Barrier-crossing rate

Haynes G R, Voth G A and Poliak E 1994 A theory for the activated barrier crossing rate constant in systems influenced by space and time dependent friction J. Chem. Phys. 101 7811... 

Ruiz-Montero M J, Frenkel D and Brey J J 1997 Efficient schemes to compute diffusive barrier crossing rates Mol. Phys. 90 925-41... 

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. 

Figure 15. Calculated values of the transmission coefficient k plotted as a function of the solvent viscosity rf for four barrier frequencies a>b at 7 = 0.85. The squares denote the calculated results for to = 3 x 1012 s I, the asterisks denote results for to = 5 x 1012 s-1, the triangles denote results for to = 1013 s I and the circles denote results for a>b = 2 x 1013 s l. The solid lines are the best-fit curves with exponents a 0.72 for wb = 3 x 1012s l, a 0.58 for wb = 5 x 1012 s-1,a 0.22 for wb = 1013 s l, and a 0.045 for cob = 2 x 10I3s-. Note here that the barrier crossing rate becomes completely decoupled from the viscosity of the solvent at wb = 2 x 10l3s-1. The transmission coefficient k is obtained by using Eq. (326). Note here that the viscosity is calculated using the procedure given in Section X and is scaled by a2/ /mkBT, and a>b is scaled by t -1. For discussion, see the text. This figure has been taken from Ref. 170.

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 the previous sections a model of the frequency-dependent collisional friction has been derived. Because the zero-frequency friction for a spherical particle in a dense fluid is well modeled by the Stokes-Einstein result, even for particles of similar size as the bath particles, there has been considerable interest in generalizing the hydrodynamic approach used to derive this result into the frequency domain in order to derive a frequency-dependent friction that takes into account collective bath motions. The theory of Zwanzig and Bixon, corrected by Metiu, Oxtoby, and Freed, has been invoked to explain deviation from the Kramers theory for unimolec-ular chemical reactions. The hydrodynamic friction can be used as input in the Grote-Hynes theory [Eq. (2.35)] to determine the reactive frequency and hence the barrier crossing rate of the molecular reaction. However, the use of sharp boundary conditions leads to an unphysical nonzero high-frequency limit to Ib(s). which compromises its utility. 

G. A. Voth, and E. Poliak, ]. Chem. Phys., 101, 7811 (1994). A Theory for the Activated Barrier Crossing Rate Constant in Systems Influenced by Space and Time Dependent Friction. 

In a similar way, the speed of exploration of the space should not be determined solely from functions of momentum. Far more reliable is to compute configurational quantities such as barrier crossing rates, passage times to reach some target region of phase space or the diffusion constant through an Einstein relation (which uses the position data). 

Schematic view of an electron travelling along the field direction in an ID system where a fraction x of the hopping sites carries a repulsive potential which a carrier has overcome either thermally or via tunnelling. Barriers are subject to a statistical distribution leading to a distribution p(W) of barrier crossing rates.

FIGURE 9.7 (a) Contour plot of the barrier-crossing rate for = 5.0. (b) Variation of the barrier-crossing rate with log(T). (c) Variation of the barrier-crossing rate with temperature (scale arbitrary). 

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