Kramers-Grote-Hynes

Berezhkovskii A M, Poliak E and Zitserman V Y 1992 Activated rate processes generalization of the Kramers-Grote-Hynes and Langer theories J. Chem. Phys. 97 2422... 

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 ... 

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. 

The free energy iv[f] must now be varied with respect to the location f as well as with respect to the transformation coefficients ao, aj j = 1,.. . , N. The details are given in Ref 107 and have been reviewed in Ref 49. The final result is that the frequency A and collective coupling parameter C are expressed in the continumn limit as functions of a generalized barrier frequency A, One then remains with a minimization problem for the free energy as a function of two variables - the location f and A, Details on the mmierical minimization may be found in Refs. 68,93. For a parabolic barrier one readily finds that the minimum is such that f = 0 and that X = In other words, in the parabolic barrier limit, optimal planar VTST reduces to the well known Kramers-Grote-Hynes expression for the rate. 

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. 

The quantum thermodynamic factor S is the quantum correction to the Kramers-Grote-Hynes classical result in the spatial diffusion limited regime, derived by Wolynes " ... 

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 third result was the establishment of a connection between the TST and GLE viewpoints by Poliak." He solved for the normal modes of the Hamiltonian Eq. (7) and then used the result in a calculation of the reaction rate through the multi-dimensional TST. Surprisingly, he recovered the Kramers-Grote-Hynes... 

These results suggest a computational strategy for the study of reactions in condensed phases. One starts from some realistic intermolecular potentials and performs a molecular-dynamics-Kramers-Grote-Hynes scheme that consists of the following steps.First, we fix the proton at the transition state and run a MD simulation. The friction kernel y(t) is calculated and along with Eqs. (7,8) enables the calculation of the Grote-Hynes rate. This scheme has also been used as a means of obtaining input for quantum calculations as well. ... 

Once the function g(s) is known, one can make the following modification to the molecular-dynamics-Kramers-Grote-Hynes scheme we outlined at the end of the Introduction. 

It is also important to note that for a system coupled linearly to a finite discrete set of harmonic oscillators, the rate of escape over the barrier is described by the TST [150] and the TST rate is exactly given by Eq. (320). In the limit of continuum of oscillators, the Kramers-Grote-Hynes result is regained [Eq. (320)] [150]. 

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