Beyond the linear approximations of the conventional approaches to the theory of chemical relaxation

The nonlinear coupling between the reacting system and its molecular bath results in a generalized Langevin equation with a memory kernel which is nonstationary as well as dependent on the reaction coordinate. In a preceding paper by Grigolini [J. Chem. Phys. 89, 4300 (1988)] a theory was developed to determine the reaction rate of a physical system characterized by a nonlinear interaction between system and bath. It is here shown that the local linearization adopted in that paper extends to this nonlinear condition the linear theory of Grote and Hynes, disregards also nonlinear effects, which does not conflict with the conservation of the Smoluchowski structure necessary to apply the standard first passage time approach. Here a clear distinction is made between the second-order local linearization (SOLL) and the infinite-order local linearization (IOLL). When deriving the Kramers equation from a microscopic description, it is possible to go beyond the SOLL approximation without contravening the basic requirement of keeping our description within a standard Fokker-Planck form. Thus, the influence of nonstationary memory kernel as well as that of the anharmonic contribution of the reaction potential can be conveniently described. The next step, of basic importance for a simple expression of the chemical reaction rate in the space diffusion regime to be found, consists of deriving the Smoluchowski equation. This must be taken in a careful way so that in the linear case the Grote and Hynes theory is recovered. The study of the simple linear case shows indeed that the contraction over the variable velocity of a Kramers equation which is not fully renormalized does not lead to a correctly renormalized Smoluchowski equation, even if the IOLL is applied. A simple rule to take into account the effects of higher-order terms is then found. In the linear case, this simple rule leads to a result coincident with the exactly renormalized structure. In the nonlinear case, at the second order in the interaction between system and bath, the novel expression coincides with the results provided by the current methods to take into account the anharmonic effects produced by colored noises. The final step of our approach consists of deriving the Smoluchowski equation from this fully renormalized Kramers equation by adopting the IOLL aproximation. The final result is more general than those previously derived by Grigolini, thereby also naturally including the Grote and Hynes theory.