We apply the recently developed Liouville space hierarchical equations of motion (HEOM) method to calculate the quantum rate dynamics for a model system of proton transfer reaction in condensed phase, which consist of a double well coupled to a harmonic bath with the Debye spectral density. The HEOM method provides a new way to directly calculate nonequilibrium reduced system dynamics, and the calculated reaction rate constants compare well with previous numerical exact results. The HEOM method also allows us to perform long time simulations, which enables systematic studies of the reaction dynamics at low frictions. The applicability of perturbative quantum master equations at various orders is also investigated by comparing with numerical exact HEOM results.
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