# 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 (2 min read)

### 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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