Electronic structure methods emerging from the combination of multiconfigurational wave functions and density functional theory (DFT) aim to take advantage of the strengths of the two nearly antagonistic theories. One of the common strategies employed to merge wave function theory (WFT) with DFT relies on the range separation of the Coulomb operator in which DFT functionals take care of the short-distance part, while long-range inter-electronic interactions are evaluated by using the chosen wave function method (WFT–srDFT). In this work, we uncover the limitations of WFT–srDFT in the characterization of open-shell systems. We show that spin polarization effects have a major impact on the (short-range) DFT exchange energy and are of vital importance in order to provide a balanced description between closed and open-shell configurations. We introduce different strategies to account for spin polarization in the short range based on the definition of a spin polarized electron density and with the use of short-range exact exchange. We test the performance of these approaches in the dissociation of the hydrogen molecule, the calculation of energy gaps in spin-triplet atoms and molecular diradicals, and the characterization of low-lying states of the gallium dimer. Our results indicate that the use of short-range DFT correlation in combination with a (full-range) multiconfigurational wave function might be an excellent approach for the study of open-shell molecules and largely improves the performance of WFT and WFT–srDFT.

1.
M.
Nakano
and
B.
Champagne
, “
Nonlinear optical properties in open-shell molecular systems
,”
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
6
,
198
210
(
2016
).
2.
G.
Gryn’ova
,
M. L.
Coote
, and
C.
Corminboeuf
, “
Theory and practice of uncommon molecular electronic configurations
,”
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
5
,
440
459
(
2015
).
3.
M.
Reiher
and
A.
Wolf
,
Relativistic Quantum Chemistry: The Fundamental Theory of Molecular Science
(
John Wiley & Sons
,
2014
).
4.
Diradicals
, edited by
W. T.
Borden
(
Wiley
,
1982
).
5.
M. S.
Platz
and
V.
Maloney
,
Kinetics and Spectroscopy of Carbenes and Biradicals
(
Springer
,
1990
).
6.
G.
Christou
,
D.
Gatteschi
,
D. N.
Hendrickson
, and
R.
Sessoli
, “
Single-molecule magnets
,”
MRS Bull.
25
,
66
71
(
2000
).
7.
L.
Salem
and
C.
Rowland
, “
The electronic properties of diradicals
,”
Angew. Chem., Int. Ed.
11
,
92
111
(
1972
).
8.
T.
Stuyver
,
B.
Chen
,
T.
Zeng
,
P.
Geerlings
,
F.
De Proft
, and
R.
Hoffmann
, “
Do diradicals behave like radicals?
,”
Chem. Rev.
119
,
11291
11351
(
2019
).
9.
X.
Hu
,
W.
Wang
,
D.
Wang
, and
Y.
Zheng
, “
The electronic applications of stable diradicaloids: Present and future
,”
J. Mater. Chem. C
6
,
11232
11242
(
2018
).
10.
T.
Stuyver
,
T.
Zeng
,
Y.
Tsuji
,
P.
Geerlings
, and
F.
De Proft
, “
Diradical character as a guiding principle for the insightful design of molecular nanowires with an increasing conductance with length
,”
Nano Lett.
18
,
7298
7304
(
2018
).
11.
Y.
Tsuji
,
R.
Hoffmann
,
M.
Strange
, and
G. C.
Solomon
, “
Close relation between quantum interference in molecular conductance and diradical existence
,”
Proc. Natl. Acad. Sci. U. S. A.
113
,
E413
E419
(
2016
).
12.
H.
Nagai
,
M.
Nakano
,
K.
Yoneda
,
R.
Kishi
,
H.
Takahashi
,
A.
Shimizu
,
T.
Kubo
,
K.
Kamada
,
K.
Ohta
,
E.
Botek
 et al., “
Signature of multiradical character in second hyperpolarizabilities of rectangular graphene nanoflakes
,”
Chem. Phys. Lett.
489
,
212
218
(
2010
).
13.
P. M.
Lahti
,
Magnetic Properties of Organic Materials
(
CRC Press
,
1999
).
14.
M. B.
Smith
and
J.
Michl
, “
Singlet fission
,”
Chem. Rev.
110
,
6891
6936
(
2010
).
15.
M. B.
Smith
and
J.
Michl
, “
Recent advances in singlet fission
,”
Annu. Rev. Phys. Chem.
