Mapping the reaction dynamics in Liouville space onto a reaction coordinate space: a first-principle- based theory

We have derived here an exact kinetic equation for the time evolution of the probability distribution for a general reaction coordinate space, starting from a multidimensional Liouville equation based on first-principles theory. To make the equation tractable we use two standard approximations, which reduce the exact equation into a Fokker-Planck-type equation with a sink term. As illustrative examples, we consider its application to two important classes of reactions, viz., the electron transfer and diffusion-controlled reactions. The kinetic equations for the reaction coordinates corresponding to these reactions become one dimensional. We also provide a scheme to find the critical value of the reaction coordinate at which the reaction takes place, and also to obtain the initial distribution function in the same space for different experimental situations. In the case of photoinduced electron transfer, we introduce the effect of excitation frequency through the use of suitable initial distribution functions.