Barrierless electronic relaxation in solution -- two state model with exact analytical solution in time domain

We propose an analytical method for solving the problem of electronic relaxation in solution in time domain, modelled by a particle undergoing diffusion under the influence of two coupled potentials. The coupling between the two potentials is assumed to be represented by a Dirac delta function of arbitrary position and strength. Smoluchowskii equation is used model the diffusion motion on both the potentials. We report an analytical expression for survival probability in time domain. This is the first time analytical solution in time domain is derived and this method can be used to solve problems involving other potentials.