Constructing matrix product operators (MPOs) is at the core of the modern density matrix renormalization group (DMRG) and its time dependent formulation. For the DMRG to be conveniently used in different problems described by different Hamiltonians, in this work, we propose a new generic algorithm to construct the MPO of an arbitrary operator with a sum-of-products form based on the bipartite graph theory. We show that the method has the following advantages: (i) it is automatic in that only the definition of the operator is required; (ii) it is symbolic thus free of any numerical error; (iii) the complementary operator technique can be fully employed so that the resulting MPO is globally optimal for any given order of degrees of freedom; and (iv) the symmetry of the system could be fully employed to reduce the dimension of MPO. To demonstrate the effectiveness of the new algorithm, the MPOs of Hamiltonians ranging from the prototypical spin–boson model and the Holstein model to the more complicated ab initio electronic Hamiltonian and the anharmonic vibrational Hamiltonian with the sextic force field are constructed. It is found that for the former three cases, our automatic algorithm can reproduce exactly the same MPOs as the optimally hand-crafted ones already known in the literature.
REFERENCES
1.
S. R.
White
, “Density matrix formulation for quantum renormalization groups
,” Phys. Rev. Lett.
69
, 2863
(1992
).2.
G. K.-L.
Chan
and S.
Sharma
, “The density matrix renormalization group in quantum chemistry
,” Annu. Rev. Phys. Chem.
62
, 465
–481
(2011
).3.
J.
Ren
, Y.
Yi
, and Z.
Shuai
, “Inner space perturbation theory in matrix product states: Replacing expensive iterative diagonalization
,” J. Chem. Theory Comput.
12
, 4871
–4878
(2016
).4.
Y.
Ma
, S.
Knecht
, S.
Keller
, and M.
Reiher
, “Second-order self-consistent-field density-matrix renormalization group
,” J. Chem. Theory Comput.
13
, 2533
–2549
(2017
).5.
T.
Yanai
, M.
Saitow
, X.-G.
Xiong
, J.
Chalupský
, Y.
Kurashige
, S.
Guo
, and S.
Sharma
, “Multistate complete-active-space second-order perturbation theory based on density matrix renormalization group reference states
,” J. Chem. Theory Comput.
13
, 4829
–4840
(2017
).6.
Z.
Li
and G. K.-L.
Chan
, “Spin-projected matrix product states: Versatile tool for strongly correlated systems
,” J. Chem. Theory Comput.
13
, 2681
–2695
(2017
).7.
Z.
Luo
, Y.
Ma
, X.
Wang
, and H.
Ma
, “Externally-contracted multireference configuration interaction method using a DMRG reference wave function
,” J. Chem. Theory Comput.
14
, 4747
–4755
(2018
).8.
A.
Baiardi
and M.
Reiher
, “The density matrix renormalization group in chemistry and molecular physics: Recent developments and new challenges
,” J. Chem. Phys.
152
, 040903
(2020
).9.
J.
Brandejs
, J.
Višňák
, L.
Veis
, M.
Maté
, Ö.
Legeza
, and J.
Pittner
, “Toward DMRG-tailored coupled cluster method in the 4c-relativistic domain
,” J. Chem. Phys.
152
, 174107
(2020
).10.
M.
Rakhuba
and I.
Oseledets
, “Calculating vibrational spectra of molecules using tensor train decomposition
,” J. Chem. Phys.
145
, 124101
(2016
).11.
A.
Baiardi
, C. J.
Stein
, V.
Barone
, and M.
Reiher
, “Vibrational density matrix renormalization group
,” J. Chem. Theory Comput.
13
, 3764
–3777
(2017
).12.
A.
Baiardi
, C. J.
Stein
, V.
Barone
, and M.
Reiher
, “Optimization of highly excited matrix product states with an application to vibrational spectroscopy
,” J. Chem. Phys.
150
, 094113
(2019
).13.
