Kramers-Grote-Hynes equation

If the potential of mean force is parabolic (w(q) = - imco q ) then the GLE (Eq. 1) may be solved using Laplace transforms. Denoting the Laplace transform of a function f(t) as f(s) = dte - f(t), taking the Laplace transform of the GLE and averaging over realizations of the random force (whose mean is 0) one finds that the time dependence of the mean position and velocity is determined by the roots of the Kramers-Grote-Hynes equation ... 

GLE and averaging over realizations of the random force (whose mean is 0) one finds that the time dependence of the mean position and velocity is determined by the roots of the Kramers-Grote-Hynes equation ... 

There is a one to one correspondence between the imperturbed fi equencies CO, C0j j = 1,. .., N,. .. appearing in the Hamiltonian equivalent of the GLE (Eq. 3) and the normal mode frequencies. The diagonalization of the potential has been carried out exphcitly in Refs. 88,90,91. One finds that the imstable mode frequency A is the positive solution of the Kramers-Grote Hynes (KGH) equation (7). This identifies the solution of the KGH equation as a physical barrier fi-equency. 

VTST has also been applied to systems with two degrees of freedom coupled to a dissipative bath." Previous results of Berezhkovskii and Zitserman which predicted strong deviations from the Kramers-Grote-Hynes expression in the presence of anisotropic friction for the two degrees of freedom " were well accounted for. Subsequent numerically exact solution of the Fokker-Planck equation further verified these results. 

A cornerstone of condensed phase reaction theory is the Kramers-Grote-Hynes theory. In a seminal paper Kramers solved the Fokker-Plank equation in two limiting cases, for high and low friction, by assiuning Markovian dynamics y(t) 5(t). He foimd that the rate is a non-monotonic function of the friction ( Kramers turnover .) Further progress was made by Grote and Hynes - who... 

The authors proceed to calculate the reaction rates by the flux correlation method. They find that the molecular dynamics results are well described by the Grote-Hynes theory [221] of activated reactions in solutions, which is based on the generalized Langevin equation, but that the simpler Kramers model [222] is inadequate and overestimates the solvent effect. Quite expectedly, the observed deviations from transition state theory increase with increasing values of T. 

The characteristic time scale for the motion of the particle in the parabolic top barrier is the inverse barrier frequency, the sharper is the barrier, the faster is the motion. Typically, atom transfer barrier are quite sharp therefore the key time scale is very short, and the short-time solvent response becomes relevant instead of the long-time overall response given by the ( used in Kramers theory (see eq.(20)). To account for this critical feature of reaction problems, Grote and Hynes (1980) introduce the generalized Langevin equation (GLE) ... 

Kramers (11) correction to the one-dimensional TST rate came from consideration of the properties of the Fokker-Planck equation in the vicinity of the barrier in the presence of ohmic friction. As noted in the previous section, if one considers only the parabolic barrier limit, the Fokker-Planck equation may be solved analytically. Grote and Hynes (23) and Hanggi and Mojtabai (65) generalized Kramers result to include the case of memory friction and the GLE. A different approach (31) would be to consider the Hamiltonian equivalent, Equation (29), for space-independent coupling (g(q) = q in Eq. (29)) in the parabolic barrier limit. 

Several remarks are in order here. First, the fact that the correlation functions need to be evaluated at the transition frequency reminds us of the earlier classical results for barrier transitions. A friction coefficient is also related to solvent fluctuations in fact, it can be written as a velocity-velocity correlation function, and as such the Redfleld theory is not much different from classical theories. This is to be expected because these can be derived from a very similar formalism based on the classical Liouville equation, albeit without the problems encountered here. In the Kramers case, we needed to evaluate the friction at zero frequency, which sometimes severely overestimates its role. It would be better to evaluate it at the barrier frequency. In the theory developed by Grote and Hynes, the friction needs to be evaluated not at the original barrier frequency but at the frequency the barrier transition actually takes place as Eq. (9.13) shows. The reason is simple in the latter case the back reaction of the system on the solvent is taken into account, something that was not allowed in the Redfield theory. 

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