The computationally expensive evaluation and storage of high-rank reduced density matrices (RDMs) has been the bottleneck in the calculation of dynamic correlation for multireference wave functions in large active spaces. We present a stochastic formulation of multireference configuration interaction and perturbation theory that avoids the need for these expensive RDMs. The algorithm presented here is flexible enough to incorporate a wide variety of active space reference wave functions, including selected configuration interaction, matrix product states, and symmetry-projected Jastrow mean field wave functions. It enjoys the usual attractive features of Monte Carlo methods, such as embarrassing parallelizability and low memory costs. We find that the stochastic algorithm is already competitive with the deterministic algorithm for small active spaces, containing as few as 14 orbitals. We illustrate the utility of our stochastic formulation using benchmark applications.
REFERENCES
1.
S.
R.
White
and R. L.
Martin
, “Ab initio
quantum chemistry using the density matrix renormalization group
,”
J. Chem. Phys.
110
, 4127
–4130
(1999
).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.
A.
A.
Holmes
, N. M.
Tubman
, and C. J.
Umrigar
, “Heat-bath configuration
interaction: An efficient selected configuration interaction algorithm inspired by
heat-bath sampling
,” J. Chem. Theory Comput.
12
, 3674
–3680
(2016
).4.
S.
Sharma
, A. A.
Holmes
, G.
Jeanmairet
, A.
Alavi
, and C. J.
Umrigar
, “Semistochastic heat-bath
configuration interaction method: Selected configuration interaction with semistochastic
perturbation theory
,” J. Chem. Theory Comput.
13
, 1595
–1604
(2017
).5.
G.
H.
Booth
, A. J. W.
Thom
, and A.
Alavi
, “Fermion Monte Carlo without
fixed nodes: A game of life, death, and annihilation in Slater determinant
space
,” J. Chem. Phys.
131
, 054106
(2009
).6.
D.
Cleland
, G. H.
Booth
, and A.
Alavi
, “Communications: Survival of the
fittest: Accelerating convergence in full configuration-interaction quantum Monte
Carlo
,” J. Chem. Phys.
132
, 041103
(2010
).7.
F.
R.
Petruzielo
, A. A.
Holmes
, H. J.
Changlani
, M. P.
Nightingale
, and C. J.
Umrigar
, “Semistochastic projector Monte
Carlo method
,” Phys. Rev. Lett.
109
, 230201
(2012
).8.
B.
O.
Roos
, P.
R.
Taylor
, P. E.
Si
et al, “A complete active space SCF method
(CASSCF) using a density matrix formulated super-CI approach
,”
Chem. Phys.
48
, 157
–173
(1980
).9.
B.
O.
Roos
, “The complete active space SCF
method in a Fock-matrix-based super-CI formulation
,” Int. J.
Quantum Chem.
18
, 175
–189
(1980
).10.
P.
E.
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
).11.
D.
Ghosh
, J.
Hachmann
, T.
Yanai
, and G. K.-L.
Chan
, “Orbital optimization in the
density matrix renormalization group, with applications to polyenes and
β-carotene
,” J. Chem. Phys.
128
, 144117
(2008
).12.
D.
Zgid
and M.
Nooijen
, “The density matrix
renormalization group self-consistent field method: Orbital optimization with the
density matrix renormalization group method in the active space
,”
J. Chem. Phys.
128
, 144116
(2008
).13.
T.
Yanai
, Y.
Kurashige
, D.
Ghosh
, and G. K.-L.
Chan
, “Accelerating convergence in
iterative solution for large-scale complete active space self-consistent-field
calculations
,” Int. J. Quantum Chem.
109
, 2178
–2190
(2009
).14.
J.
E.
Smith
, B.
Mussard
, A. A.
Holmes
, and S.
Sharma
, “Cheap and near exact CASSCF
with large active spaces
,” J. Chem. Theory Comput.
13
, 5468
–5478
(2017
).15.
