Optimization of highly excited matrix product states

The wave function of a density matrix renormalization group (DMRG) calculation is represented as a matrix product state (MPS). Because DMRG is a ground-state method, excited states have to be calculated by a ground-state search in the space orthogonal to the already optimized lower-lying states. This is very impractical, especially for vibrational wave functions, where for large molecules the modes with high intensity are high in energy.

In our current paper we show how a combination of root-homing algorithms and shift-and-invert Hamiltonians or folded Hamiltonians allows us to efficiently optimize high-lying states directly. As the optimization is now independent from the optimzation of lower-lying states, these calculations are trivially parallelizable. Besides a thorough investigation of the performance of these methods, we apply them to the calculation of highly-excited vibrational states of ethylene and a dipeptide. We also demonstrate how a sampling reconstruction scheme for vibrational MPSs can aid the classification of a vibrational state as fundamental, overtone or combination band and how it allows us detect anharmonic couplings.

Many thanks go to Alberto Baiardi who did by far the biggest part of the implementation as well as our PIs Vincenzo Barone and Markus Reiher for their great and encouraging supervision.

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