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Quantum Electronic Structure
A Perturbative Density Matrix Renormalization
Group Algorithm for Large Active Spaces
Sheng Guo, Zhendong Li, and Garnet KinLic Chan
J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b00273 • Publication Date (Web): 21 Jun 2018
Downloaded from http://pubs.acs.org on June 21, 2018
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A Perturbative Density Matrix Renormalization
Group Algorithm for Large Active Spaces
Sheng Guo, Zhendong Li, and Garnet KinLic Chan
∗
Division of Chemistry and Chemical Engineering, California Institute of Technology,
Pasadena, CA 91125, USA
Email: gkc1000@gmail.com
Abstract
We describe a low cost alternative to the standard variational DMRG (density ma
trix renormalization group) algorithm that is analogous to the combination of selected
conﬁguration interaction plus perturbation theory (SCI+PT). We denote the resulting
method pDMRG (perturbative DMRG) to distinguish it from the standard variational
DMRG. pDMRG is expected to be useful for systems with very large active spaces, for
which variational DMRG becomes too expensive. Similar to SCI+PT, in pDMRG a
zerothorder wavefunction is ﬁrst obtained by a standard DMRG calculation, but with
a small bond dimension. Then, the residual correlation is recovered by a secondorder
perturbative treatment. We discuss the choice of partitioning for the perturbation the
ory, which is crucial for its accuracy and robustness. To circumvent the problem of a
large bond dimension in the ﬁrstorder wavefunction, we use a sum of matrix product
states (MPS) to expand the ﬁrstorder wavefunction, yielding substantial savings in
computational cost and memory. We also propose extrapolation schemes to reduce
the errors in the zeroth and ﬁrstorder wavefunctions. Numerical results for Cr
2
with
a (28e,76o) active space and 1,3butadiene with a (22e,82o) active space reveal that
pDMRG provides ground state energies of a similar quality to variational DMRG
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with very large bond dimensions, but at a signiﬁcantly lower computational cost. This
suggests that pDMRG will be an eﬃcient tool for benchmark studies in the future.
1 Introduction
Achieving chemical accuracy (ca. 1mE
h
) in systems with a mix of multireference and dy
namic correlations remains a challenging problem in molecular quantum chemistry. While
complete active spaces (CAS) with tens of partially ﬁlled orbitals can be reliably treated by
techniques such as the density matrix renormalization group (DMRG),
1–11
reaching chemical
accuracy in the subsequent description of the dynamic correlation is diﬃcult. The most com
mon technique to treat dynamical correlation in the multireference setting is secondorder
perturbation theory (PT).
12–23
However, one often ﬁnds that a secondorder perturbative
treatment is not powerful enough to accurately describe correlations involving some of the
moderately correlated nonvalence orbitals in a complex system. For example, in 3d transi
tion metal systems, binding energies and exchange couplings can be substantially in error if
the virtual 4d, semicore 3s3p, or valence ligand orbitals, are treated only at the secondorder
perturbative level. The standard remedy is to include these additional moderately correlated
orbitals in the multireference active space treatment. However, for complex systems this can
create enormous active spaces that are inaccessible or otherwise impractical even for current
DMRG methods.
Recently, selected conﬁguration interaction (SCI) methods
24–26
have experienced a signif
icant revival.
27–32
The general idea of selected conﬁguration interaction is quite old, dating
back to the CIPSI method,
24
and before that, to the handselected conﬁguration interaction
calculations carried out in the earliest days of quantum chemistry.
33,34
Although modern day
SCI methods diﬀer in how they select determinants, they all share a similar basic strategy.
In particular, a small number of determinants are ﬁrst selected for a variational treatment
 in modern calculations, typically 10
6
10
8
determinants  and the residual correlation is
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treated by secondorder PT, most commonly using the EpsteinNesbet (EN) partitioning.