64
,
361
386
(
2013
).
16.
D.
Casanova
, “
Theoretical modeling of singlet fission
,”
Chem. Rev.
118
,
7164
7207
(
2018
).
17.
V. V.
Zhivonitko
,
J.
Bresien
,
A.
Schulz
, and
I. V.
Koptyug
, “
Parahydrogen-induced polarization with a metal-free P–P biradicaloid
,”
Phys. Chem. Chem. Phys.
21
,
5890
5893
(
2019
).
18.
E.
Mendez-Vega
,
M.
Maehara
,
A. H.
Raut
,
J.
Mieres-Perez
,
M.
Tsuge
,
Y. P.
Lee
, and
W.
Sander
, “
Activation of molecular hydrogen by arylcarbenes
,”
Chem. - Eur. J.
24
,
18801
18808
(
2018
).
19.
J.
Hinze
, “
MC-SCF. I. The multi-configuration self-consistent-field method
,”
J. Chem. Phys.
59
,
6424
6432
(
1973
).
20.
B. O.
Roos
,
P. R.
Taylor
, and
P. E. M.
Sigbahn
, “
A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach
,”
Chem. Phys.
48
,
157
173
(
1980
).
21.
P.
Siegbahn
,
A.
Heiberg
,
B.
Roos
, and
B.
Levy
, “
A comparison of the super-CI and the Newton-Raphson scheme in the complete active space SCF method
,”
Phys. Scr.
21
,
323
(
1980
).
22.
P. E. M.
Siegbahn
,
J.
Almlöf
,
A.
Heiberg
, and
B. O.
Roos
, “
The complete active space SCF (CASSCF) method in a Newton–Raphson formulation with application to the HNO molecule
,”
J. Chem. Phys.
74
,
2384
2396
(
1981
).
23.
R. J.
Buenker
and
S. D.
Peyerimhoff
, “
Individualized configuration selection in CI calculations with subsequent energy extrapolation
,”
Theor. Chim. Acta
35
,
33
58
(
1974
).
24.
R. J.
Buenker
,
S. D.
Peyerimhoff
, and
W.
Butscher
, “
Applicability of the multi-reference double-excitation CI (MRD-CI) method to the calculation of electronic wavefunctions and comparison with related techniques
,”
Mol. Phys.
35
,
771
791
(
1978
).
25.
E. R.
Davidson
and
C. F.
Bender
, “
Perturbation theory for multiconfiguration reference states
,”
Chem. Phys. Lett.
59
,
369
374
(
1978
).
26.
B.
Kirtman
, “
Simultaneous calculation of several interacting electronic states by generalized Van Vleck perturbation theory
,”
J. Chem. Phys.
75
,
798
808
(
1981
).
27.
B. O.
Roos
,
P.
Linse
,
P. E. M.
Siegbahn
, and
M. R. A.
Blomberg
, “
A simple method for the evaluation of the second-order-perturbation energy from external double-excitations with a CASSCF reference wavefunction
,”
Chem. Phys.
66
,
197
207
(
1982
).
28.
K.
Andersson
, “
Different forms of the zeroth-order Hamiltonian in second-order perturbation theory with a complete active space self-consistent field reference function
,”
Theor. Chim. Acta
91
,
31
46
(
1995
).
29.
K.
Andersson
,
P. A.
Malmqvist
,
B. O.
Roos
,
A. J.
Sadlej
, and
K.
Wolinski
, “
Second-order perturbation theory with a CASSCF reference function
,”
J. Phys. Chem.
94
,
5483
5488
(
1990
).
30.
K.
Andersson
,
P. Å.
Malmqvist
, and
B. O.
Roos
, “
Second-order perturbation theory with a complete active space self-consistent field reference function
,”
J. Chem. Phys.
96
,
1218
1226
(
1992
).
31.
W.
Kohn
and
L. J.
Sham
, “
Self-consistent equations including exchange and correlation effects
,”
Phys. Rev.
140
,
A1133
(
1965
).
32.
W.
Kohn
, “
Nobel lecture: Electronic structure of matter—Wave functions and density functionals
,”
Rev. Mod. Phys.
71
,
1253
(
1999
).
33.
A. J.
Cohen
,
P.
Mori-Sánchez
, and
W.
Yang
, “
Challenges for density functional theory
,”
Chem. Rev.