E.
Ronca
, Z.
Li
, C. A.
Jimenez-Hoyos
, and G. K.-L.
Chan
, “Time-step targeting time-dependent and dynamical density matrix renormalization group algorithms with ab initio Hamiltonians
,” J. Chem. Theory Comput.
13
, 5560
–5571
(2017
).14.
L.-H.
Frahm
and D.
Pfannkuche
, “Ultrafast ab initio quantum chemistry using matrix product states
,” J. Chem. Theory Comput.
15
, 2154
–2165
(2019
).15.
S. M.
Greene
and V. S.
Batista
, “Tensor-train split-operator Fourier transform (TT-soft) method: Multidimensional nonadiabatic quantum dynamics
,” J. Chem. Theory Comput.
13
, 4034
–4042
(2017
).16.
Y.
Yao
, K.-W.
Sun
, Z.
Luo
, and H.
Ma
, “Full quantum dynamics simulation of a realistic molecular system using the adaptive time-dependent density matrix renormalization group method
,” J. Phys. Chem. Lett.
9
, 413
–419
(2018
).17.
J.
Ren
, Z.
Shuai
, and G. K.-L.
Chan
, “Time-dependent density matrix renormalization group algorithms for nearly exact absorption and fluorescence spectra of molecular aggregates at both zero and finite temperature
,” J. Chem. Theory Comput.
14
, 5027
–5039
(2018
).18.
Y.
Kurashige
, “Matrix product state formulation of the multiconfiguration time-dependent Hartree theory
,” J. Chem. Phys.
149
, 194114
(2018
).19.
A.
Baiardi
and M.
Reiher
, “Large-scale quantum dynamics with matrix product states
,” J. Chem. Theory Comput.
15
, 3481
–3498
(2019
).20.
X.
Xie
, Y.
Liu
, Y.
Yao
, U.
Schollwöck
, C.
Liu
, and H.
Ma
, “Time-dependent density matrix renormalization group quantum dynamics for realistic chemical systems
,” J. Chem. Phys.
151
, 224101
(2019
).21.
W.
Li
, J.
Ren
, and Z.
Shuai
, “Numerical assessment for accuracy and GPU acceleration of TD-DMRG time evolution schemes
,” J. Chem. Phys.
152
, 024127
(2020
).22.
W.
Li
, J.
Ren
, and Z.
Shuai
, “Finite-temperature TD-DMRG for the carrier mobility of organic semiconductors
,” J. Phys. Chem. Lett.
11
, 4930
–4936
(2020
).23.
H.-D.
Meyer
, U.
Manthe
, and L. S.
Cederbaum
, “The multi-configurational time-dependent Hartree approach
,” Chem. Phys. Lett.
165
, 73
–78
(1990
).24.
M. H.
Beck
, A.
Jäckle
, G. A.
Worth
, and H.-D.
Meyer
, “The multiconfiguration time-dependent Hartree (MCTDH) method: A highly efficient algorithm for propagating wavepackets
,” Phys. Rep.
324
, 1
–105
(2000
).25.
H.
Wang
and M.
Thoss
, “Multilayer formulation of the multiconfiguration time-dependent Hartree theory
,” J. Chem. Phys.
119
, 1289
–1299
(2003
).26.
H. R.
Larsson
, “Computing vibrational eigenstates with tree tensor network states (TTNS)
,” J. Chem. Phys.
151
, 204102
(2019
).27.
S.
Östlund
and S.
Rommer
, “Thermodynamic limit of density matrix renormalization
,” Phys. Rev. Lett.
75
, 3537
(1995
).28.
G. M.
Crosswhite
and D.
Bacon
, “Finite automata for caching in matrix product algorithms
,” Phys. Rev. A
78
, 012356
(2008
).29.
U.
Schollwöck
, “The density-matrix renormalization group in the age of matrix product states
,” Ann. Phys.
