  We describe a low cost alternative to the standard variational DMRG (density
matrix 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 theory, 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 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.
