  In the context of Frenkel exciton dynamics in aggregated molecules the
polaron transformation technique facilitates a treatment where diagonal
elements attributed to electronic excited-state populations are decoupled from
fluctuations associated with vibrational degrees-of-freedom. In this article we
describe for the first time how the polaron transformation can be applied in
the context of the "Hierarchical Equations of Motion" (HEOM) technique for
treatment of open quantum systems with all vibrational components attributed to
an environment. By using a generating function approach to introduce a shift in
the excited state potential energy surface, we derive hierarchical equations
for polaron transformation in analogy to those for time propagation. We
demonstrate the applicability of the developed approach by calculating the
dynamics of underdamped and overdamped oscillators coupled to electronic
excitation of a monomer without and with previous polaron transformation and
study the dynamics of the expectation value of the respective vibrational
coordinates. Furthermore, we investigate the dynamics of a dimer with a barrier
comparable to the thermal energy between the minima of the lower excitonic
potential energy surface. It turns out that the assumption of localization at
the monomer unit with energetically higher potential minimum, introduced via
polaron transformation, has a substantial influence on the transfer dynamics.
Here, it makes a clear difference whether the polaron transformation is
performed in the local or exciton basis. This reflects the fact that the
polaron transformation only accounts for equilibration of the vibrational, but
not of the excitonic dynamics. We sketch an approach to compensate this
shortcoming in view of obtaining an initial state for the calculation of
emission spectra of molecular aggregates.
