Turnover theory

The main difference between the two approaches is that PGH consider the dynamics in the normal modes coordinate system. At any value of the damping, if the particle reaches the parabolic barrier with positive momentum i n the unstable mode p, it will immediately cross it. The same is not true when considering the dynamics in the system coordinate for which the motion is not separable even in the barrier region, as done by Mel nikov and Meshkov. In PGH theory the 

For E 0, let f(E)dEdt denote the probability to find the system within the time interval dt, with a mode energy between E and E + dE at the barrier of the p mode. For a thermal distribution W, near the barrier top feq(E) = 

The boundary condition for this integral equation is that deep in the well, equilibrium is maintained. If the barrier height is large with respect to keT, this allows one to replace the lower limit of the integration by - == . 

The important quantity here, is A which is the average energy lost by the unstable p mode as it traverses from the barrier to the well and back. The equation of motion for the imperturbed unstable mode is p + V (p) = 0 and this defines the trajectory p(t) which at time —is initiated at the barrier top, moves to the well, reaches a turning point and then comes back to the barrier top at the time + °o. The force exerted by the imstable mode on the bath comes from the nonlinearity F(t) = -w([uooP(t)]- The average energy loss A, to first order in Ui is then found to be (see also Eq. 10)  

For many one dimensional potentials, the infinite period trajectory is known analytically so that also the Fourier transformed force F(A) is known analytically. Finding the energy loss reduces then to a single quadrature. 

The force exerted by the unstable mode on the bath comes from the nonlinearity F(t) = - [u00p(t). The average energy loss A, to first order in U is then found to be (see also Eq. 10)  

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Hershkovitz E and Poliak E 1997 Multidimensional generalization of the PGH turnover theory for activated rate processes J. Chem. Phys. 106 7678... 

These include the Rayleigh quotient method" and variational transition state theory (VTST).46 9 xhg 0 called PGH turnover theory and its semiclassical analog/ which presents an explicit expression for the rate of reaction for almost arbitrary values of the friction function is reviewed in Section IV. Quantum rate theories are discussed in Section V and the review ends with a Discussion of some open questions and problems. 

The expressions for the depopulation factor as given in Eqs. 29 and 30 for the single and double well potential cases respectively, remain unchanged. This version of the turnover theory for space and time dependent friction has been tested successfully against numerical simulation data, in Refs. 68,137. 

This semiclassical turnover theory differs significantly from the semiclassical turnover theory suggested by Mel nikov, who considered the motion along the system coordinate, and quantized the original bath modes and did not consider the bath of stable normal modes. In addition, Mel nikov considered only Ohmic friction. The turnover theory was tested by Topaler and Makri, who compared it to exact quantum mechanical computations for a double well potential. Remarkably, the results of the semiclassical turnover theory were in quantitative agreement with the quantum mechanical results. 

There are two main ingredients that go into the semiclassical turnover theory, which differ from the classical limit.51 In the latter case, a particle which has energy E > 0 crosses the barrier while if the energy is lower it is reflected. In a semiclassical theory, at any energy E there is a transmission probability T(E) for the particle to be transmitted through the barrier. The second difference is that the bath, which is harmonic, may be treated as a quantum mechanical bath. Within first order perturbation theory, the equations of motion for the bath are those of a forced oscillator, and so their formally exact quantum solution is known. 

Kramers model is simplistic. It is one-dimensional and assumes that the friction is Markovian—uncorrelated in time. A multidimensional generalization of Kramers problem in the spatial diffusion limit was proposed and solved by Langer (18). The multidimensional energy diffusion limit was solved by Matkowsky, Schuss, and coworkers (19,20). A multidimensional turnover theory has been recently formulated (21). 

Kramers approach to rate theory in the underdamped and spatial-diffusion-limited regimes spurred extensions which were applicable to the much more complex STGLE. Grote and Hynes (23) used a parabolic barrier approximation to derive the rate expression for the GLE in the spatial diffusion limit. Carmeli and Nitzan derived expressions for the rate of the GLE (24) and the STGLE (25) in the underdamped limit. The overdamped limit for the rate in the presence of delta correlated friction was solved using the mean first passage time expression (26,27). A turnover theory, valid for space- and time-dependent friction, has only been recently presented by Haynes, Voth, and Poliak... 

We have discussed here in detail the aspects of the turnover theory that lead to an expression for the rate constant. The same methodology may be also used to obtain more detailed information on the distribution f E) of particles hitting the barrier. Specifically... 

D. Turnover Theory for Space- and Time-Dependent Friction... 

In the Kramers turnover theory (12,67), the expression for the rate, valid for all values of the damping is still given as a product of three factors as in Eq. (6). As discussed in the previous section, the depopulation factor is determined uniquely by the reduced energy loss parameter 8 = (3A (12,67), the explicit dependence is given in Eq. 

One of the main differences between the one-dimensional and the multidimensional turnover theory comes from the change between the one-dimensional and multidimensional energy loss mechanism. 

Note that the present formulation of the turnover theory has used the weak coupling perturbation theory and has not based itself on the normal modes of the system and the... 

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