326
, 96
–192
(2011
).30.
N.
Nakatani
and G. K.-L.
Chan
, “Efficient tree tensor network states (TTNS) for quantum chemistry: Generalizations of the density matrix renormalization group algorithm
,” J. Chem. Phys.
138
, 134113
(2013
).31.
K.
Gunst
, F.
Verstraete
, S.
Wouters
, Ö.
Legeza
, and D.
Van Neck
, “T3NS: Three-legged tree tensor network states
,” J. Chem. Theory Comput.
14
, 2026
–2033
(2018
).32.
Z.
Li
, M. J.
O’Rourke
, and G. K.-L.
Chan
, “Generalization of the exponential basis for tensor network representations of long-range interactions in two and three dimensions
,” Phys. Rev. B
100
, 155121
(2019
).33.
S.
Keller
, M.
Dolfi
, M.
Troyer
, and M.
Reiher
, “An efficient matrix product operator representation of the quantum chemical Hamiltonian
,” J. Chem. Phys.
143
, 244118
(2015
).34.
J.
Haegeman
, C.
Lubich
, I.
Oseledets
, B.
Vandereycken
, and F.
Verstraete
, “Unifying time evolution and optimization with matrix product states
,” Phys. Rev. B
94
, 165116
(2016
).35.
L.
Raff
, R.
Komanduri
, M.
Hagan
, and S.
Bukkapatnam
, Neural Networks in Chemical Reaction Dynamics
(OUP USA
, 2012
).36.
B.
Jiang
and H.
Guo
, “Permutation invariant polynomial neural network approach to fitting potential energy surfaces
,” J. Chem. Phys.
139
, 054112
(2013
).37.
V.
Barone
, M.
Biczysko
, and J.
Bloino
, “Fully anharmonic IR and Raman spectra of medium-size molecular systems: Accuracy and interpretation
,” Phys. Chem. Chem. Phys.
16
, 1759
–1787
(2014
).38.
V.
Barone
, “Anharmonic vibrational properties by a fully automated second-order perturbative approach
,” J. Chem. Phys.
122
, 014108
(2005
).39.
V.
Barone
, M.
Biczysko
, J.
Bloino
, M.
Borkowska-Panek
, I.
Carnimeo
, and P.
Panek
, “Toward anharmonic computations of vibrational spectra for large molecular systems
,” Int. J. Quantum Chem.
112
, 2185
–2200
(2012
).40.
A.
Jäckle
and H. D.
Meyer
, “Product representation of potential energy surfaces
,” J. Chem. Phys.
104
, 7974
–7984
(1996
).41.
D.
Peláez
and H.-D.
Meyer
, “The multigrid POTFIT (MGPF) method: Grid representations of potentials for quantum dynamics of large systems
,” J. Chem. Phys.
138
, 014108
(2013
).42.
F.
Otto
, “Multi-layer POTFIT: An accurate potential representation for efficient high-dimensional quantum dynamics
,” J. Chem. Phys.
140
, 014106
(2014
).43.
S.
Manzhos
and T.
Carrington
, Jr, “Using neural networks to represent potential surfaces as sums of products
,” J. Chem. Phys.
125
, 194105
(2006
).44.
S.
Carter
, S. J.
Culik
, and J. M.
Bowman
, “Vibrational self-consistent field method for many-mode systems: A new approach and application to the vibrations of CO adsorbed on Cu(100)
,” J. Chem. Phys.
107
, 10458
–10469
(1997
).45.
E. L.
Klinting
, B.
Thomsen
, I. H.
Godtliebsen
, and O.
Christiansen
, “Employing general fit-bases for construction of potential energy surfaces with an adaptive density-guided approach
,” J. Chem. Phys.
148
, 064113
(2018
).46.
U.
Manthe
, “A time-dependent discrete variable representation for (multiconfiguration) Hartree methods
,” J. Chem. Phys.