112
,
289
320
(
2012
).
34.
S.
Ghosh
,
P.
Verma
,
C. J.
Cramer
,
L.
Gagliardi
, and
D. G.
Truhlar
, “
Combining wave function methods with density functional theory for excited states
,”
Chem. Rev.
118
,
7249
7292
(
2018
).
35.
A.
Savin
and
H.-J.
Flad
, “
Density functionals for the Yukawa electron-electron interaction
,”
Int. J. Quantum Chem.
56
,
327
332
(
1995
).
36.
W.
Yang
, “
Generalized adiabatic connection in density functional theory
,”
J. Chem. Phys.
109
,
10107
10110
(
1998
).
37.
R.
Pollet
,
F.
Colonna
,
T.
Leininger
,
H.
Stoll
,
H.-J.
Werner
, and
A.
Savin
, “
Exchange-correlation energies and correlation holes for some two- and four-electron atoms along a nonlinear adiabatic connection in density functional theory
,”
Int. J. Quantum Chem.
91
,
84
93
(
2003
).
38.
A.
Savin
,
F.
Colonna
, and
R.
Pollet
, “
Adiabatic connection approach to density functional theory of electronic systems
,”
Int. J. Quantum Chem.
93
,
166
190
(
2003
).
39.
J.
Toulouse
,
F.
Colonna
, and
A.
Savin
, “
Long-range–short-range separation of the electron-electron interaction in density-functional theory
,”
Phys. Rev. A
70
,
062505
(
2004
).
40.
E.
Rebolini
,
J.
Toulouse
,
A. M.
Teale
,
T.
Helgaker
, and
A.
Savin
, “
Excitation energies along a range-separated adiabatic connection
,”
J. Chem. Phys.
141
,
044123
(
2014
).
41.
T.
Leininger
,
H.
Stoll
,
H.-J.
Werner
,
A.
Savin
 et al., “
Combining long-range configuration interaction with short-range density functionals
,”
Chem. Phys. Lett.
275
,
151
160
(
1997
).
42.
R.
Pollet
,
A.
Savin
,
T.
Leininger
, and
H.
Stoll
, “
Combining multideterminantal wave functions with density functionals to handle near-degeneracy in atoms and molecules
,”
J. Chem. Phys.
116
,
1250
1258
(
2002
).
43.
P.
Gori-Giorgi
and
A.
Savin
, “
Properties of short-range and long-range correlation energy density functionals from electron-electron coalescence
,”
Phys. Rev. A
73
,
032506
(
2006
).
44.
E.
Fromager
,
J.
Toulouse
, and
H. J. A.
Jensen
, “
On the universality of the long-/short-range separation in multiconfigurational density-functional theory
,”
J. Chem. Phys.
126
,
074111
(
2007
).
45.
E.
Fromager
,
F.
Réal
,
P.
Wåhlin
,
U.
Wahlgren
, and
H. J. A.
Jensen
, “
On the universality of the long-/short-range separation in multiconfigurational density-functional theory. II. Investigating f0 actinide species
,”
J. Chem. Phys.
131
,
054107
(
2009
).
46.
D.
Casanova
, “
Short-range density functional correlation within the restricted active space CI method
,”
J. Chem. Phys.
148
,
124118
(
2018
).
47.
J. G.
Angyán
,
I. C.
Gerber
,
A.
Savin
, and
J.
Toulouse
, “
van der Waals forces in density functional theory: Perturbational long-range electron-interaction corrections
,”
Phys. Rev. A
72
,
012510
(
2005
).
48.
E.
Goll
,
H.-J.
Werner
, and
H.
Stoll
, “
A short-range gradient-corrected density functional in long-range coupled-cluster calculations for rare gas dimers
,”
Phys. Chem. Chem. Phys.
7
,
3917
3923
(
2005
).
49.
E.
Fromager
and
H. J. A.
Jensen
, “
Self-consistent many-body perturbation theory in range-separated density-functional theory: A one-electron reduced-density-matrix-based formulation
,”
Phys. Rev. A
78
,
022504
(
2008
).
50.
E.
Fromager
,
R.
Cimiraglia
, and
H. J. A.
Jensen
, “
Merging multireference perturbation and density-functional theories by means of range separation: Potential curves for Be2, Mg2, and Ca2
,”
Phys. Rev. A
81
,
024502
(
2010
).
