Reactive frequency

This self-consistent equation has a simple message the relevant friction for the reaction is determined by the (Laplace) frequency component of the time dependent friction at the reactive frequency X. This frequency sets the basic time scale for the microscopic events affecting k. 

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Hanggi and Mojtabai189 based their treatment on the generalized Fokker-Planck expression and demonstrated that the steady-state escape rate is given by the Grote and Hynes relationship (4.200) where the reactive frequency Ar is defined as the long-time limit given by... 

The diffusion coelficient in the barrier region is affected by the curvature of the potential, and is expressed in terms of the reactive frequency Ay, which was shown by Grote and Hynes to be the solution to... 

The frequency dependence of the flux is found from solution of the reactive frequency Aj in Eq. (2.35), which presupposes a knowledge of Ibm(s)- In the spirit of this work, this is given by Eq. (2.34) in terms of b( S) and Im( ) although the contribution from the region R > R as discus.sed m Section IIC may have to be taken into account. The form for 7t( ) is... 

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. 

With the assumption of a parabolic reaction barrier in the top region, the reaction transmission coefficient, i.e. the ratio of the actual rate constant to its TST value, k = k/kTST, is found to be the ratio of the reactive frequency Ar to the mean barrier frequency ... 

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)). 

BMI DCA is not significant. We attribute this intriguing finding to fast ET dynamics in the barrier region we noted above. Since barrier crossing occurs very rapidly, it is governed not by the long-time hydrodynamic friction related to viseosity, but by the short-time frietion evaluated at the reactive frequency. 

Where X is a reactive frequency found from the iterative solution to the equation ... 

The poles (zeroes of the denominator) of the propagator G(time behavior of the position and momentum of the Brownian particle on the barrier. In this case, there are two poles, one with a negative real part and one with a positive real part. The one with the negative real part corresponds to the damping of momentum. The solution with positive real part corresponds to motion in the direction oo, and can therefore be called the reactive frequency. 

We have seen in Section 9.3 that tautomerization involves changing the charge distribution in the molecule, and the above considerations imply that a polarizable solvent will exert a frictional force on this process. If this is viewed as a barrier process - but we will argue that this is only partly the case - then there is an insightful way to find the reactive frequency. The solution to Eq. (9.9), namely a>, can formally also be written as... 

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