118
, 7164
–7207
(2018
).55.
M. H.
Farag
and A. I.
Krylov
, “Singlet fission in perylenediimide dimers
,” J. Phys. Chem. C
122
, 25753
–25763
(2018
).56.
M. E.
Sandoval-Salinas
, A.
Carreras
, J.
Casado
, and D.
Casanova
, “Singlet fission in spiroconjugated dimers
,” J. Chem. Phys.
150
, 204306
(2019
).57.
S.
Matsika
, X.
Feng
, A. V.
Luzanov
, and A. I.
Krylov
, “What we can learn from the norms of one-particle density matrices, and what we can’t: Some results for interstate properties in model singlet fission systems
,” J. Phys. Chem. A
118
, 11943
–11955
(2014
).58.
N. V.
Korovina
, J.
Joy
, X.
Feng
, C.
Feltenberger
, A. I.
Krylov
, S. E.
Bradforth
, and M. E.
Thompson
, “Linker-dependent singlet fission in tetracene dimers
,” J. Am. Chem. Soc.
140
, 10179
–10190
(2018
).59.
J.
Kim
, H. T.
Teo
, Y.
Hong
, J.
Oh
, H.
Kim
, C.
Chi
, and D.
Kim
, “Multiexcitonic triplet pair generation in oligoacene dendrimers as amorphous solid-state miniatures
,” Angew. Chem., Int. Ed.
59
, 20956
–20964
(2020
).60.
A. D.
Chien
, A. R.
Molina
, N.
Abeyasinghe
, O. P.
Varnavski
, T.
Goodson
, and P. M.
Zimmerman
, “Structure and dynamics of the 1(TT) state in a quinoidal bithiophene: Characterizing a promising intramolecular singlet fission candidate
,” J. Phys. Chem. C
119
, 28258
–28268
(2015
).61.
H.
Kim
and P. M.
Zimmerman
, “Coupled double triplet state in singlet fission
,” Phys. Chem. Chem. Phys.
20
, 30083
–30094
(2018
).62.
H.
Kim
, B.
Keller
, R.
Ho-Wu
, N.
Abeyasinghe
, R. J.
Vázquez
, T.
Goodson
, and P. M.
Zimmerman
, “Enacting two-electron transfer from a double-triplet state of intramolecular singlet fission
,” J. Am. Chem. Soc.
140
, 7760
–7763
(2018
).63.
H.
Jiang
and P. M.
Zimmerman
, “Charge transfer via spin flip configuration interaction: Benchmarks and application to singlet fission
,” J. Chem. Phys.
153
, 064109
(2020
).64.
J.
Olsen
, B. O.
Roos
, P.
Jørgensen
, and H. J. A.
Jensen
, “Determinant based configuration interaction algorithms for complete and restricted configuration interaction spaces
,” J. Chem. Phys.
89
, 2185
(1988
).65.
D.
Casanova
, “Avoided crossings, conical intersections, and low-lying excited states with a single reference method: The restricted active space spin-flip configuration interaction approach
,” J. Chem. Phys.
137
, 084105
(2012
).66.
P. M.
Zimmerman
, F.
Bell
, M.
Goldey
, A. T.
Bell
, and M.
Head-Gordon
, “Restricted active space spin-flip configuration interaction: Theory and examples for multiple spin flips with odd numbers of electrons
,” J. Chem. Phys.
137
, 164110
(2012
).67.
A.
Pérez-Guardiola
, R.
Ortiz-Cano
, M. E.
Sandoval-Salinas
, J.
Fernández-Rossier
, D.
Casanova
, A. J.
Pérez-Jiménez
, and J. C.
Sancho-García
, “From cyclic nanorings to single-walled carbon nanotubes: Disclosing the evolution of their electronic structure with the help of theoretical methods
,” Phys. Chem. Chem. Phys.
21
, 2547
–2557
(2019
).68.
Y.
Ni
, M. E.
Sandoval-Salinas
, T.
Tanaka
, H.
Phan
, T. S.
Herng
, T. Y.
Gopalakrishna
, J.
Ding
, A.
