Barrier-crossing dynamics

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

The key quantity in barrier crossing processes in tiiis respect is the barrier curvature Mg which sets the time window for possible influences of the dynamic solvent response. A sharp barrier entails short barrier passage times during which the memory of the solvent environment may be partially maintained. This non-Markov situation may be expressed by a generalized Langevin equation including a time-dependent friction kernel y(t) [ ]... 

Schroeder J and Troe J 1993 Soivent effects in the dynamics of dissociation, recombination and isomerization reactions Activated Barrier Crossing ed G R Fieming and P Hanggi (Singapore Worid Scientific) p 206... 

Agmon N and Kosloff R 1987 Dynamics of two-dimensional diffusional barrier crossing J. Phys. Chem. 91 1988-96... 

Haynes G R and Voth G A 1995 Reaction coordinate dependent friction in classical activated barrier crossing dynamics when it matters and when it doesn t J. Chem. Phys. 103 10 176... 

In HMC the momenta are constantly being refreshed with the consequence that the accompanying dynamics will generate a spatial diffusion process superposed on the ini rtial dynamics, as in BGK or Smoluchowski dynamics. It is well known from the theory of barrier crossing that this added spatial... 

In the situation where the transformation involved barrier crossing, e.g., associated with a nonpolar to polar transformation, the computational time was substantially reduced using the X-dynamics formalism, compared with a standard FEP method. This is because X-dynamics searches for alternative lower free energy pathways the coupling parameters (A/ and A2) evolve in the canonical ensemble independently and find a smoother path then when constrained to move as A = A2. Furthermore, a biasing potential in the form... 

Charutz, D. M. and Levine, R. D. Dynamics of barrier crossing in solution simulations and a hard-sphere model, J.Chem.Phys., 98 (1993), 1979-1988... 

J. Hicks, M. Vandersall, Z. Babarogic, and K. B. Eisenthal, The dynamics of barrier crossings in solution The effect of a solvent polarity-dependent barrier, Chem. Phys. Lett. 116, 18-24(1985). 

Nonadiabatic effects of course occur. When they are not too important, they can be treated by perturbation theory. When they are strong, SACM breaks down. Barrier crossing rates may be less sensitive to nonadiabatic effects than detailed product-state distributions. The classical trajectory treatments from refs. [3], [34], and [36] of my article quantitatively describe the transition from adiabatic to non-adi-abatic dynamics, where the former corresponds to small and the latter to large relative kinetic energies between the reactants. 

In the limit of very large viscosity, such as the one observed near the glass transition temperature, it is expected that rate of isomerization will ultimately go to zero. It is shown here that in this limit the barrier crossing dynamics itself becomes irrelevant and the Grote-Hynes theory continues to give a rate close to the transition theory result. However, there is no paradox or difficulty here. The existing theories already predict an interpolation scheme that can explain the crossover to inverse viscosity dependence of the rate... 

As can be seen from the numbers, the exponent a is clearly a function of barrier frequency (cob) and its value is decreasing with increase in a>b- For cob — 2 x 1013 s-1, its value almost goes to zero (a < 0.05), which clearly indicates that beyond this frequency the barrier crossing rate is entirely decoupled from solvent viscosity so that one recovers the well-known TST result that neglects the dynamic solvent effects. 

In order to complete the above analysis, one needs to solve the full non-Markovian Langevin equation (NMLE) with the frequency-dependent friction for highly viscous liquids to obtain the rate. This requires extensive numerical solution because now the barrier crossing dynamics and the diffusion cannot be treated separately. However, one may still write phenomenologically the rate as [172],... 

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