This work deals with the variational determination of the two-particle reduced density matrix (2-RDM) and the energy corresponding to the ground state of N-particle systems within the doubly occupied configuration interaction (DOCI) space. Here, we impose for the first time up to four-particle N-representability constraint conditions in the variational determination of the 2-RDM matrix elements using the standard semidefinite programming algorithms. The energies and 2-RDMs obtained from this treatment and the corresponding computational costs are compared with those arisen from previously reported less restrictive variational methods [D. R. Alcoba et al., J. Chem. Phys. 149, 194105 (2018)] as well as with the exact DOCI values. We apply the different approximations to the one-dimensional XXZ model of quantum magnetism, which has a rich phase diagram with one critical phase and constitutes a stringent test for the method. The numerical results show the usefulness of our treatment to achieve a high degree of accuracy.
REFERENCES
1.
A. J.
Coleman
and V. I.
Yukalov
, Reduced Density Matrices: Coulson’s Challenge
, Lecture Notes in Chemistry (Springer-Verlag
, New York
, 2000
), p. 282
.2.
Reduced-Density-Matrix Mechanics: With Application to Many-Electron Atoms and Molecules
, Advances in Chemical Physics Vol. 134, edited by D. A.
Mazziotti
(John Wiley & Sons, Inc.
, Hoboken, NJ, USA
, 2007
), pp. 21
–59
.3.
W.
Kohn
, in Nobel Lectures in Chemistry
, edited by I.
Grethe
(World Scientific
, Singapore
, 2003
), pp. 213
–237
.4.
F.
Colmenero
and C.
Valdemoro
, Phys. Rev. A
47
, 979
(1993
).5.
H.
Nakatsuji
and K.
Yasuda
, Phys. Rev. Lett.
76
, 1039
(1996
).6.
D. A.
Mazziotti
, Phys. Rev. A
57
, 4219
(1998
).7.
C.
Valdemoro
, L. M.
Tel
, E.
Pérez-Romero
, and A.
Torre
, J. Mol. Struct.: THEOCHEM
537
, 1
(2001
).8.
M.
Nakata
, H.
Nakatsuji
, M.
Ehara
, M.
Fukuda
, K.
Nakata
, and K.
Fujisawa
, J. Chem. Phys.
114
, 8282
(2001
).9.
D. A.
Mazziotti
, Phys. Rev. Lett.
93
, 213001
(2004
).10.
D. A.
Mazziotti
, Phys. Rev. Lett.
97
, 143002
(2006
).11.
C.
Valdemoro
, L. M.
Tel
, D. R.
Alcoba
, and E.
Pérez-Romero
, Theor. Chem. Acc.
118
, 503
(2007
).12.
D. R.
Alcoba
, C.
Valdemoro
, L. M.
Tel
, and E.
Pérez-Romero
, Int. J. Quantum Chem.
109
, 3178
(2009
).13.
C.
Garrod
and J. K.
Percus
, J. Math. Phys.
5
, 1756
(1964
).14.
15.
16.
D. A.
Mazziotti
, Phys. Rev. Lett.
108
, 263002
(2012
).17.
P.
Ring
and P.
Schuck
, The Nuclear Many-Body Problem
(Springer-Verlag
, 1980
).18.
D. S.
Koltun
and J. M.
Eisenberg
, Quantum Mechanics of Many Degrees of Freedom
(Wiley
, 1988
).19.
T.
Perez
and P.
Cassam-Chenaï
, J. Math. Chem.
56
, 1428
(2018
).20.
L.
Bytautas
, T. M.
Henderson
, C. A.
Jiménez-Hoyos
, J. K.
Ellis
, and G. E.
Scuseria
, J. Chem. Phys.
135
, 044119
(2011
).21.
L.
Bytautas
and J.
Dukelsky
, Comput. Theor. Chem.
1141
, 74
(2018
).22.
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
).23.
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
).24.
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
).25.
A.
Rubio-García
, D. R.
Alcoba
, P.
Capuzzi
, and J.
Dukelsky
, J. Chem. Theory Comput.