Osuka
, D.
Casanova
, and J.
Wu
, “[n]Cyclo-para-biphenylmethine polyradicaloids: [n]annulene analogs and unusual valence tautomerization
,” Chem
5
, 108
–121
(2019
).69.
W.
Pauli
, “Zur Quantenmechanik des Magnetischen Elektrons
,” Z. Phys.
43
, 601
–623
(1927
).70.
in front of the two-electron SOMF part. The two-particle part of the Breit–Pauli spin–orbit Hamiltonian does not have a prefactor. Consequently, it should not be present in the second-quantized expression. In our implementation, we use spin-integrated expressions from Appendix B of Ref. 71, which are correct and do not contain this prefactor.
71.
E.
Epifanovsky
, K.
Klein
, S.
Stopkowicz
, J.
Gauss
, and A. I.
Krylov
, “Spin-orbit couplings within the equation-of-motion coupled-cluster framework: Theory, implementation, and benchmark calculations
,” J. Chem. Phys.
143
, 064102
(2015
).72.
B. A.
Heß
, C. M.
Marian
, U.
Wahlgren
, and O.
Gropen
, “A mean-field spin-orbit method applicable to correlated wavefunctions
,” Chem. Phys. Lett.
251
, 365
–371
(1996
).73.
R.
McWeeny
, “On the origin of spin-Hamiltonian parameters
,” J. Chem. Phys.
42
, 1717
–1725
(1965
).74.
D. G.
Fedorov
, “Theoretical study of spin-orbit coupling in molecules
,” Ph.D. thesis, Iowa State University
, 1999
.75.
D. G.
Fedorov
, S.
Koseki
, M. W.
Schmidt
, and M. S.
Gordon
, “Spin-orbit coupling in molecules: Chemistry beyond the adiabatic approximation
,” Int. Rev. Phys. Chem.
22
, 551
–592
(2003
).76.
D. G.
Fedorov
and M. S.
Gordon
, “A study of the relative importance of one and two-electron contributions to spin–orbit coupling
,” J. Chem. Phys.
112
, 5611
–5623
(2000
).77.
A.
Landé
, “Termstruktur und Zeemaneffekt der Multipletts
,” Z. Phys.
15
, 189
–205
(1923
).78.
E. U.
Condon
and G. H.
Shortley
, The Theory of Atomic Spectra
(Cambridge University Press
, 1935
).79.
A.
Berning
, M.
Schweizer
, H.-J.
Werner
, P. J.
Knowles
, and P.
Palmieri
, “Spin-orbit matrix elements for internally contracted multireference configuration interaction wavefunctions
,” Mol. Phys.
98
, 1823
–1833
(2000
).80.
H. F.
King
and T. R.
Furlani
, “Computation of one and two electron spin-orbit integrals
,” J. Comput. Chem.
9
, 771
–778
(1988
).81.
F.
Plasser
, M.
Wormit
, and A.
Dreuw
, “New tools for the systematic analysis and visualization of electronic excitations. I. Formalism
,” J. Chem. Phys.
141
, 024106
–024113
(2014
).82.
T. H.
Dunning
, Jr., “Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen
,” J. Chem. Phys.
90
, 1007
–1023
(1989
).83.
D. E.
Woon
and T. H.
Dunning
, Jr., “Gaussian basis sets for use in correlated molecular calculations. III. The atoms aluminum through argon
,” J. Chem. Phys.
98
, 1358
–1371
(1993
).84.
A. K.
Wilson
, D. E.
Woon
, K. A.
Peterson
, and T. H.
Dunning
, “Gaussian basis sets for use in correlated molecular calculations. IX. The atoms gallium through krypton
,” J. Chem. Phys.
110
, 7667
–7676
(1999
).85.
J.
Koput
and K. A.
Peterson
, “Ab initio potential energy surface and vibrational–rotational energy levels of X2Σ+ CaOH
,” J. Phys. Chem. A
106
, 9595
–9599
(2002
).86.
N. B.
Balabanov
and K. A.
Peterson
, “Systematically convergent basis sets for transition metals. I. All-electron correlation consistent basis sets for the 3d elements Sc–Zn
,” J. Chem. Phys.