R.
E.
Thomas
, Q.
Sun
, A.
Alavi
, and G. H.
Booth
, “Stochastic multiconfigurational
self-consistent field theory
,” J. Chem. Theory Comput.
11
, 5316
(2015
).16.
G.
Li Manni
, S. D.
Smart
, and A.
Alavi
, “Combining the complete active
space self-consistent field method and the full configuration interaction quantum Monte
Carlo within a super-CI framework, with application to challenging
metal-porphyrins
,” J. Chem. Theory Comput.
12
, 1245
–1258
(2016
).17.
J.
Hachmann
, J. J.
Dorando
, M.
Avilés
, and G. K.-L.
Chan
, “The radical character of the
acenes: A density matrix renormalization group study
,” J. Chem.
Phys.
127
, 134309
(2007
).18.
K.
H.
Marti
, I. M.
Ondík
, G.
Moritz
, and M.
Reiher
, “Density matrix renormalization
group calculations on relative energies of transition metal complexes and
clusters
,” J. Chem. Phys.
128
, 014104
(2008
).19.
Y.
Kurashige
and T.
Yanai
, “High-performance ab
initio density matrix renormalization group method: Applicability to
large-scale multireference problems for metal compounds
,” J.
Chem. Phys.
130
, 234114
(2009
).20.
Y.
Kurashige
, G. K.-L.
Chan
, and T.
Yanai
, “Entangled quantum electronic
wavefunctions of the Mn4CaO5 cluster in photosystem
II
,” Nat. Chem.
5
, 660
(2013
).21.
S.
Sharma
, K.
Sivalingam
, F.
Neese
, and G. K.-L.
Chan
, “Low-energy spectrum of
iron–sulfur clusters directly from many-particle quantum mechanics
,”
Nat. Chem.
6
, 927
(2014
).22.
R.
Olivares-Amaya
, W.
Hu
, N.
Nakatani
, S.
Sharma
, J.
Yang
, and G. K.-L.
Chan
, “The ab-initio
density matrix renormalization group in practice
,” J. Chem.
Phys.
142
, 034102
(2015
).23.
B.
Mussard
and S.
Sharma
, “One-step treatment of
spin–orbit coupling and electron correlation in large active spaces
,”
J. Chem. Theory Comput.
14
, 154
–165
(2017
).24.
J.
Li
, M.
Otten
, A. A.
Holmes
, S.
Sharma
, and C. J.
Umrigar
, “Fast semistochastic heat-bath
configuration interaction
,” J. Chem. Phys.
149
, 214110
(2018
).25.
G.
H.
Booth
, A.
Grüneis
, G.
Kresse
, and A.
Alavi
, “Towards an exact description of
electronic wavefunctions in real solids
,” Nature
493
, 365
(2013
).26.
G.
Li Manni
and A.
Alavi
, “Understanding the mechanism
stabilizing intermediate spin states in Fe(II)-porphyrin
,” J.
Phys. Chem. A
122
, 4935
–4947
(2018
).27.
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
).28.
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
).29.
C.
Angeli
, R.
Cimiraglia
, S.
Evangelisti
, T.
Leininger
, and J.-P.
Malrieu
, “Introduction of
n-electron valence states for multireference perturbation
theory
,” J. Chem. Phys.
114
, 10252
–10264
(2001
).30.
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
).31.
C.
Angeli
, S.
Borini
, M.
Cestari
, and R.
Cimiraglia
, “A quasidegenerate
formulation of the second order n-electron valence state perturbation
theory approach
,” J. Chem. Phys.
121
, 4043
–4049
(2004
).32.
H.-J.
Werner
and P. J.
Knowles
, “An efficient internally
contracted multiconfiguration–reference configuration interaction
method
,” J. Chem. Phys.
89
, 5803
–5814
(1988
).33.
P.
J.
Knowles
and H.-J.
