In this work, we formulate the equations of motion corresponding to the Hermitian operator method in the framework of the doubly occupied configuration interaction space. The resulting algorithms turn out to be considerably simpler than the equations provided by that method in more conventional spaces, enabling the determination of excitation energies in N-electron systems under an affordable polynomial computational cost. The implementation of this technique only requires to know the elements of low-order reduced density matrices of an N-electron reference state, which can be obtained from any approximate method. We contrast our procedure against the reduced Bardeen–Cooper–Schrieffer and Richardson–Gaudin–Kitaev integrable models, pointing out the reliability of our proposal.
REFERENCES
1.
2.
3.
4.
J. E.
Mayer
, Phys. Rev.
100
, 1579
(1955
).5.
Reduced-Density-Matrix Mechanics: With Applications to Many-Electron Atoms and Molecules
, Advances in Chemical Physics Vol. 134
, edited by D. A.
Mazziotti
(John Wiley & Sons, Inc.
, 2007
).6.
C.
Garrod
, M. V.
Mihailović
, and M.
Rosina
, J. Math. Phys.
16
, 868
(1975
).7.
M. V.
Mihailović
and M.
Rosina
, Nucl. Phys. A
237
, 221
(1975
).8.
J.
Karwowski
, W.
Duch
, and C.
Valdemoro
, Phys. Rev. A
33
, 2254
(1986
).9.
L.
Lain
, A.
Torre
, J.
Karwowski
, and C.
Valdemoro
, Phys. Rev. A
38
, 2721
(1988
).10.
R. J.
Bartlett
, J. Phys. Chem.
93
, 1697
(1989
).11.
R.
Parr
, Density-Functional Theory of Atoms and Molecules
(Oxford University Press, Clarendon Press
, New York, Oxford
, 1989
).12.
D. A.
Mazziotti
, Phys. Rev. A
57
, 4219
(1998
).13.
C.
Valdemoro
, L. M.
Tel
, E.
Perez-Romero
, and A.
Torre
, J. Mol. Struct.: THEOCHEM
537
, 1
(2001
).14.
M.
Nakata
, H.
Nakatsuji
, M.
Ehara
, M.
Fukuda
, K.
Nakata
, and K.
Fujisawa
, J. Chem. Phys.
114
, 8282
(2001
).15.
D. A.
Mazziotti
, Phys. Rev. A
65
, 062511
(2002
).16.
Z.
Zhao
, B. J.
Braams
, M.
Fukuda
, M. L.
Overton
, and J. K.
Percus
, J. Chem. Phys.
120
, 2095
(2004
).17.
J. R.
Hammond
and D. A.
Mazziotti
, Phys. Rev. A
73
, 062505
(2006
).18.
B.
Verstichel
, H.
van Aggelen
, D.
Van Neck
, P.
Bultinck
, and S.
De Baerdemacker
, Comput. Phys. Commun.
182
, 1235
(2011
).19.
D. A.
Mazziotti
, Chem. Rev.
112
, 244
(2012
).20.
B.
Verstichel
, H.
van Aggelen
, W.
Poelmans
, S.
Wouters
, and D.
Van Neck
, Comput. Theor. Chem.
1003
, 12
(2013
).21.
J. S. M.
Anderson
, M.
Nakata
, R.
Igarashi
, K.
Fujisawa
, and M.
Yamashita
, Comput. Theor. Chem.
1003
, 22
(2013
).22.
A. J.
Coleman
, Rev. Mod. Phys.
35
, 668
(1963
).23.
C.
Garrod
and J. K.
Percus
, J. Math. Phys.
5
, 1756
–1776
(1964
).24.
A. J.
Coleman
and V. I.
Yukalov
, Reduced Density Matrices: Coulson’s Challenge
(Springer-Verlag
, New York
, 2000
).25.
D. A.
Mazziotti
, Phys. Rev. A
72
, 032510
(2005
).26.
D. A.
Mazziotti
, Phys. Rev. A
85
, 062507
(2012
).27.
F.
Weinhold
and E. B.
Wilson
, J. Chem. Phys.
46
, 2752
(1967
).28.
F.
Weinhold
and E. B.
Wilson
, J. Chem. Phys.
47
, 2298
(1967
).29.
T.
Matsubara
and H.
Matsuda
, Prog. Theor. Phys.
16
, 569
(1956
).30.
H.
Matsuda
and T.
Matsubara
, Prog. Theor. Phys.
17
, 19
(1957
).31.
W.
Poelmans
, M.
Van Raemdonck
, B.