123
, 064107
(2005
).87.
R. S.
Mulliken
, “Report on notation for the spectra of polyatomic molecules
,” J. Chem. Phys.
23
, 1997
–2011
(1955
).88.
L. A.
Mück
, “Highly accurate quantum chemistry: Spin-orbit splittings via multireference coupled-cluster methods and applications in heavy-atom main-group chemistry
,” Ph.D. thesis, Johannes-Gutenberg Universität Mainz
, 2013
.89.
K.
Huber
and G.
Herzberg
, Constants of Diatomic Molecules
(Van Nostrand Reinhold
, New York
, 1979
).90.
P.
Bollmark
, B.
Lindgren
, B.
Rydh
, and U.
Sassenberg
, “The vacuum ultraviolet spectrum of selenium hydride I. Determination of the ground state spin-orbit coupling constant
,” Phys. Scr.
17
, 561
–564
(1978
).91.
A.
Kramida
, Yu.
Ralchenko
, J.
Reader
, and NIST ASD Team
, NIST Atomic Spectra Database (version 5.7.1), available online: https://physics.nist.gov/asd, March 27, 2020, National Institute of Standards and Technology
, Gaithersburg, MD
, 2019
.92.
N.
Wells
and I. C.
Lane
, “Electronic states and spin-forbidden cooling transitions of AlH and AlF
,” Phys. Chem. Chem. Phys.
13
, 19018
–19025
(2011
).93.
R. J.
Hendricks
, D. A.
Holland
, S.
Truppe
, B. E.
Sauer
, and M. R.
Tarbutt
, “Vibrational branching ratios and hyperfine structure of 11BH and its suitability for laser cooling
,” Front. Phys.
2
, 51
(2014
).94.
Y.-F.
Gao
and T.
Gao
, “Laser cooling of BH and GaF: Insights from an ab initio study
,” Phys. Chem. Chem. Phys.
17
, 10830
–10837
(2015
).95.
M. V.
Ivanov
, T.-C.
Jagau
, G.-Z.
Zhu
, E. R.
Hudson
, and A. I.
Krylov
, “In search of molecular ions for optical cycling: A difficult road
,” Phys. Chem. Chem. Phys.
22
, 17075
–17090
(2020
).96.
C. R.
Brazier
, “Emission spectroscopy of the triplet system of the BH radical
,” J. Mol. Spectrosc.
177
, 90
–105
(1996
).97.
W.-T.
Luh
and W. C.
Stwalley
, “The X1Σ+, A1Π, and B1Σ+ potential energy curves and spectroscopy of BH
,” J. Mol. Spectrosc.
102
, 212
–223
(1983
).98.
W.
Szajna
and M.
Zachwieja
, “The emission spectrum of the C1Σ+–X1Σ+ system of AlH
,” J. Mol. Spectrosc.
260
, 130
–134
(2010
).99.
O.
Christiansen
, J.
Gauss
, and B.
Schimmelpfennig
, “Spin-orbit coupling constants from coupled-cluster response theory
,” Phys. Chem. Chem. Phys.
2
, 965
–971
(2000
).100.
C. D.
Sherrill
, M. L.
Leininger
, T. J.
Van Huis
, and H. F.
Schaefer
III, “Structures and vibrational frequencies in the full configuration interaction limit: Predictions for four electronic states of methylene using triple-zeta plus double polarization (TZ2P) basis
,” J. Chem. Phys.
108
, 1040
–1049
(1998
).101.
J.
Stefens
, Y.
Yamaguchi
, C.
Sherrill
, and H. F.
Schaefer
III, “The B1, A1, B1, and electronic states of NH2+
,” J. Phys. Chem.
102
, 3999
–4006
(1998
).102.
Y.
Yamaguchi
, T.
Van Huis
, C.
Sherrill
, and H. F.
Schaefer
III, “The A1, B1, B1 and A1 electronic states of SiH2
,” Theor. Chem. Acc.
97
, 341
–349
(1997
).103.
T.
Van Huis
, Y.
