  During the past 15 years, the density matrix renormalization group (DMRG) has
become increasingly important for ab initio quantum chemistry. Its underlying
wavefunction ansatz, the matrix product state (MPS), is a low-rank
decomposition of the full configuration interaction tensor. The virtual
dimension of the MPS, the rank of the decomposition, controls the size of the
corner of the many-body Hilbert space that can be reached with the ansatz. This
parameter can be systematically increased until numerical convergence is
reached. The MPS ansatz naturally captures exponentially decaying correlation
functions. Therefore DMRG works extremely well for noncritical one-dimensional
systems. The active orbital spaces in quantum chemistry are however often far
from one-dimensional, and relatively large virtual dimensions are required to
use DMRG for ab initio quantum chemistry (QC-DMRG). The QC-DMRG algorithm, its
computational cost, and its properties are discussed. Two important aspects to
reduce the computational cost are given special attention: the orbital choice
and ordering, and the exploitation of the symmetry group of the Hamiltonian.
With these considerations, the QC-DMRG algorithm allows to find numerically
exact solutions in active spaces of up to 40 electrons in 40 orbitals.
