An exact solution in the theory of fluorescence resonance energy transfer with vibrational relaxation

The elegant expression of Förster that predicts the well-known 1/R⁶ distance (R) dependence of the rate of energy transfer, although widely used, was derived using several approximations. Notable among them is the neglect of the vibrational relaxation in the reactant (donor) and product (acceptor) manifolds. Vibrational relaxation can play an important role when the energy transfer rate is faster than the vibrational relaxation rate. Under such conditions, donor to acceptor energy transfer can occur from the excited vibrational states. This phenomenon is not captured by the usual formulation based on the overlap of donor emission and acceptor absorption spectra. Here, we develop a Green's function-based generalized formalism and obtain an exact solution for the excited state population relaxation and the rate of energy transfer in the presence of vibrational relaxation. We find that the application of the well-known Förster's expression might lead to overestimation of R.