51.
E.
Fromager
and
H. J. A.
Jensen
, “
Analysis of self-consistency effects in range-separated density-functional theory with Møller-Plesset perturbation theory
,”
J. Chem. Phys.
135
,
034116
(
2011
).
52.
C.
Angeli
,
R.
Cimiraglia
, and
J.-P.
Malrieu
, “
n-electron valence state perturbation theory: A spinless formulation and an efficient implementation of the strongly contracted and of the partially contracted variants
,”
J. Chem. Phys.
117
,
9138
9153
(
2002
).
53.
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
).
54.
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
).
55.
D.
Casanova
, “
Efficient implementation of restricted active space configuration interaction with the hole and particle approximation
,”
J. Comput. Chem.
34
,
720
730
(
2013
).
56.
D.
Casanova
, “
Second-order perturbative corrections to the restricted active space configuration interaction with the hole and particle approach
,”
J. Chem. Phys.
140
,
144111
(
2014
).
57.
S. H.
Vosko
,
L.
Wilk
, and
M.
Nusair
, “
Accurate spin-dependent electron liquid correlation energies for local spin density calculations: A critical analysis
,”
Can. J. Phys.
58
,
1200
1211
(
1980
).
58.
S.
Paziani
,
S.
Moroni
,
P.
Gori-Giorgi
, and
G. B.
Bachelet
, “
Local-spin-density functional for multideterminant density functional theory
,”
Phys. Rev. B
73
,
155111
(
2006
).
59.
E.
Goll
,
H.-J.
Werner
,
H.
Stoll
,
T.
Leininger
,
P.
Gori-Giorgi
, and
A.
Savin
, “
A short-range gradient-corrected spin density functional in combination with long-range coupled-cluster methods: Application to alkali-metal rare-gas dimers
,”
Chem. Phys.
329
,
276
282
(
2006
).
60.
J.
Toulouse
,
A.
Savin
, and
H.-J.
Flad
, “
Short-range exchange-correlation energy of a uniform electron gas with modified electron–electron interaction
,”
Int. J. Quantum Chem.
100
,
1047
1056
(
2004
).
61.
J. K.
Pedersen
, “
Description of correlation and relativistic effects in calculations of molecular properties
,”
Ph.D. thesis
,
University of Southern Denmark
,
2004
.
62.
M.
Fuchs
,
Y.-M.
Niquet
,
X.
Gonze
, and
K.
Burke
, “
Describing static correlation in bond dissociation by Kohn–Sham density functional theory
,”
J. Chem. Phys.
122
,
094116
(
2005
).
63.
W.
Heitler
and
F.
London
, “
Interaction between neutral atoms and homopolar binding according to quantum mechanics
,” in
Quantum Chemistry: Classic Scientific Papers
(
World Scientific
,
2000
), pp.
140
155
.
64.
Á. J.
Pérez-Jiménez
,
J. M.
Pérez-Jordá
, and
F.
Illas
, “
Density functional theory with alternative spin densities: Application to magnetic systems with localized spins
,”
J. Chem. Phys.
120
,
18
25
(
2004
).
65.
M.-C.
Kim
,
E.
Sim
, and
K.
Burke
, “
Understanding and reducing errors in density functional calculations
,”
Phys. Rev. Lett.
111
,
073003
(
2013
).
66.
L.
Hedin
, “
New method for calculating the one-particle Green’s function with application to the electron-gas problem
,”
Phys. Rev.
139
,
A796
(
1965
).
67.
L.
Hedin
, “
On correlation effects in electron spectroscopies and the GW approximation
,”
J. Phys.: Condens. Matter
11
,
R489
(
1999
).
68.
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
2192
(
1988
).
69.
P. Å.
Malmqvist
,
A.
Rendell
, and
B. O.
Roos
, “
The restricted active space self-consistent-field method, implemented with a split graph unitary group approach
,”
J. Phys. Chem.
94
,
5477
5482
(
1990
).
70.
D.
Casanova
and
A. I.
Krylov
, “
Spin-flip methods in quantum chemistry
,”
Phys. Chem. Chem. Phys.
22
,
4326
4342
(
2020
).
71.
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
).
72.
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.
129
,
16430
16435
(
2017
).
73.
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
).
74.
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
).
75.
C.
Liu
,
M. E.
Sandoval-Salinas
,
Y.
