Optimization of highly excited matrix product states with an application to vibrational spectroscopy

An efficient approximation to the full configuration interaction solution can be obtained with the density matrix renormalization group (DMRG) algorithm without a restriction to a predefined excitation level. In a standard DMRG implementation, however, excited states are calculated with a ground-state optimization in the space orthogonal to all lower lying wave function solutions. A trivial parallelization is therefore not possible, and the calculation of highly excited states becomes prohibitively expensive, especially in regions with a high density of states. Here, we introduce two variants of the DMRG algorithm that allows us to target directly specific energy regions and therefore highly excited states. The first one, based on shift-and-invert techniques, is particularly efficient for low-lying states but is not stable in regions with a high density of states. The second one, based on the folded auxiliary operator, is less efficient but more accurate in targeting high-energy states. We apply the algorithm to the solution of the nuclear Schrödinger equation but emphasize that it can be applied to the diagonalization of general Hamiltonians as well, such as the electronic Coulomb Hamiltonian to address X-ray spectra. In combination with several root-homing algorithms and a stochastic sampling of the determinant space, excited states of interest can be adequately tracked and analyzed during the optimization. We validate these algorithms by calculating several highly excited vibrational states of ethylene and demonstrate that we can accurately calculate prominent spectral features of large molecules such as the sarcosine-glycine dipeptide.