Verstichel
, S.
De Baerdemacker
, A.
Torre
, L.
Lain
, G. E.
Massaccesi
, D. R.
Alcoba
, P.
Bultinck
, and D.
Van Neck
, J. Chem. Theory Comput.
11
, 4064
(2015
).32.
L.
Bytautas
, T. M.
Henderson
, C. A.
Jiménez-Hoyos
, J. K.
Ellis
, and G. E.
Scuseria
, J. Chem. Phys.
135
, 044119
(2011
).33.
P. A.
Limacher
, T. D.
Kim
, P. W.
Ayers
, P. A.
Johnson
, S.
De Baerdemacker
, D.
Van Neck
, and P.
Bultinck
, Mol. Phys.
112
, 853
(2014
).34.
M.
Van Raemdonck
, D. R.
Alcoba
, W.
Poelmans
, S.
De Baerdemacker
, A.
Torre
, L.
Lain
, G. E.
Massaccesi
, D.
Van Neck
, and P.
Bultinck
, J. Chem. Phys.
143
, 104106
(2015
).35.
D. R.
Alcoba
, A.
Torre
, L.
Lain
, G. E.
Massaccesi
, O. B.
Oña
, E. M.
Honoré
, W.
Poelmans
, D.
Van Neck
, P.
Bultinck
, and S.
De Baerdemacker
, J. Chem. Phys.
148
, 024105
(2018
).36.
D. R.
Alcoba
, P.
Capuzzi
, A.
Rubio-García
, J.
Dukelsky
, G. E.
Massaccesi
, O. B.
Oña
, A.
Torre
, and L.
Lain
, J. Chem. Phys.
149
, 194105
(2018
).37.
G. E.
Massaccesi
, A.
Rubio-García
, P.
Capuzzi
, E.
Ríos
, O. B.
Oña
, J.
Dukelsky
, L.
Lain
, A.
Torre
, and D. R.
Alcoba
, J. Stat. Mech.: Theory Exp.
2021
, 013110
.38.
D. R.
Alcoba
, O. B.
Oña
, L.
Lain
, A.
Torre
, P.
Capuzzi
, G. E.
Massaccesi
, E.
Ríos
, A.
Rubio-García
, and J.
Dukelsky
, J. Chem. Phys.
154
, 224104
(2021
).39.
M.
Bouten
, P.
Van Leuven
, M.
Mihailović
, and M.
Rosina
, Nucl. Phys. A
202
, 127
(1973
).40.
M.
Bouten
, P.
van Leuven
, M.
Mihailović
, and M.
Rosina
, Nucl. Phys. A
221
, 173
(1974
).41.
M.
Rosina
, Int. J. Quantum Chem.
13
, 737
(1978
).42.
D. J.
Rowe
, Rev. Mod. Phys.
40
, 153
(1968
).43.
D. A.
Mazziotti
, Phys. Rev. A
68
, 052501
(2003
).44.
C.
Valdemoro
, D. R.
Alcoba
, L. M.
Tel
, and E.
Pérez-Romero
, Int. J. Quantum Chem.
111
, 245
(2011
).45.
C.
Valdemoro
, D. R.
Alcoba
, O. B.
Oña
, L. M.
Tel
, and E.
Pérez-Romero
, J. Math. Chem.
50
, 492
(2012
).46.
S.
Hemmatiyan
, M.
Sajjan
, A. W.
Schlimgen
, and D. A.
Mazziotti
, J. Phys. Chem. Lett.
9
, 5373
(2018
).47.
R.
Dutta
, T. M.
Henderson
, and G. E.
Scuseria
, J. Chem. Theory Comput.
16
, 6358
(2020
).48.
J.
Bardeen
, L. N.
Cooper
, and J. R.
Schrieffer
, Phys. Rev.
108
, 1175
(1957
).49.
R. W.
Richardson
, Phys. Rev.
141
, 949
–956
(1966
).50.
J.
von Delft
, A. D.
Zaikin
, D. S.
Golubev
, and W.
Tichy
, Phys. Rev. Lett.
77
, 3189
(1996
).51.
A.
Rubio-García
, D. R.
Alcoba
, P.
Capuzzi
, and J.
Dukelsky
, J. Chem. Theory Comput.
14
, 4183
(2018
).52.
A. Y.
Kitaev
, Phys.-Usp.
44
, 131
(2001
).53.
M.
Ibañez
, J.
Links
, G.
Sierra
, and S.
Zhao
, Phys. Rev. B
79
, 180501(R)
(2009
).54.