Werner
, “An efficient method for the
evaluation of coupling coefficients in configuration interaction
calculations
,” Chem. Phys. Lett.
145
, 514
–522
(1988
).34.
P.
J.
Knowles
and H.-J.
Werner
, “Internally contracted
multiconfiguration-reference configuration interaction calculations for excited
states
,” Theor. Chim. Acta
84
, 95
–103
(1992
).35.
R.
J.
Bartlett
and M.
Musiał
, “Coupled-cluster theory in
quantum chemistry
,” Rev. Mod. Phys.
79
, 291
(2007
).36.
T.
Yanai
and G. K.-L.
Chan
, “Canonical transformation theory
for multireference problems
,” J. Chem. Phys.
124
, 194106
(2006
).37.
E.
Neuscamman
, T.
Yanai
, and G. K.-L.
Chan
, “A review of canonical
transformation theory
,” Int. Rev. Phys. Chem.
29
, 231
–271
(2010
).38.
F.
A.
Evangelista
, “A driven similarity
renormalization group approach to quantum many-body problems
,”
J. Chem. Phys.
141
, 054109
(2014
).39.
Y.
Kurashige
and T.
Yanai
, “Second-order perturbation theory
with a density matrix renormalization group self-consistent field reference function:
Theory and application to the study of chromium dimer
,” J. Chem.
Phys.
135
, 094104
(2011
).40.
S.
Guo
, M.
A.
Watson
, W.
Hu
, Q.
Sun
, and G. K.-L.
Chan
, “N-electron
valence state perturbation theory based on a density matrix renormalization group
reference function, with applications to the chromium dimer and a trimer model of poly
(p-phenylenevinylene)
,” J. Chem. Theory
Comput.
12
, 1583
–1591
(2016
).41.
M.
Roemelt
, S.
Guo
, and G. K.-L.
Chan
, “A projected approximation to
strongly contracted N-electron valence perturbation theory for DMRG
wavefunctions
,” J. Chem. Phys.
144
, 204113
(2016
).42.
D.
Zgid
, D.
Ghosh
, E.
Neuscamman
, and G. K.-L.
Chan
, “A study of cumulant
approximations to n-electron valence multireference perturbation
theory
,” J. Chem. Phys.
130
, 194107
(2009
).43.
M.
Saitow
, Y.
Kurashige
, and T.
Yanai
, “Multireference configuration
interaction theory using cumulant reconstruction with internal contraction of density
matrix renormalization group wave function
,” J. Chem.
Phys.
139
, 044118
(2013
).44.
Y.
Kurashige
, J.
Chalupskỳ
, T. N.
Lan
, and T.
Yanai
, “Complete active space
second-order perturbation theory with cumulant approximation for extended active-space
wavefunction from density matrix renormalization group
,” J.
Chem. Phys.
141
, 174111
(2014
).45.
M.
Saitow
, Y.
Kurashige
, and T.
Yanai
, “Fully internally contracted
multireference configuration interaction theory using density matrix renormalization
group: A reduced-scaling implementation derived by computer-aided tensor
factorization
,” J. Chem. Theory Comput.
11
, 5120
–5131
(2015
).46.
S.
Shirai
, Y.
Kurashige
, and T.
Yanai
, “Computational evidence of
inversion of 1La and 1Lb-derived excited
states in naphthalene excimer formation from ab initio multireference
theory with large active space: DMRG-CASPT2 study
,” J. Chem.
Theory Comput.
12
, 2366
–2372
(2016
).47.
Q.
M.
Phung
, S.
Wouters
, and K.
Pierloot
, “Cumulant approximated
second-order perturbation theory based on the density matrix renormalization group for
transition metal complexes: A benchmark study
,” J. Chem. Theory
Comput.
12
, 4352
–4361
(2016
).48.
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
).49.
N.
Nakatani
and S.
Guo
, “Density matrix renormalization
group (DMRG) method as a common tool for large active-space CASSCF/CASPT2
calculations
,” J. Chem. Phys.