Hong
,
T. Y.
Gopalakrishna
,
H.
Phan
,
N.
Aratani
,
T. S.
Herng
,
J.
Ding
,
H.
Yamada
,
D.
Kim
 et al., “
Macrocyclic polyradicaloids with unusual super-ring structure and global aromaticity
,”
Chem
4
,
1586
1595
(
2018
).
76.
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
).
77.
D.
Casanova
and
A. I.
Krylov
, “
Quantifying local exciton, charge resonance, and multiexciton character in correlated wave functions of multichromophoric systems
,”
J. Chem. Phys.
144
,
014102
(
2016
).
78.
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
).
79.
D.
Casanova
, “
Electronic structure study of singlet fission in tetracene derivatives
,”
J. Chem. Theory Comput.
10
,
324
334
(
2014
).
80.
X.
Feng
,
D.
Casanova
, and
A. I.
Krylov
, “
Intra- and intermolecular singlet fission in covalently linked dimers
,”
J. Phys. Chem. C
120
,
19070
19077
(
2016
).
81.
M. E.
Sandoval-Salinas
,
A.
Carreras
,
J.
Casado
, and
D.
Casanova
, “
Singlet fission in spiroconjugated dimers
,”
J. Chem. Phys.
150
,
204306
(
2019
).
82.
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
).
83.
M.
Hubert
,
H. J. A.
Jensen
, and
E. D.
Hedegård
, “
Excitation spectra of nucleobases with multiconfigurational density functional theory
,”
J. Phys. Chem. A
120
,
36
43
(
2016
).
84.
Y.
Shao
,
Z.
Gan
,
E.
Epifanovsky
,
A. T.
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
).
85.
M.
Head-Gordon
, “
Characterizing unpaired electrons from the one-particle density matrix
,”
Chem. Phys. Lett.
372
,
508
511
(
2003
).
86.
R. M.
Martin
,
Electronic Structure: Basic Theory and Practical Methods
(
Cambridge University Press
,
2004
).
87.
G.
Onida
,
L.
Reining
, and
A.
Rubio
, “
Electronic excitations: Density-functional versus many-body Green’s-function approaches
,”
Rev. Mod. Phys.
74
,
601
659
(
2002
).
88.
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 a triple-zeta plus double polarization (TZ2P) basis
,”
J. Chem. Phys.
108
,
1040
1049
(
1998
).
89.
J. C.
Stephens
,
Y.
Yamaguchi
,
C. D.
Sherrill
, and
H. F.
Schaefer
, “
.X̃3B1,ã1A1,b̃1B1,c̃1Σg+ electronic states of NH2
,”
J. Phys. Chem. A
102
,
3999
4006
(
1998
).
90.
Y.
Yamaguchi
,
T. J.
Van Huis
,
C. D.
Sherrill
, and
H. F.
Schaefer
 III
, “
The x̃1A1,ã3B1,ã1B̃1, and B̃1A1 electronic states of SiH2
,”
Theor. Chem. Acc.
97
,
341
349
(
1997
).
91.
T. J.
Van Huis
,
Y.
Yamaguchi
,
C. D.
Sherrill
, and
H. F.
Schaefer
, “
.X̃1A1,ã3B1,Ã1B1, and B̃1A11A1 electronic states of PH2+.
,”
J. Phys. Chem. A
101
,
6955
6963
(
1997
).
92.
L. V.
Slipchenko
and
A. I.
Krylov
, “
Singlet-triplet gaps in diradicals by the spin-flip approach: A benchmark study
,”
J. Chem. Phys.
117
,
4694
4708
(
2002
).
93.
P.
Jensen
and
P. R.
Bunker
, “
The potential surface and stretching frequencies of X̃3B1 methylene (CH2) determined from experiment using the morse oscillator-rigid bender internal dynamics Hamiltonian
,”
J. Chem. Phys.
89
,
1327
1332
(
1988
).
94.
Y.
Yamaguchi
,
C. D.
Sherrill
, and
H. F.
Schaefer
, “
The X̃3B1,ã1A1,b̃1B1, and c̃1A1 electronic states of CH2
,”
J. Phys. Chem.
100
,
7911
7918
(
1996
).
95.
S. T.
Gibson
,
J. P.
Greene
, and
J.
