  Linear response theory for the density matrix renormalization group
(DMRG-LRT) was first presented in terms of the DMRG renormalization projectors
[Dorando et al., J. Chem. Phys. 130, 184111 (2009)]. Later, with an
understanding of the manifold structure of the matrix product state (MPS)
ansatz, which lies at the basis of the DMRG algorithm, a way was found to
construct the linear response space for general choices of the MPS gauge in
terms of the tangent space vectors [Haegeman et al., Phys. Rev. Lett. 107,
070601 (2011)]. These two developments led to the formulation of the
Tamm-Dancoff and random phase approximations (TDA and RPA) for MPS. This work
describes how these LRTs may be efficiently implemented through minor
modifications of the DMRG sweep algorithm, at a computational cost which scales
the same as the ground-state DMRG algorithm. In fact, the mixed canonical MPS
form implicit to the DMRG sweep is essential for efficient implementation of
the RPA, due to the structure of the second-order tangent space. We present ab
initio DMRG-TDA results for excited states of polyenes, the water molecule, and
a [2Fe-2S] iron-sulfur cluster.