105
, 6989
–6994
(1996
).47.
U.
Manthe
, “A multilayer multiconfigurational time-dependent Hartree approach for quantum dynamics on general potential energy surfaces
,” J. Chem. Phys.
128
, 164116
(2008
).48.
G.
Avila
and T.
Carrington
, Jr, “Solving the Schrödinger equation using Smolyak interpolants
,” J. Chem. Phys.
139
, 134114
(2013
).49.
G. K.-L.
Chan
, A.
Keselman
, N.
Nakatani
, Z.
Li
, and S. R.
White
, “Matrix product operators, matrix product states, and ab initio density matrix renormalization group algorithms
,” J. Chem. Phys.
145
, 014102
(2016
).50.
T.
Xiang
, “Density-matrix renormalization-group method in momentum space
,” Phys. Rev. B
53
, R10445
(1996
).51.
C.
Hubig
, I.
McCulloch
, and U.
Schollwöck
, “Generic construction of efficient matrix product operators
,” Phys. Rev. B
95
, 035129
(2017
).52.
See https://github.com/shuaigroup/Renormalizer for Renormalizer.
53.
J. A.
Bondy
, U. S. R.
Murty
et al, Graph Theory with Applications
(Macmillan
, London
, 1976
), Vol. 290.54.
H. W.
Kuhn
, “The Hungarian method for the assignment problem
,” Nav. Res. Logist. Q.
2
, 83
–97
(1955
).55.
J. E.
Hopcroft
and R. M.
Karp
, “An n^5/2 algorithm for maximum matchings in bipartite graphs
,” SIAM J. Comput.
2
, 225
–231
(1973
).56.
I. V.
Oseledets
, “Tensor-train decomposition
,” SIAM J. Sci. Comput.
33
, 2295
–2317
(2011
).57.
L.
Lovász
, “On determinants, matchings, and random algorithms
,” in FCT
(Akademie-Verlag
, 1979
), Vol. 79, pp. 565
–574
.58.
Ö.
Legeza
and J.
Sólyom
, “Optimizing the density-matrix renormalization group method using quantum information entropy
,” Phys. Rev. B
68
, 195116
(2003
).59.
G.
Moritz
, B. A.
Hess
, and M.
Reiher
, “Convergence behavior of the density-matrix renormalization group algorithm for optimized orbital orderings
,” J. Chem. Phys.
122
, 024107
(2005
).60.
M.
Schröter
, S. D.
Ivanov
, J.
Schulze
, S. P.
Polyutov
, Y.
Yan
, T.
Pullerits
, and O.
Kühn
, “Exciton–vibrational coupling in the dynamics and spectroscopy of Frenkel excitons in molecular aggregates
,” Phys. Rep.
567
, 1
–78
(2015
).61.
T.
Jiang
, W.
Li
, J.
Ren
, and Z.
Shuai
, “Finite temperature dynamical density matrix renormalization group for spectroscopy in frequency domain
,” J. Phys. Chem. Lett.
11
, 3761
–3768
(2020
).62.
P.
Jordan
and E. P.
Wigner
, “About the Pauli exclusion principle
,” Z. Phys.
47
, 631
–651
(1928
).63.
See https://github.com/zhendongli2008/QCMPO for Qcmpo.
64.
Q.
Sun
, T. C.
Berkelbach
, N. S.
Blunt
, G. H.
Booth
, S.
Guo
, Z.
Li
, J.
Liu
, J. D.
McClain
, E. R.
Sayfutyarova
, S.
Sharma
et al, “PySCF: The Python-based simulations of chemistry framework
,” Wiley Interdiscip. Rev.: Comput. Mol. Sci.
8
, e1340
(2018
).65.
O.
Christiansen
and J. M.
Luis
, “Beyond vibrational self-consistent-field methods: Benchmark calculations for the fundamental vibrations of ethylene
,” Int. J. Quantum Chem.
