Broken symmetry solutions—solutions to the independent particle model that do not obey all symmetries required by the Hamiltonian—have attracted significant interest for capturing multireference properties with mean-field scaling. However, identification and optimization of broken-symmetry solutions is difficult owing to the non-linear nature of the self-consistent field (SCF) equations, particularly for solutions belonging to low-symmetry subgroups and where multiple broken symmetry solutions are sought. Linearization of SCF solution space results in the Lie algebra, which this work utilizes as a framework for elucidation of the set of solutions that exist at the desired symmetry. To demonstrate that searches constructed in the Lie algebra yield the set of broken symmetry solutions, a grid-based search of real-restricted, real-unrestricted, complex-restricted, paired-unrestricted, and real-general solutions of the C2v (nearly D4h) H4 molecule is performed.

1.
C. A.
Jiménez-Hoyos
,
T. M.
Henderson
,
T.
Tsuchimochi
, and
G.
Scuseria
,
J. Chem. Phys.
136
,
164109
(
2012
).
2.
I.
Mayer
,
Chem. Phys. Lett.
11
,
397
(
1971
).
3.
P.
Pulay
and
T. P.
Hamilton
,
J. Chem. Phys.
88
,
4926
(
1988
).
4.
L. M.
Thompson
and
H. P.
Hratchian
,
J. Chem. Phys.
142
,
054106
(
2015
).
5.
L. M.
Thompson
and
H. P.
Hratchian
,
J. Chem. Phys.
141
,
034108
(
2014
).
6.
A. J. W.
Thom
and
M.
Head-Gordon
,
Phys. Rev. Lett.
101
,
193001
(
2008
).
7.
K.
Kowalski
and
K.
Jankowski
,
Phys. Rev. Lett.
81
,
1195
(
1998
).
8.
H.
Fukutome
,
Prog. Theor. Phys.
52
,
115
(
1974
).
9.
D. W.
Small
,
E. J.
Sundstrom
, and
M.
Head-Gordon
,
J. Chem. Phys.
142
,
024104
(
2015
).
10.
C. A.
Jiménez-Hoyos
,
R.
Rodríguez-Guzmán
, and
G.
Scuseria
,
J. Phys. Chem. A
118
,
9925
(
2014
).
11.
H. B.
Schlegel
and
J. J. W.
McDouall
, “
Do you have SCF stability and convergence problems?
,” in
Computational Advances in Organic Chemistry: Molecular Structure and Reactivity
, edited by
C.
Ögretir
and
I. G.
Csizmadia
(
Kluwer Academic Publishers
,
Dordrecht
,
1991
), pp.
167
185
.
12.
J. J.
Goings
,
F.
Ding
,
M. J.
Frisch
, and
X.
Li
,
J. Chem. Phys.
142
,
154109
(
2015
).
13.
Z.
Tóth
and
P.
Pulay
,
J. Chem. Phys.
145
,
164102
(
2016
).
14.
L. M.
Thompson
and
H. P.
Hratchian
,
J. Phys. Chem. A
119
,
8744
(
2015
).
15.
L. M.
Thompson
,
C. C.
Jarrold
, and
H. P.
Hratchian
,
J. Chem. Phys.
146
,
104301
(
2017
).
16.
K. T.
Jensen
,
R. L.
Benson
,
S.
Cardamone
, and
A. J. W.
Thom
,
J. Chem. Theory Comput.
14
,
4629
(
2018
).
17.
L. C.
Jake
,
T. M.
Henderson
, and
G.
Scuseria
,
J. Chem. Phys.
148
,
024109
(
2018
).
18.
K. J.