14
, 4183
(2018
).26.
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
).27.
D. R.
Alcoba
, A.
Torre
, L.
Lain
, G. E.
Massaccesi
, O. B.
Oña
, and E.
Ríos
, J. Chem. Phys.
150
, 164106
(2019
).28.
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
).29.
M.
Yamashita
, K.
Fujisawa
, M.
Fukuda
, K.
Kobayashi
, K.
Nakata
, and M.
Maho Nakata
, in Semidefinite, Cone Polynomial Optimization
, edited by M. F.
Anjos
and J. B.
Lasserre
(Springer
, New York
, 2011
), p. 687
.30.
P.
Jorgensen
, Second Quantization-Based Methods in Quantum Chemistry
(Academic Press
, 1981
).31.
32.
33.
C. A.
Coulson
, Rev. Mod. Phys.
32
, 170
(1960
).34.
A. J.
Coleman
, Rev. Mod. Phys.
35
, 668
(1963
).35.
D. A.
Mazziotti
, Phys. Rev. A
65
, 062511
(2002
).36.
D. A.
Mazziotti
and R. M.
Erdahl
, Phys. Rev. A
63
, 042113
(2001
).37.
F.
Weinhold
and E. B.
Wilson
, J. Chem. Phys.
46
, 2752
(1967
).38.
F.
Weinhold
and E. B.
Wilson
, J. Chem. Phys.
47
, 2298
(1967
).39.
W.
Poelmans
, “Variational determination of the two-particle density matrix: The case of doubly-occupied space
,” Ph.D. thesis, Ghent University
, 2015
.40.
N. C.
Rubin
and D. A.
Mazziotti
, J. Phys. Chem. C
119
, 14706
(2015
).41.
K.
Head-Marsden
and D. A.
Mazziotti
, J. Chem. Phys.
147
, 084101
(2017
).42.
I.
Talmi
, Simple Models of Complex Nuclei
(Harwood Academic Publishers
, Chur, Switzerland; Langhorne, PA, USA
, 1993
).43.
D. R.
Alcoba
, A.
Torre
, L.
Lain
, O. B.
Oña
, G. E.
Massaccesi
, and P.
Capuzzi
, in Advances in Quantum Chemistry
(Academic Press
, 2018
), Vol. 76, pp. 315
–332
.44.
A.
Auerbach
, Interacting Electrons and Quantum Magnetism
, Graduate Texts in Contemporary Physics (Springer New York
, New York, NY
, 1994
).45.
C. N.
Yang
and C. P.
Yang
, Phys. Rev.
147
, 303
(1966
).46.
P.
Jordan
and E.
Wigner
, Z. Phys.
47
, 631
(1928
).47.
R. M.
Erdahl
and M.
Rosina
, in Reduced Density Operators with Applications to Physical and Chemical Systems-II
, Queen’s Papers in Pure and Applied Mathematics Vol. 40, edited by R. M.
Erdahl
(Queens University
, Kingston, Ontario
, 1974
), p. 36
.48.
Y.
Nesterov
and A.
Nemirovskii
, Interior-Point Polynomial Algorithms in Convex Programming
(Society for Industrial and Applied Mathematics
, 1994
).49.
L.
Vandenberghe
and S.
Boyd
, SIAM Rev.
38
, 49
(1996
).50.
S.
Wright
, Handbook of Semidefinite Programming
, International Series in Operations Research and Management Science Vol. 27, 1st ed., edited by H.
Wolkowicz
, R.
Saigal
, and L.
Vandenberghe
(Springer US
, 2000
).51.
S.
Wright
, Primal-Dual Interior-Point Methods
(Society for Industrial and Applied Mathematics
, 1997
).52.
R. M.
Erdahl
and B.
Jin
, J. Mol. Struct.: THEOCHEM
527
, 207
(2000
).53.
T.
Barthel
and R.
Hübener
, Phys. Rev. Lett.
108
, 200404
(2012
).© 2019 Author(s).
2019
Author(s)
You do not currently have access to this content.