S. M. A.
Rombouts
, J.
Dukelsky
, and G.
Ortiz
, Phys. Rev. B
82
, 224510
(2010
).55.
M.
Van Raemdonck
, S.
De Baerdemacker
, and D.
Van Neck
, Phys. Rev. B
89
, 155136
(2014
).56.
G.
Ortiz
, J.
Dukelsky
, E.
Cobanera
, C.
Esebbag
, and C.
Beenakker
, Phys. Rev. Lett.
113
, 267002
(2014
).57.
P. R.
Surjan
, Second Quantized Approach to Quantum Chemistry. An Elementary Introduction
(Springer-Verlag
, Berlin
, 1989
).58.
J.
Dukelsky
, C.
Esebbag
, and P.
Schuck
, Phys. Rev. Lett.
87
, 066403
(2001
).59.
J.
Dukelsky
, S.
Pittel
, and G.
Sierra
, Rev. Mod. Phys.
76
, 643
(2004
).60.
G.
Ortiz
, R.
Somma
, J.
Dukelsky
, and S.
Rombouts
, Nucl. Phys. B
707
, 421
(2005
).61.
J. A.
Gomez
, T. M.
Henderson
, and G. E.
Scuseria
, J. Chem. Phys.
150
, 144108
(2019
).62.
J.
Dukelsky
and G.
Sierra
, Phys. Rev. B
61
, 12302
(2000
).63.
H.-Q.
Zhou
, J.
Links
, R. H.
McKenzie
, and M. D.
Gould
, Phys. Rev. B
65
, 060502(R)
(2002
).64.
J.
Dukelsky
, G. G.
Dussel
, J. G.
Hirsch
, and P.
Schuck
, Nucl. Phys. A
714
, 63
(2003
).65.
O.
El Araby
, V.
Gritsev
, and A.
Faribault
, Phys. Rev. B
85
, 115130
(2012
).66.
M.
Henderson
, G. E.
Scuseria
, J.
Dukelsky
, A.
Signoracci
, and T.
Duguet
, Phys. Rev. C
89
, 054305
(2014
).67.
P. W.
Claeys
, S.
De Baerdemacker
, M.
Van Raemdonck
, and D.
Van Neck
, Phys. Rev. B
91
, 155102
(2015
).68.
P. A.
Johnson
, C.-É.
Fecteau
, F.
Berthiaume
, S.
Cloutier
, L.
Carrier
, M.
Gratton
, P.
Bultinck
, S.
De Baerdemacker
, D.
Van Neck
, P.
Limacher
, and P. W.
Ayers
, J. Chem. Phys.
153
, 104110
(2020
).69.
C.-É.
Fecteau
, S.
Cloutier
, J.-D.
Moisset
, J.
Boulay
, P.
Bultinck
, A.
Faribault
, and P. A.
Johnson
, J. Chem. Phys.
156
, 194103
(2022
).70.
71.
D. A.
Corvalan
, O. B.
Oña
, D. R.
Alcoba
, A.
Torre
, L.
Lain
, and G. E.
Massaccesi
, J. Chem. Phys.
157
, 204103
(2022
).72.
M.
Yamashita
, K.
Fujisawa
, K.
Nakata
, M.
Nakata
, M.
Fukuda
, K.
Kobayashi
, and K.
Goto
, “A high-performance software package for semidefinite programs: SDPA 7
,” Technical Report No. B-460, Department of Mathematical and Computing Science, Tokyo Institute of Technology
, 2010
.73.
M.
Nakata
, “A numerical evaluation of highly accurate multiple-precision arithmetic version of semidefinite programming solver: SDPA-GMP, -QD and -DD
,” in Proceedings of 2010 IEEE Multi-Conference on Systems and Control
(IEEE
, 2010
), pp. 29
–34
.74.
M.
Yamashita
, K.
Fujisawa
, M.
Fukuda
, K.
Kobayashi
, K.
Nakata
, and M.
Nakata
, in Semidefinite, Cone and Polynomial Optimization
, edited by M. F.
Anjos
and J. B.
Lasserre
(Springer
, New York
, 2011
), p. 687
.75.
J.
Dukelsky
, S.
Pittel
, and C.
Esebbag
, Phys. Rev. C
93
, 034313
(2016
).76.
K.
Dietrich
, H. J.
Mang
, and J. H.
Pradal
, Phys. Rev.
135
, B22
(1964
).77.
G.
Moore
and N.
Read
, Nucl. Phys. B
360
, 362
(1991
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
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