146
, 094102
(2017
).50.
S.
Wouters
, V.
Van Speybroeck
, and D.
Van Neck
, “DMRG-CASPT2 study of the
longitudinal static second hyperpolarizability of all-trans polyenes
,”
J. Chem. Phys.
145
, 054120
(2016
).51.
P.
Celani
and H.-J.
Werner
, “Multireference perturbation
theory for large restricted and selected active space reference wave
functions
,” J. Chem. Phys.
112
, 5546
–5557
(2000
).52.
K.
Shamasundar
, G.
Knizia
, and H.-J.
Werner
, “A new internally contracted
multi-reference configuration interaction method
,” J. Chem.
Phys.
135
, 054101
(2011
).53.
S.
Sharma
and G. K.-L.
Chan
, “Communication: A flexible
multi-reference perturbation theory by minimizing the Hylleraas functional with matrix
product states
,” J. Chem. Phys.
141
, 111101
(2014
).54.
S.
Sharma
, G.
Knizia
, S.
Guo
, and A.
Alavi
, “Combining internally contracted
states and matrix product states to perform multireference perturbation
theory
,” J. Chem. Theory Comput.
13
, 488
–498
(2017
).55.
A.
Y.
Sokolov
, S.
Guo
, E.
Ronca
, and G. K.-L.
Chan
, “Time-dependent
N-electron valence perturbation theory with matrix product state
reference wavefunctions for large active spaces and basis sets: Applications to the
chromium dimer and all-trans polyenes
,” J.
Chem. Phys.
146
, 244102
(2017
).56.
S.
Sharma
and A.
Alavi
, “Multireference linearized
coupled cluster theory for strongly correlated systems using matrix product
states
,” J. Chem. Phys.
143
, 102815
(2015
).57.
S.
Sharma
, G.
Jeanmairet
, and A.
Alavi
, “Quasi-degenerate perturbation
theory using matrix product states
,” J. Chem. Phys.
144
, 034103
(2016
).58.
D.
Tahara
and M.
Imada
, “Variational Monte Carlo method
combined with quantum-number projection and multi-variable
optimization
,” J. Phys. Soc. Jpn.
77
, 114701
(2008
).59.
E.
Neuscamman
, “Size consistency error in
the antisymmetric geminal power wave function can be completely
removed
,” Phys. Rev. Lett.
109
, 203001
(2012
).60.
E.
Neuscamman
, “The Jastrow antisymmetric
geminal power in Hilbert space: Theory, benchmarking, and application to a novel
transition state
,” J. Chem. Phys.
139
, 194105
(2013
).61.
A.
Mahajan
and S.
Sharma
, “Symmetry-projected Jastrow
mean-field wave function in variational Monte Carlo
,” J. Phys.
Chem. A
123
, 3911
–3921
(2019
).62.
A.
McLean
and B.
Liu
, “Classification of configurations
and the determination of interacting and noninteracting spaces in configuration
interaction
,” J. Chem. Phys.
58
, 1066
–1078
(1973
).63.
64.
P.
E.
Siegbahn
, “Direct configuration
interaction with a reference state composed of many reference
configurations
,” Int. J. Quantum Chem.
18
, 1229
–1242
(1980
).65.
K.
Sivalingam
, M.
Krupicka
, A. A.
Auer
, and F.
Neese
, “Comparison of fully internally
and strongly contracted multireference configuration interaction
procedures
,” J. Chem. Phys.
145
, 054104
(2016
).66.
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
).67.
A.
Bortz
, M.
Kalos
, and J.
Lebowitz
, “A new algorithm for Monte
Carlo simulation of Ising spin systems
,” J. Comput.
Phys.
17
, 10
–18
(1975
).68.
D.
T.
Gillespie
, “A general method for
numerically simulating the stochastic time evolution of coupled chemical
reactions
,” J. Comput. Phys.