104
, 667
–680
(2005
).66.
O.
Christiansen
, “Selected new developments in vibrational structure theory: Potential construction and vibrational wave function calculations
,” Phys. Chem. Chem. Phys.
14
, 6672
–6687
(2012
).67.
C.
Qu
and J. M.
Bowman
, “Quantum approaches to vibrational dynamics and spectroscopy: Is ease of interpretation sacrificed as rigor increases?
,” Phys. Chem. Chem. Phys.
21
, 3397
–3413
(2019
).68.
W.
Mizukami
and D. P.
Tew
, “A second-order multi-reference perturbation method for molecular vibrations
,” J. Chem. Phys.
139
, 194108
(2013
).69.
F.
Pfeiffer
and G.
Rauhut
, “Multi-reference vibration correlation methods
,” J. Chem. Phys.
140
, 064110
(2014
).70.
L. J.
Doriol
, F.
Gatti
, C.
Iung
, and H.-D.
Meyer
, “Computation of vibrational energy levels and eigenstates of fluoroform using the multiconfiguration time-dependent Hartree method
,” J. Chem. Phys.
129
, 224109
(2008
).71.
P.
Carbonniere
and V.
Barone
, “Coriolis couplings in variational computations of vibrational spectra beyond the harmonic approximation: Implementation and validation
,” Chem. Phys. Lett.
392
, 365
–371
(2004
).72.
M.
Sibaev
and D. L.
Crittenden
, “PyVCI: A flexible open-source code for calculating accurate molecular infrared spectra
,” Comput. Phys. Commun.
203
, 290
–297
(2016
).73.
M.
Sibaev
and D. L.
Crittenden
, “The PyPES library of high quality semi-global potential energy surfaces
,” J. Comput. Chem.
36
, 2200
–2207
(2015
).74.
T.
Delahaye
, A.
Nikitin
, M.
Rey
, P. G.
Szalay
, and V. G.
Tyuterev
, “A new accurate ground-state potential energy surface of ethylene and predictions for rotational and vibrational energy levels
,” J. Chem. Phys.
141
, 104301
(2014
).75.
S.
Wouters
, N.
Nakatani
, D.
Van Neck
, and G. K.-L.
Chan
, “Thouless theorem for matrix product states and subsequent post density matrix renormalization group methods
,” Phys. Rev. B
88
, 075122
(2013
).76.
N.
Nakatani
, S.
Wouters
, D.
Van Neck
, and G. K.-L.
Chan
, “Linear response theory for the density matrix renormalization group: Efficient algorithms for strongly correlated excited states
,” J. Chem. Phys.
140
, 024108
(2014
).77.
J. J.
Dorando
, J.
Hachmann
, and G. K.-L.
Chan
, “Targeted excited state algorithms
,” J. Chem. Phys.
127
, 084109
(2007
).78.
X.
Yu
, D.
Pekker
, and B. K.
Clark
, “Finding matrix product state representations of highly excited eigenstates of many-body localized Hamiltonians
,” Phys. Rev. Lett.
118
, 017201
(2017
).79.
W.
Hu
and G. K.-L.
Chan
, “Excited-state geometry optimization with the density matrix renormalization group, as applied to polyenes
,” J. Chem. Theory Comput.
11
, 3000
–3009
(2015
).80.
F.
Gatti
and C.
Iung
, “Exact and constrained kinetic energy operators for polyatomic molecules: The polyspherical approach
,” Phys. Rep.
484
, 1
–69
(2009
).81.
M.
Ndong
, L.
Joubert-Doriol
, H.-D.
Meyer
, A.
Nauts
, F.
Gatti
, and D.
Lauvergnat
, “Automatic computer procedure for generating exact and analytical kinetic energy operators based on the polyspherical approach
,” J. Chem. Phys.
136
, 034107
(2012
).© 2020 Author(s).
2020
Author(s)
You do not currently have access to this content.
