Implications and shortcomings of the Kramers theory

The rate of a barrier crossing reaction decreases like y in the y 0 limit and decreases hke y when y oo, where the friction y measures the strength of the molecule-solvent interaction, or rather the interaction between the reaction coordinate and its environment. 

As a function of y the rate goes through a maximum which is smaller than the TST rate. 

The last observation concerning the difference between small and large molecules points out one direction in which the one-dimensional Kramers theory 

Problem 14.6. Examine the theory of vibrational energy relaxation of Chapter 13 in order to explain the above statement that a Markovian theory of vibrational energy relaxation can overestimate the rate of this process by many orders of 

Another obvious shortcoming of the theory is its classical nature. The need for quantum mechanics can arise in two ways. First and obvious is the possibility that the transition of interest is affected by tunneling or by nonadiabatic curve-crossing transitions. We have discussed the TST aspects of these phenomena in Sections 14.3.5 and 14.3.6. Less obvious is the fact that, as discussed in Sections 13.4.1 and 13.6, quantum mechanical effects in the vibrational energy relaxation of small molecules can be very large. Both these manifestations of quantum effects in barrier crossing become important, in particular, at low temperatures. 

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