A Perturbative Density Matrix Renormalization Group Algorithm for Large Active Spaces.

TL;DR: Numerical results for Cr2 with a (28e, 76o) active space and 1,3-butadiene with an (22e, 82o)active space reveal that p-DMRG provides ground state energies of a similar quality to variational DMRG with very large bond dimensions but at a significantly lower computational cost, which suggests that the method will be an efficient tool for benchmark studies in the future.

Abstract: We describe a low cost alternative to the standard variational DMRG (density matrix renormalization group) algorithm that is analogous to the combination of the selected configuration interaction plus perturbation theory (SCI+PT). We denote the resulting method p-DMRG (perturbative DMRG) to distinguish it from the standard variational DMRG. p-DMRG is expected to be useful for systems with very large active spaces, for which variational DMRG becomes too expensive. Similar to SCI+PT, in p-DMRG, a zeroth-order wave function is first obtained by a standard DMRG calculation but with a small bond dimension. Then, the residual correlation is recovered by a second-order perturbative treatment. We discuss the choice of partitioning for perturbation theory, which is crucial for its accuracy and robustness. To circumvent the problem of a large bond dimension in the first-order wave function, we use a sum of matrix product states to expand the first-order wave function, yielding substantial savings in computational cost and memory. We also propose extrapolation schemes to reduce the errors in the zeroth- and first-order wave functions. Numerical results for Cr2 with a (28e, 76o) active space and 1,3-butadiene with an (22e, 82o) active space reveal that p-DMRG provides ground state energies of a similar quality to variational DMRG with very large bond dimensions but at a significantly lower computational cost. This suggests that p-DMRG will be an efficient tool for benchmark studies in the future.

Summary

1 Introduction

• Achieving chemical accuracy (ca. 1mEh) in systems with a mix of multireference and dynamic 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.
• 33,34 Although modern day SCI methods differ in how they select determinants, they all share a similar basic strategy.
• Note that there are advantages to using a MPS representation, rather than a determinantal expansion, of the variational reference wavefunction.
• Two particular pieces needed to establish p-DMRG as an accurate and efficient alternative to variational DMRG are then discussed in the following sections.

2.1 Perturbative density matrix renormalization group (p-DMRG)

• Here the authors first recapitulate the DMRG algorithm in the MPS language.
• The DMRG algorithm provides an efficient way to variationally optimize an MPS that optimizes the tensors site-by-site.
• The correlation treatment offered by the MPS, where every orbital is treated on an equal footing, is too flexible.

2.2 Choices of zeroth-order Hamiltonian Ĥ0

• There are several criteria that a good partitioning of Ĥ must satisfy.
• The Fock operator or the diagonal part of Ĥ in the determinant space used in the EN partition both satisfy this criterion, while the simplest projective definition Ĥ0 = PĤP +QĤQ does not.
• For this form of Ĥ0, when solving Eq. (9) using the DMRG sweep algorithm, the Hamiltonian and wavefunction multiplication on the left hand side (LHS) scales as O(K2M31 ) instead of O(K3M3) in the standard variational DMRG.
• Thus, it is much lower than the lowest energy of the perturbers, which is the lowest eigenvalue of QĤdQ, whose eigenstates are relatively uncorrelated.
• In contrast, the PT3 energy varies more slowly.

2.3 Splitting the first order wavefunction

• A similar computational obstacle arises also in SCI+PT, which gives rise to the memory bottleneck associated with storing all determinants contributing to the first-order wavefunction.
• One way to remove this bottleneck is to use a stochastic computation of the perturbation correction, as proposed in31,32 for SCI+.
• In the current work, the authors will use a deterministic approach, where they represent the first-order wavefunction as a linear combination of MPS,45 each with a modest bond dimension.
• The form of the LHS is the same for each i, but the RHS becomes more costly as N increases.
• Thus, in practice, the authors try to use an M1 as large as possible given the computational resources, and only then use Eq. (12) to continue the calculations to a larger effective M1, which would otherwise be too costly within a single MPS representation.

