Beyond Kramers, Grote-Hynes theory and MD

In order to evaluate the validity of Kramers theory, it is natural to focus on uni-molecular reactions since in this case diffusive motion does not come into play. 

This discrepancy can be due to a breakdown in the hydrodynamics friction law, Eq. (11.45), i.e., Stokes law, and/or a breakdown of the basic assumptions of Kramers theory. As we will see in the following section, a problem with Kramers theory is that the Langevin equation does not provide a sufficiently accurate description of the dynamics associated with the reaction coordinate. 

The Langevin equation, Eq. (11.5), that was used in Kramers calculation of the dynamical effects on the rate constant, is only valid in the limit of long times, where an equilibrium situation may be established. The reaction coordinate undergoes many collisions with the atoms in the solvent due to thermal agitation. From the Langevin equation of motion and Eq. (11.9), we obtained an expression for the autocorrelation function of the velocity  

This result cannot hold for small times t. The function in Eq. (11.52) has a discon- 

To overcome this difficulty Kubo, Mori, Zwanzig, and others introduced a time-dependent friction coefficient, writing, instead of the simple Langevin equation, the 

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