Regime Kramer

According to Kramers model, for flat barrier tops associated with predominantly small barriers, the transition from the low- to the high-damping regime is expected to occur in low-density fluids. This expectation is home out by an extensively studied model reaction, the photoisomerization of tran.s-stilbene and similar compounds [70, 71] involving a small energy barrier in the first excited singlet state whose decay after photoexcitation is directly related to the rate coefficient of tran.s-c/.s-photoisomerization and can be conveniently measured by ultrafast laser spectroscopic teclmiques. 

This ensures the correct connection between the one-dimensional Kramers model in the regime of large friction and multidimensional imimolecular rate theory in that of low friction, where Kramers model is known to be incorrect as it is restricted to the energy diflfiision limit. For low damping, equation (A3.6.29) reduces to the Lindemann-Flinshelwood expression, while in the case of very large damping, it attains the Smoluchowski limit... 

According to Kramers model, for flat barrier tops associated with predominantly small barriers, the transition from the low- to the high-damping regime is expected to occur in low-density fluids. This expectation is home... 

Borkoveo M and Berne B J 1985 Reaotion dynamios in the iow pressure regime the Kramers modei and ooiiision modeis of moieouies with many degrees of freedom J. Chem. Phys. 82 794-9... 

This result reflects the Kramers relation (Gardiner, 1985). A millisecond time of unbinding, i.e.. Tact 1 ms, corresponds in this case to a rupture force of 155 pN. For such a force the potential barrier AU is not abolished completely in fact, a residual barrier of 9 kcal/mol is left for the ligand to overcome. The AFM experiments with an unbinding time of 1 ms are apparently functioning in the thermally activated regime. 

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 " ... 

Assuming that occe2lh in the metallic regime, many observations support v = 1 both above and below the transition (cf. pp. 150, 157). On the other hand, calculations by Kramer and Mackinnon (cf. Chapter 1, Section 6.3) give v = 1.6, of course without electron-electron interactions. Mackinnon (private communication) suggests that interactions may result in v=l. The only... 

The situation is analogous to the so-called energy diffusion regime in the Kramers picture of reactions in solution except that here the molecule acts as its own solvent [E. W. Schlag, J. Grotemeyer, and R. D. Levine, Chem. Phys. Lett. 190, 521 (1992)]. 

Equation (320) predicts the TST result for very weak friction (Ar to ) and predicts the Kramers result for low barrier frequency (i.e., (ob —> 0) so that (2r) can be replaced by (0) in Eq. (322). If die barrier frequency is large (ia>b > 1013 s 1) and the friction is not negligible ( (0)/fi — cob), then the situation is not so straightforward. In this regime, which often turns out to be the relevant one experimentally, the effective friction (2r) can be quite small even if the zero frequency (i.e., the macroscopic) friction (proportional to viscosity) is very large. The non-Markovian effects can play a very important role in this regime. 

The second part (sections H and I) is devoted to a detailed discussion of the dynamics of unimolecular reactions in the presence and the absence of a potential barrier. Section H presents a critical examination of the Kramers approach. It is stressed that the expressions of the reaction rates in the low-, intermediate-, and high-friction limits are subjected to restrictive conditions, namely, the high barrier case and the quasi-stationary regime. The dynamics related to one-dimensional diffusion in a bistable potential is analyzed, and the exactness of the time dependence of the reaction rate is emphasized. The essential results of the non-Markovian theory extending the Kramers conclusions are also discussed. The final section investigates in detail the time evolution of an unimolecular reaction in the absence of a potential barrier. The formal treatment makes evident a two-time-scale description of the dynamics. 

KRAMERS EQUATION—ARBITRARY FRICTION REGIME. In the presence Of inertial effects, the one-dimensional motion is determined by a bivariate Fokker-Planck equation. 

Different attempts have been elaborated178-182 to find a general expression that accounts for the limiting forms (4.171) and (4.176) obtained in the high- and extreme-low-friction regimes. A new approach183 made conspicuous the connection between the diverse Kramers limits. 

Fig. 4.6. Influence of the temperature on the Kramers reaction rate [intermediate friction regime Eq. (4.174)] for o>bt° = 5x10j (curves 1) or 104 (curves 2). The straight line represents e p(-QJkeT) with ( B = 2 kcal M. ...

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