  Frenkel exciton population dynamics of an excitonic dimer is studied by
comparing results from a quantum master equation (QME) involving rates from
second-order perturbative treatment with respect to the excitonic coupling with
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 cumulant expansion.
The coupling of electronic transitions to bath modes is modeled either as
overdamped oscillators for description of thermal bath components or as
underdamped oscillators to account for intramolecular vibrations. Cases of
initial nonequilibrium and equilibrium vibrations are discussed. In 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, also this approach can
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.