Oosterbaan
,
A. F.
White
, and
M.
Head-Gordon
,
J. Chem. Phys.
149
,
044116
(
2018
).
19.
H.
Fukutome
,
Prog. Theor. Phys.
45
,
1382
(
1971
).
20.
D. J.
Thouless
,
Nucl. Phys.
21
,
225
(
1960
).
21.
J. L.
Stuber
and
J.
Paldus
, “
Symmetry breaking in the independent particle model
,” in
Fundamental World of Quantum Chemistry: A Tribute to the Memory of Per-Olov Löwdin
, edited by
E.
Brandas
and
E. S.
Kryachko
(
Kluwer Academic Publishers
,
2003
), pp.
67
140
.
22.
A.
Yershova
,
S.
Jain
,
S. M.
LaValle
, and
J. C.
Mitchell
,
Int. J. Rob. Res.
29
,
801
(
2010
).
23.
J. C.
Mitchell
,
SIAM J. Sci. Comput.
30
,
525
(
2008
).
24.
F.
Neese
,
J. Phys. Chem. Solids
65
,
781
(
2004
).
25.
G. M. J.
Barca
,
A. T. B.
Gilbert
, and
P. M. W.
Gill
,
J. Chem. Theory Comput.
14
,
1501
(
2018
).
26.
L. M.
Thompson
,
X.
Sheng
,
A.
Mahler
,
D.
Mullally
, and
H. P.
Hratchian
(
2018
). “
MQCpack v0.1-beta
,” Zenodo.
27.
M. J.
Frisch
,
G. W.
Trucks
,
H. B.
Schlegel
,
G. E.
Scuseria
,
M. A.
Robb
,
J. R.
Cheeseman
,
G.
Scalmani
,
V.
Barone
,
G. A.
Petersson
,
H.
Nakatsuji
,
X.
Li
,
M.
Caricato
,
A. V.
Marenich
,
J.
Bloino
,
B. G.
Janesko
,
R.
Gomperts
,
B.
Mennucci
,
H. P.
Hratchian
,
J. V.
Ortiz
,
A. F.
Izmaylov
,
J. L.
Sonnenberg
,
D.
Williams-Young
,
F.
Ding
,
F.
Lipparini
,
F.
Egidi
,
J.
Goings
,
B.
Peng
,
A.
Petrone
,
T.
Henderson
,
D.
Ranasinghe
,
V. G.
Zakrzewski
,
J.
Gao
,
N.
Rega
,
G.
Zheng
,
W.
Liang
,
M.
Hada
,
M.
Ehara
,
K.
Toyota
,
R.
Fukuda
,
J.
Hasegawa
,
M.
Ishida
,
T.
Nakajima
,
Y.
Honda
,
O.
Kitao
,
H.
Nakai
,
T.
Vreven
,
K.
Throssell
,
J. A.
Montgomery
, Jr.
,
J. E.
Peralta
,
F.
Ogliaro
,
M. J.
Bearpark
,
J. J.
Heyd
,
E. N.
Brothers
,
K. N.
Kudin
,
V. N.
Staroverov
,
T. A.
Keith
,
R.
Kobayashi
,
J.
Normand
,
K.
Raghavachari
,
A. P.
Rendell
,
J. C.
Burant
,
S. S.
Iyengar
,
J.
Tomasi
,
M.
Cossi
,
J. M.
Millam
,
M.
Klene
,
C.
Adamo
,
R.
Cammi
,
J. W.
Ochterski
,
R. L.
Martin
,
K.
Morokuma
,
O.
Farkas
,
J. B.
Foresman
, and
D. J.
Fox
, gaussian 16, Revision A.03,
Gaussian, Inc.
,
Wallingford, CT
,
2016
.
28.
S.
Huzinaga
,
J. Chem. Phys.
42
,
1293
(
1965
).
29.
T. M.
Henderson
,
C. A.
Jiménez-Hoyos
, and
G.
Scuseria
,
J. Chem. Theory Comput.
14
,
649
(
2018
).
30.
D. J.
Wales
and
J. P. K.
Doye
,
J. Phys. Chem. A
101
,
5111
(
1997
).
31.
J.
Paldus
,
J. Chem. Phys.
61
,
5321
(
1974
).
32.
M. V.
Berry
,
Proc. R. Soc. A
392
,
45
(
1984
).
33.
H.
Fukutome
,
Int. J. Quantum Chem.
20
,
955
(
1981
).
34.
T.
Helgaker
,
P.
Jørgensen
, and
J.
Olsen
,
Molecular Electronic-Structure Theory
(
Wiley
,
2012
).

Supplementary Material

You do not currently have access to this content.