Frenkel exciton population dynamics of an excitonic dimer is studied by comparing the results from a quantum master equation involving rates from second-order perturbative treatment with respect to the excitonic coupling with the non-perturbative results from "Hierarchical Equations of Motion" (HEOM). By formulating generic Liouville-space expressions for the rates, we can choose to evaluate them either via HEOM propagations or by applying the cumulant expansion. The coupling of electronic transitions to bath modes is modeled either as overdamped oscillators for the description of thermal bath components or as underdamped oscillators to account for intramolecular vibrations. Cases of initial nonequilibrium and equilibrium vibrations are discussed. In the case of HEOM, initial equilibration enters via a polaron transformation. Pointing out the differences between the nonequilibrium and equilibrium approach in the context of the projection operator formalism, we identify a further description, where the transfer dynamics is driven only by fluctuations without involvement of dissipation. Despite this approximation, this approach can also yield meaningful results in certain parameter regimes. While for the chosen model, HEOM has no technical advantage for evaluation of the rate expressions compared to cumulant expansion, there are situations where only evaluation with HEOM is applicable. For instance, a separation of reference and interaction Hamiltonian via a polaron transformation to account for the interplay between Coulomb coupling and vibrational oscillations of the bath at the level of a second-order treatment can be adjusted for a treatment with HEOM.
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