  During the past 15 years, the density matrix renormalization group (DMRG) has
become increasingly important for ab initio quantum chemistry. The underlying
matrix product state (MPS) ansatz is a low-rank decomposition of the full
configuration interaction tensor. The virtual dimension of the MPS controls the
size of the corner of the many-body Hilbert space that can be reached.
  Whereas the MPS ansatz will only yield an efficient description for
noncritical one-dimensional systems, it can still be used as a variational
ansatz for other finite-size systems. Rather large virtual dimensions are then
required. The two most important aspects to reduce the corresponding
computational cost are a proper choice and ordering of the active space
orbitals, and the exploitation of the symmetry group of the Hamiltonian. By
taking care of both aspects, DMRG becomes an efficient replacement for exact
diagonalization in quantum chemistry.
  DMRG and Hartree-Fock theory have an analogous structure. The former can be
interpreted as a self-consistent mean-field theory in the DMRG lattice sites,
and the latter in the particles. It is possible to build upon this analogy to
introduce post-DMRG methods. Based on an approximate MPS, these methods provide
improved ans\"atze for the ground state, as well as for excitations.
Exponentiation of the single-particle (single-site) excitations for a Slater
determinant (an MPS with open boundary conditions) leads to the Thouless
theorem for Hartree-Fock theory (DMRG), an explicit nonredundant
parameterization of the entire manifold of Slater determinants (MPS
wavefunctions). This gives rise to the configuration interaction expansion for
DMRG. The Hubbard-Stratonovich transformation lies at the basis of auxiliary
field quantum Monte Carlo for Slater determinants. An analogous transformation
for spin-lattice Hamiltonians allows to formulate a promising variant for MPSs.