Yamaguchi
, C.
Sherrill
, and H. F.
Schaefer
III, “The A1, B1, B1 and A1 electronic states of PH2+
,” J. Phys. Chem. A
101
, 6955
–6963
(1997
).104.
M.
Montalti
, A.
Credi
, L.
Prodi
, and M. T.
Gandolfi
, Handbook of Photochemistry
(CRC Press
, 2006
).105.
B.
de Souza
, G.
Farias
, F.
Neese
, and R.
Izsák
, “Predicting phosphorescence rates of light organic molecules using time-dependent density functional theory and the path integral approach to dynamics
,” J. Chem. Theory Comput.
15
, 1896
–1904
(2019
).106.
S.
Hirata
and M.
Head-Gordon
, “Time-dependent density functional theory within the Tamm–Dancoff approximation
,” Chem. Phys. Lett.
314
, 291
–299
(1999
).107.
J.
Sous
, P.
Goel
, and M.
Nooijen
, “Similarity transformed equation of motion coupled cluster theory revisited: A benchmark study of valence excited states
,” Mol. Phys.
112
, 616
–638
(2014
).108.
M.
Schreiber
, M. R.
Silva-Junior
, S. P. A.
Sauer
, and W.
Thiel
, “Benchmarks for electronically excited states: CASPT2, CC2, CCSD, and CC3
,” J. Chem. Phys.
128
, 134110
(2008
).109.
R.
Izsák
, “Single-reference coupled cluster methods for computing excitation energies in large molecules: The efficiency and accuracy of approximations
,” Wiley Interdiscip. Rev.: Comput. Mol. Sci.
10
, e1445
(2020
).110.
Q.
Ou
and J. E.
Subotnik
, “Electronic relaxation in benzaldehyde evaluated via TD-DFT and localized diabatization: Intersystem crossings, conical intersections, and phosphorescence
,” J. Phys. Chem. C
117
, 19839
–19849
(2013
).111.
112.
J. A.
Berson
, “A new class of non-Kekule molecules with tunable singlet–triplet energy spacings
,” Acc. Chem. Res.
30
, 238
–244
(1997
).113.
T.
Matsumoto
, T.
Ishida
, N.
Koga
, and H.
Iwamura
, “Intramolecular magnetic coupling between two nitrene or two nitroxide units through 1,1-diphenylethylene chromophores. Isomeric dinitrenes and dinitroxides related in connectivity to trimethylenemethane, tetramethyleneethane, and pentamethylenepropane
,” J. Am. Chem. Soc.
114
, 9952
–9959
(1992
).114.
W.
Lin
, S. R.
Wilson
, and G. S.
Girolami
, “Carbon–carbon bond formation promoted by organoruthenium complexes. The first unsubstituted π-metallabenzene complex, Cp*2Ru2(η2:η5-C5H5)(SiMe3), and synthesis of the tetramethyleneethane complex Cp*2Ru2(η3:η3-C6H8)Cl4
,” Organometallics
16
, 2356
–2361
(1997
).115.
Z. D.
Pozun
, X.
Su
, and K. D.
Jordan
, “Establishing the ground state of the disjoint diradical tetramethyleneethane with quantum Monte Carlo
,” J. Am. Chem. Soc.
135
, 13862
–13869
(2013
).116.
A. D.
Chien
and P. M.
Zimmerman
, “Recovering dynamic correlation in spin flip configuration interaction through a difference dedicated approach
,” J. Chem. Phys.
146
, 014103
(2017
).117.
L.
Veis
, A.
Antalík
, Ö.
Legeza
, A.
Alavi
, and J.
Pittner
, “The intricate case of tetramethyleneethane: A full configuration interaction quantum Monte Carlo benchmark and multireference coupled cluster studies
,” J. Chem. Theory Comput.
14
, 2439
–2445
(2018
).118.
S. K.
Lower
and M. A.
El-Sayed
, “The triplet state and molecular electronic processes in organic molecules
,” Chem. Rev.
66
, 199
–241
(1966
).© 2020 Author(s).
2020
Author(s)
You do not currently have access to this content.
