Recrossing Dynamics and the Grote-Hynes Equation

The random force is due to the effect of all the solvent molecules as well as the nonreactive modes of the reactants. 

If one assumes that G takes on the form of an inverted parabola at the top of the barrier, G x) = — (where the transition state barrier is assumed 

the frequency-dependent friction is the Laplace transform of the time-dependent friction The presence of the Laplace transform means that the time-dependence of the friction must be known in order to determine the Laplace transform. This friction can be readily determined from molecular dynamics simulations in the approximation where the motion along the reaction coordinate is fixed at x = 0. (A discussion of some subtle, but important, aspects of this approximation is given by Carter et al. ) In that case, the random force R(t) can be calculated from equilibrium dynamics in the presence of this one constraint. From R(t), the time-dependent friction (t) can be calculated and the implicit Eq. [7] solved. The result gives the Grote—Hynes value of the transmission coefficient for that system. 

The calculation of the Grote-Hynes transmission coefficient has been performed for a wide variety of reaction systems in both weakly interacting and strongly interacting solvents atom exchange in rare gas solvents, and in water, electron transfer in dipolar solvents, ion pair 

Another use for standard models is as a target. It is important to determine at what point the model breaks down and whether that point is significant in realistic chemical dynamics. Some of the more important developments in the tests of Grote-Hynes theory have been in the application of variational transition state theory (VTST) to models of solution reaction dynamics. The origin of the use of VTST in solution dynamics is in the observation that the GLE can be equivalently formulated in Hamiltonian terms by a reaction coordinate coupled to a bath of harmonic oscillators. It has been shown by van der 

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