  The similarities between Hartree-Fock (HF) theory and the density-matrix
renormalization group (DMRG) are explored. Both methods can be formulated as
the variational optimization of a wave-function ansatz. Linearization of the
time-dependent variational principle near a variational minimum allows to
derive the random phase approximation (RPA). We show that the non-redundant
parametrization of the matrix product state (MPS) tangent space [J. Haegeman et
al., Phys. Rev. Lett. 107, 070601 (2011)] leads to the Thouless theorem for
MPS, i.e. an explicit non-redundant parametrization of the entire MPS manifold,
starting from a specific MPS reference. Excitation operators are identified,
which extends the analogy between HF and DMRG to the Tamm-Dancoff approximation
(TDA), the configuration interaction (CI) expansion, and coupled cluster
theory. For a small one-dimensional Hubbard chain, we use a CI-MPS ansatz with
single and double excitations to improve on the ground state and to calculate
low-lying excitation energies. For a symmetry-broken ground state of this
model, we show that RPA-MPS allows to retrieve the Goldstone mode. We also
discuss calculations of the RPA-MPS correlation energy. With the long-range
quantum chemical Pariser-Parr-Pople Hamiltonian, low-lying TDA-MPS and RPA-MPS
excitation energies for polyenes are obtained.
