  Fluorescence resonance energy transfer (FRET) is widely used as a
'spectroscopic ruler' to measure fluctuations in macromolecules because of the
strong dependence of the rate on the separation (R) between the donor (D) and
acceptor (A). However, the well-known Forster rate expression that predicts an
$R^{-6}$ dependence, is limited by several approximations. Notable among them
is the neglect of the vibronic relaxation in the reactant (donor) and product
(acceptor) manifolds. Vibronic relaxation can play an important role when the
energy transfer rate is faster than the vibronic relaxation rate. Under such
conditions, donor to acceptor energy transfer can occur from the excited
vibronic states. This phenomenon is not captured by the usual formulation based
on the overlap of donor emission and acceptor absorption spectra. Here, we
attempt to eliminate this lacuna, by allowing relaxation in the vibronic energy
levels and adopting a relaxation model to account for vibronic cascading down
in the donor manifold. We develop a Green's function based generalized
formalism and provide an exact solution for the excited state population
relaxation and the rate of energy transfer in the presence of vibronic
relaxation. We find and verify that the neglect of vibronic relaxations can
significantly alter the energy transfer rate and overestimates the distance
between D and A.