Berkowitz
, “
Photoionization of the amidogen radical
,”
J. Chem. Phys.
83
,
4319
4328
(
1985
).
96.
J.
Berkowitz
,
J. P.
Greene
,
H.
Cho
, and
B.
Ruščić
, “
Photoionization mass spectrometric studies of SiHn (n = 1–4)
,”
J. Chem. Phys.
86
,
1235
1248
(
1987
).
97.
R.
Escribano
and
A.
Campargue
, “
Absorption spectroscopy of SiH2 near 640 nm
,”
J. Chem. Phys.
108
,
6249
6257
(
1998
).
98.
J.
Berkowitz
and
H.
Cho
, “
A photoionization study of PH: PH2 revisited
,”
J. Chem. Phys.
90
,
1
6
(
1989
).
99.
A.-M. C.
Cristian
,
Y.
Shao
, and
A. I.
Krylov
, “
Bonding patterns in benzene triradicals from structural, spectroscopic, and thermochemical perspectives
,”
J. Phys. Chem. A
108
,
6581
6588
(
2004
).
100.
N.
Orms
,
D. R.
Rehn
,
A.
Dreuw
, and
A. I.
Krylov
, “
Characterizing bonding patterns in diradicals and triradicals by density-based wave function analysis: A uniform approach
,”
J. Chem. Theory Comput.
14
,
638
648
(
2018
).
101.
P. G.
Wenthold
,
R. R.
Squires
, and
W. C.
Lineberger
, “
Ultraviolet photoelectron spectroscopy of the o-, m-, and p-benzyne negative ions. Electron affinities and singlet-triplet splittings for o-, m-, and p-benzyne
,”
J. Am. Chem. Soc.
120
,
5279
5290
(
1998
).
102.
P.
Dowd
, “
Trimethylenemethane
,”
Acc. Chem. Res.
5
,
242
248
(
1972
).
103.
D. H.
Ess
,
E. R.
Johnson
,
X.
Hu
, and
W.
Yang
, “
Singlet-triplet energy gaps for diradicals from fractional-spin density-functional theory
,”
J. Phys. Chem. A
115
,
76
83
(
2011
).
104.
L. V.
Slipchenko
and
A. I.
Krylov
, “
Electronic structure of the trimethylenemethane diradical in its ground and electronically excited states: Bonding, equilibrium geometries, and vibrational frequencies
,”
J. Chem. Phys.
118
,
6874
6883
(
2003
).
105.
H.
Dong
,
D. A.
Hrovat
,
H.
Quast
, and
W. T.
Borden
, “
Calculations of the relative energies of the low-lying electronic states of 2-methylenedihydrophenalene-1,3-diyl: Effects of a 1,8-naphtho bridging group on trimethylenemethane and of a vinylidene bridging group on 1,8-naphthoquinodimethane
,”
J. Phys. Chem. A
113
,
895
901
(
2009
).
106.
C. J.
Cramer
and
B. A.
Smith
, “
Trimethylenemethane. Comparison of multiconfiguration self-consistent field and density functional methods for a non-Kekulé hydrocarbon
,”
J. Phys. Chem.
100
,
9664
9670
(
1996
).
107.
D. R.
Yarkony
and
H. F.
Schaefer
 III
, “
Triplet electronic ground state of trimethylenemethane
,”
J. Am. Chem. Soc.
96
,
3754
3758
(
1974
).
108.
S. B.
Auster
,
R. M.
Pitzer
, and
M. S.
Platz
, “
Excitation energies in trimethylenemethane derivatives
,”
J. Am. Chem. Soc.
104
,
3812
3815
(
1982
).
109.
X.
Li
and
J.
Paldus
, “
Electronic structure of organic diradicals: Evaluation of the performance of coupled-cluster methods
,”
J. Chem. Phys.
129
,
174101
(
2008
).
110.
D.
Casanova
,
L. V.
Slipchenko
,
A. I.
Krylov
, and
M.
Head-Gordon
, “
Double spin-flip approach within equation-of-motion coupled cluster and configuration interaction formalisms: Theory, implementation, and examples
,”
J. Chem. Phys.
130
,
044103
(
2009
).
111.
D. S.
Ginter
,
M. L.
Ginter
, and
K. K.
Innes
, “
Electronic spectra of the Ga2, In2, and Tl2 molecules
,”
J. Phys. Chem.
69
,
2480
2483
(
1965
).
