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 ... 

Here we will give a simple derivation of the Grote-Hynes equation based on the GLE [7], Alternative derivations can be found in the literature [4,5]. 

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... 

Before we continue with the derivation of the Grote-Hynes expression for the transmission coefficient, it may be instructive to study the GLE, if not from the basic linear response theory point of view, then for a simple system where the GLE can be derived from the Hamiltonian of the system. For the special case where all forces are linear, that is, a parabolic reaction barrier and a harmonic solvent, it is possible to derive the GLE directly from the Hamiltonian. This allows us to identify and express the various terms in the GLE by system parameters, which helps to clarify the origin of the various terms in the equation. 

There are in principle dynamical recrossing corrections to the normal mode TST rate constant Equation (3.127) itself due to solvent frictional effects, which can be calculated via Grote-Hynes theory [81,82] when the time-dependent frictions on the three... 

Equations (25) and (26) are Grote-Hynes key results. They show that k is determined by Xr and that, in its turn, this reactive frequency Ar is determined both by the barrier frequency Wb and by the Laplace transform frequency component of the friction (see eq.(27)). 

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. 

---