This thesis describes the development of perturbation theories based on the density matrix renormalization group (DMRG) and their applications to strongly correlated electronic systems. We introduce two types of perturbation theories based on DMRG for systems with a mixture of static and dynamic correlation.

The first one uses the framework of multi-reference perturbation theory. We present a combination of the DMRG and the strongly-contracted variant of second order N-electron valence state perturbation theory (SC-NEVPT2) that uses an efficient algorithm to compute high order reduced density matrices from DMRG wave functions. To demonstrate its capability, we apply it to calculations of the chromium dimer potential energy curve at the basis set limit, and the excitation energies of the poly-p-phenylene vinylene trimer.

The second one, perturbative DMRG (p-DMRG), is designed for systems with very large active spaces. In p-DMRG, a zeroth-order wavefunction with a small number of variational parameters is first obtained by a standard DMRG calculation. Then, the residual correlation is recovered by a second-order perturbative treatment. To circumvent the problem of a large bond dimension in the first order wavefunction, we use a sum of several DMRG wavefunctions and extrapolation schemes to expand the first-order wavefunction, yielding substantial savings in computational cost and memory.

We also present an efficient stochastic algorithm for p-DMRG, which bypasses the difficulty in solving the first-order equation in the deterministic algorithm. The key part of the algorithm is to represent the right hand side of the first-order equation, QVˆ