22
, 403
–434
(1976
).69.
I.
Sabzevari
and S.
Sharma
, “Improved speed and scaling in
orbital space variational Monte Carlo
,” J. Chem. Theory
Comput.
14
, 6276
–6286
(2018
).70.
M.
Nightingale
and V.
Melik-Alaverdian
, “Optimization of
ground-and excited-state wave functions and van der Waals clusters
,”
Phys. Rev. Lett.
87
, 043401
(2001
).71.
C.
Umrigar
, J.
Toulouse
, C.
Filippi
, S.
Sorella
, and R. G.
Hennig
, “Alleviation of the Fermion-sign
problem by optimization of many-body wave functions
,” Phys. Rev.
Lett.
98
, 110201
(2007
).72.
J.
Toulouse
and C. J.
Umrigar
, “Optimization of quantum Monte
Carlo wave functions by energy minimization
,” J. Chem.
Phys.
126
, 084102
(2007
).73.
J.
Toulouse
and C.
Umrigar
, “Full optimization of
Jastrow–Slater wave functions with application to the first-row atoms and homonuclear
diatomic molecules
,” J. Chem. Phys.
128
, 174101
(2008
).74.
L.
Zhao
and E.
Neuscamman
, “A blocked linear method for
optimizing large parameter sets in variational Monte Carlo
,” J.
Chem. Theory Comput.
13
, 2604
–2611
(2017
).75.
I.
Sabzevari
, A.
Mahajan
, and S.
Sharma
, “An accelerated linear method
for optimizing non-linear wavefunctions in variational Monte Carlo
,”
preprint arXiv:1908.04423 (2019
).76.
S.
J.
Reddi
, S.
Kale
, and S.
Kumar
, “On the convergence of Adam and
beyond
,” e-print arXiv:1904.09237.77.
L.
R.
Schwarz
, A.
Alavi
, and G. H.
Booth
, “Projector quantum Monte Carlo
method for nonlinear wave functions
,” Phys. Rev. Lett.
118
, 176403
(2017
).78.
L.
Otis
and E.
Neuscamman
, “Complementary first and
second derivative methods for ansatz optimization in variational Monte
Carlo
,” Phys. Chem. Chem. Phys.
21
, 14491
(2019
).79.
K.
G.
Dyall
, “The choice of a zeroth-order
Hamiltonian for second-order perturbation theory with a complete active space
self-consistent-field reference function
,” J. Chem.
Phys.
102
, 4909
–4918
(1995
).80.
H.-J.
Werner
, P. J.
Knowles
, G.
Knizia
, F. R.
Manby
, and M.
Schütz
, “Molpro: A general-purpose
quantum chemistry program package
,” Wiley Interdiscip. Rev.:
Comput. Mol. Sci.
2
, 242
–253
(2012
).81.
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
, S.
Wouters
, and K.-L. G.
Chan
, “PySCF: The python-based
simulations of chemistry framework
,” Wiley Interdiscip. Rev.:
Comput. Mol. Sci.
8
, e1340
(2018
).82.
L.
Zhao
and E.
Neuscamman
, “Equation of motion theory
for excited states in variational Monte Carlo and the Jastrow antisymmetric geminal
power in Hilbert space
,” J. Chem. Theory Comput.
12
, 3719
–3726
(2016
).83.
N.
S.
Blunt
, A.
Alavi
, and G. H.
Booth
, “Nonlinear biases, stochastically
sampled effective Hamiltonians, and spectral functions in quantum Monte Carlo
methods
,” Phys. Rev. B
98
, 085118
(2018
).84.
C.
Angeli
, R.
Cimiraglia
, and M.
Pastore
, “A comparison of various
approaches in internally contracted multireference configuration interaction: The carbon
dimer as a test case
,” Mol. Phys.
110
, 2963
–2968
(2012
).©2019 Authors. Published by AIP Publishing.
2019
Author(s)
You do not currently have access to this content.