112.
K.
Balasubramanian
, “
Spectroscopic constants and potential energy curves of gallium molecules (Ga2, Ga2, and Ga2+)
,”
J. Phys. Chem.
94
,
7764
7768
(
1990
).
113.
K. K.
Das
, “
Ab initio MRD-CI study of the electronic states of the gallium dimer
,”
J. Phys. B: At., Mol. Opt. Phys.
30
,
803
(
1997
).
114.
G.
Balducci
,
G.
Gigli
, and
G.
Meloni
, “
Dissociation energies of the Ga2, In2, and GaIn molecules
,”
J. Chem. Phys.
109
,
4384
4388
(
1998
).
115.
T. K.
Ghosh
,
K.
Tanaka
, and
Y.
Mochizuki
, “
Theoretical study of the spectroscopic constants of low-lying states of Ga2
,”
J. Mol. Struct.: THEOCHEM
451
,
61
71
(
1998
).
116.
X.
Tan
and
P. J.
Dagdigian
, “
Electronic spectrum of the gallium dimer
,”
J. Phys. Chem. A
107
,
2642
2649
(
2003
).
117.
A.
Köhn
,
H.-J.
Himmel
, and
B.
Gaertner
, “
Why does a Ga2 dimer react spontaneously with H2, but a Ga atom does not?—A detailed quantum chemical investigation of the differences in reactivity between Ga atoms and Ga2 dimers, in combination with experimental results
,”
Chem. - Eur. J.
9
,
3909
3919
(
2003
).
118.
N.
Gaston
and
A. J.
Parker
, “
On the bonding of Ga2, structures of Gan clusters and the relation to the bulk structure of gallium
,”
Chem. Phys. Lett.
501
,
375
378
(
2011
).
119.
R.
Tonner
and
N.
Gaston
, “
The dimeric nature of bonding in gallium: From small clusters to the α-gallium phase
,”
Phys. Chem. Chem. Phys.
16
,
24244
24249
(
2014
).
120.
N.
Drebov
,
F.
Weigend
, and
R.
Ahlrichs
, “
Structures and properties of neutral gallium clusters: A theoretical investigation
,”
J. Chem. Phys.
135
,
044314
(
2011
).
121.
Z. L.
Xiao
,
R. H.
Hauge
, and
J. L.
Margrave
, “
Cryogenic reactions of gallium with molecular hydrogen and methane
,”
Inorg. Chem.
32
,
642
646
(
1993
).
122.
H.-J.
Himmel
,
L.
Manceron
,
A. J.
Downs
, and
P.
Pullumbi
, “
Formation and characterization of the gallium and indium subhydride molecules Ga2H2 and In2H2: A matrix isolation study
,”
J. Am. Chem. Soc.
124
,
4448
4457
(
2002
).
123.
P.
Pullumbi
,
C.
Mijoule
,
L.
Manceron
, and
Y.
Bouteiller
, “
Aluminium, gallium and indium dihydrides. An IR matrix isolation and ab initio study
,”
Chem. Phys.
185
,
13
24
(
1994
).
124.
L. B.
Knight
, Jr.
,
J. J.
Banisaukas
 III
,
R.
Babb
, and
E. R.
Davidson
, “
Electron spin resonance matrix isolation and ab initio theoretical investigations of 69,71GaH2, 69,71GaD2, H69,71GaCH3, and D69,71GaCD3
,”
J. Chem. Phys.
105
,
6607
6615
(
1996
).
125.
F.
Weigend
and
R.
Ahlrichs
, “
Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy
,”
Phys. Chem. Chem. Phys.
7
,
3297
3305
(
2005
).
126.
Y.
Zhao
,
W.
Xu
,
Q.
Li
,
Y.
Xie
, and
H. F.
Schaefer
, “
Gallium clusters Gan (n = 1−6): Structures, thermochemistry, and electron affinities
,”
J. Phys. Chem. A
108
,
7448
7459
(
2004
).
127.
B. O.
Roos
,
R.
Lindh
,
P.-Å.
Malmqvist
,
V.
Veryazov
, and
P.-O.
Widmark
, “
Main group atoms and dimers studied with a new relativistic ANO basis set
,”
J. Phys. Chem. A
108
,
2851
2858
(
2004
).

Supplementary Material

You do not currently have access to this content.