3 Results

• To test the performance of p-DMRG for various choices of Ĥ0, the authors examined two diatomic molecules: C2 and Cr2, for which variational DMRG results are available in the literature.
• All electrons were correlated corresponding to an orbital space of (12e,60o).
• Energy (E+2099 in Eh) of Cr2 obtained with standard variational DMRG in the cc-pVDZ-DK basis set, also known as Table 1.
• 2 = ∑i j=1E2,j represents the accumulated second-order perturbation energy for the sum of the first i first-order MPS.
• Using the extrapolated E2, the p-DMRG energies with every M0 are lower than the variational DMRG results with M=12000, the largest M used.

3.3 Butadiene with (22e, 82o) active space

• The final system the authors consider is 1,3-butadiene.
• The authors used the same basis ANO-L-VDZP[3s2p1d]/[2s1p]60 as used in previous studies.
• All electrons except for a frozen 1s core were correlated, leading to an orbital space with (22e, 82o).
• The authors used the same extrapolation procedures as used for Cr2 in the previous section.
• A similar extrapolation using the previous DMRG energies gives -155.5578Eh, which is consistent with the extrapolated p-DMRG energy.

4 Conclusion

• The authors defined a p-DMRG method that uses perturbation theory within the DMRG framework to efficiently target exact energies in large orbital spaces where not all orbitals are strongly correlated.
• Using a carefully defined zeroth order Hamiltonian, and with extrapo- lation procedures, the authors found that p-DMRG can indeed provide benchmark quality energies as accurate as those obtained in far more expensive standard variational DMRG calculation.
• The zeroth-order DMRG energies are slightly different from the previous DMRG energies in Ref. 8 due to the use of different orbitals and optimization schedules.
• Split-localized canonical orbitals were used in this work, and E (0) DMRG are for converged one-site MPS calculations without any noise.

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Quantum Electronic Structure
A Perturbative Density Matrix Renormalization
Group Algorithm for Large Active Spaces
Sheng Guo, Zhendong Li, and Garnet Kin-Lic Chan
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A Perturbative Density Matrix Renormalization
Group Algorithm for Large Active Spaces
Sheng Guo, Zhendong Li, and Garnet Kin-Lic Chan
∗
Division of Chemistry and Chemical Engineering, California Institute of Technology,
Pasadena, CA 91125, USA
E-mail: 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 p-DMRG (perturbative DMRG) to distinguish it from the standard variational
DMRG. p-DMRG is expected to be useful for systems with very large active spaces, for
which variational DMRG becomes too expensive. Similar to SCI+PT, in p-DMRG a
zeroth-order wavefunction is first obtained by a standard DMRG calculation, but with
a small bond dimension. Then, the residual correlation is recovered by a second-order
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 first-order wavefunction, we use a sum of matrix product
states (MPS) to expand the first-order wavefunction, yielding substantial savings in
computational cost and memory. We also propose extrapolation schemes to reduce
the errors in the zeroth- and first-order wavefunctions. Numerical results for Cr
2
with
a (28e,76o) active space and 1,3-butadiene with a (22e,82o) active space reveal that
p-DMRG provides ground state energies of a similar quality to variational DMRG
1
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with very large bond dimensions, but at a significantly lower computational cost. This
suggests that p-DMRG 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 second-order
perturbation theory (PT).
12–23
However, one often finds that a second-order perturbative
treatment is not powerful enough to accurately describe correlations involving some of the
moderately correlated non-valence 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, semi-core 3s3p, or valence ligand orbitals, are treated only at the second-order
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 hand-selected 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 second-order PT, most commonly using the Epstein-Nesbet (EN) partitioning.
Some important recent improvements include the use of stochastic methods to evaluate the
second-order 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 heat-bath 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 second-order perturbation correction in selected CI stands
in stark contrast to the accuracy of second-order 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 second-order 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 p-DMRG. In p-DMRG, 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 p-DMRG 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 Epstein-Nesbet 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 second-order PT,
4
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