Some important recent improvements include the use of stochastic methods to evaluate the
secondorder energies (E
2
) in order to handle large basis sets,
31,32
as well as the develop
ment of more systematic extrapolations with respect to the thresholds in the method. One
ﬁnds that SCI methods achieve chemical accuracy in the total energy for a variety of small
molecule problems using a remarkably small number of variational determinants. However,
it is important to observe that the variational CI energy alone is itself usually quite poor.
For example, in a heatbath CI calculation on the chromium dimer (48e, 42o) active space
30
popularized in DMRG benchmarks,
8
the variational CI energy was more than 60 mE
h
above
the the DMRG benchmark result. Instead, it is the second order PT correction that yields
the ﬁnal high accuracy result. In the above case, the total energy error after using the per
turbation theory correction is reduced to less than 1 mE
h
, a reduction by a factor of almost
one hundred. In other cases, corrections from PT reduce the total energy error by a factor
of 10 or more.
The remarkable accuracy of the secondorder perturbation correction in selected CI stands
in stark contrast to the accuracy of secondorder perturbation corrections when used with
complete active spaces. The physical reason for the diﬀerence is that even if the reference
wavefunction is determined exactly (within the complete active space) it is unbalanced due to
the lack of dynamical correlation. In contrast, although the variational selected CI computes
only a quite approximate reference wavefunction, it is determined in a full, or at least large,
space of orbitals, leading to a more balanced reference state. This suggests that the key
to an accurate secondorder correlation contribution involves balancing the diﬀerent orbital
correlations, rather than describing only the strongest correlations exactly, as in a valence
CAS. This observation is independent of choosing selected CI for the reference wavefunction,
and it is the motivation for this work.
In the current paper, we will explore how we can use quite approximate, but balanced,
variational DMRG reference wavefunctions computed in large active spaces, and correct
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them eﬃciently and to high accuracy, with second order PT within the same orbital space.
We name this technique “perturbatively corrected DMRG” or pDMRG. In pDMRG, we
represent both the zeroth order variational reference wavefunction Ψ
(0)
i as well as the ﬁrst
order perturbative correction Ψ
(1)
i in terms of matrix product states (MPS). Note that there
are advantages to using a MPS representation, rather than a determinantal expansion, of the
variational reference wavefunction. The MPS representation allows us to construct compact
strongly correlated wavefunctions even where there is little to no determinantal sparsity, for
example in systems with many coupled spins, where there is little sparsity in the coupled low
spin conﬁgurations of the system. A second reason is that volume extensivity of the energy
is achieved by a matrix product state with a cost ∝ e
V
2/3
rather than ∝ e
V
in conﬁguration
interaction. Asymptotically, this makes the variational MPS representation exponentially
more compact than a variational determinant expansion, and in practice, allows for a larger
number of spatially separated orbitals to be treated.
35
Relative to a standard variational DMRG calculation, the cost savings in pDMRG arise
from two sources. First, as described above, the zeroth order wavefunction can be computed
using a bond dimension M
0
much smaller than is needed to fully converge the variational
DMRG calculation. Second, although the bond dimension M
1
for the ﬁrst order wavefunction
still needs to be quite large, the ﬁrst order wavefunction is determined by minimizing the
Hylleraas functional,
18,36
L[Ψ
1
i] = hΨ
1
 (
ˆ
H
0
− E
0
) Ψ
1
i + 2 hΨ
1

ˆ
V Ψ
0
i ,
ˆ
V =
ˆ
H −
ˆ
H
0
. (1)
which is less expensive than minimizing the variational DMRG energy, because the zeroth
order Hamiltonian
ˆ
H
0
can be chosen to be simpler than the full Hamiltonian
ˆ
H. For example,
if
ˆ
H
0
is the Fock operator or the EpsteinNesbet Hamiltonian, then the computational cost
to evaluate the Hylleraas functional is a factor of K (where K is the number of orbitals) less
than that to evaluate the variational DMRG energy. In addition, since in secondorder PT,
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