Barrier crossings starting equations

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

The starting point of a non-Markovian theory of barrier crossing is the generalized Langevin equation (cf. Eqs (8.61) and (8.62))... 

To solve the time-dependent Schrodinger equation with the boundary condition for (jc, r = 0) in the form of a wave packet. The wave packet may be directed into the entrance channel toward the reaction barrier (from various starting conditions). In the barrier range, the wave packet splits into a wave packet crossing the barrier and a wave packet reflected from the barrier (cf., p. 180). 

Starting with the molecular movement, the rate of a molecular process crossing an activation energy barrier can be related to temperature through the Arrhenius equation. 

---