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During the past 15 years, the density matrix renormalization group (DMRG) hasbecome increasingly important for ab initio quantum chemistry. The underlyingmatrix product state (MPS) ansatz is a low-rank decomposition of the fullconfiguration interaction tensor. The virtual dimension of the MPS controls thesize of the corner of the many-body Hilbert space that can be reached.  Whereas the MPS ansatz will only yield an efficient description fornoncritical one-dimensional systems, it can still be used as a variationalansatz for other finite-size systems. Rather large virtual dimensions are thenrequired. The two most important aspects to reduce the correspondingcomputational cost are a proper choice and ordering of the active spaceorbitals, and the exploitation of the symmetry group of the Hamiltonian. Bytaking care of both aspects, DMRG becomes an efficient replacement for exactdiagonalization in quantum chemistry.  DMRG and Hartree-Fock theory have an analogous structure. The former can beinterpreted 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 tointroduce post-DMRG methods. Based on an approximate MPS, these methods provideimproved ans\"atze for the ground state, as well as for excitations.Exponentiation of the single-particle (single-site) excitations for a Slaterdeterminant (an MPS with open boundary conditions) leads to the Thoulesstheorem for Hartree-Fock theory (DMRG), an explicit nonredundantparameterization of the entire manifold of Slater determinants (MPSwavefunctions). This gives rise to the configuration interaction expansion forDMRG. The Hubbard-Stratonovich transformation lies at the basis of auxiliaryfield quantum Monte Carlo for Slater determinants. An analogous transformationfor spin-lattice Hamiltonians allows to formulate a promising variant for MPSs.

Accurate variational electronic structure calculations with the density matrix renormalization group