Path probability method

K. Masuda-Jindo, R. Kikuchi and R. Thomson, "Theory and Applications of the Cluster Variation and Path Probability Method" (Plenum, 1996) in press. 

T. Mohri, Y. Ichikawa, T. Nakahara and T. Suzuki, Theory and Applications of Cluster Vajiation and Path Probability Methods, ed. by J. M. Sanchez et al. Plenum, New York, in press. 

R. Kikuch, The path probability method. Prog. Theor. Phys. Svppl, 35,1-64(1966). 

These methods, which probably deserve more attention than they have received to date, simultaneously optimize the positions of a number of points along the reaction path. The method of Elber and Karpins [91] was developed to find transition states. It fiimishes, however, an approximation to the reaction path. In this method, a number (typically 10-20) equidistant points are chosen along an approximate reaction path coimecting two stationary points a and b, and the average of their energies is minimized under the constraint that their spacing remains equal. This is obviously a numerical quadrature of the integral s f ( (.v)where... 

The basis of the transition path sampling method is the statistical description of dynamical pathways in terms of a probability distribution. To define such a distribution consider a molecular system evolving in time and imagine that we take snapshots of this system at regularly spaced times fj separated by the time step At. Each of these snapshots, or states, consists of a complete description z of the system in terms of the positions q = <71, <72, , [Pg.252]

In the transition path sampling method we are interested in trajectories that start in a certain region of configuration space, which we will call region si, and end in another region, 38. We call such trajectories reactive. Accordingly, we restrict the probability distribution from (7.3) to reactive trajectories only (see Fig. 7.2)... 

J.N.L. Connor, W. Jakubetz, J. Manz, Exact quantum transition-probabilities by state path sum method—CoUinear F -F H2 reaction. Mot Phys. 29 (2) (1975) 347-355. 

It is obvious that the results of the two numerical methods are indistinguishable. Note that the procedure we outlined requires separate calculation at different times. An efficient method for calculating the probability at any other time f from the probability at a single time t is the transition path sampling method, which was developed by Dellago et al. [24]. We have employed that procedure to complete graph in Fig. 11. 

We develop in Chapter 3 a time dependent calculation of the ABC Green s function on a grid, called the power series Green s function (PSG). We compute the cumulative reaction probability for the collinear H+H2 test problem. The similarity of our approach in Chapter 3 to modern path integral methods is also discussed. 

At still shorter time scales other techniques can be used to detenuiue excited-state lifetimes, but perhaps not as precisely. Streak cameras can be used to measure faster changes in light intensity. Probably the most iisellil teclmiques are pump-probe methods where one intense laser pulse is used to excite a sample and a weaker pulse, delayed by a known amount of time, is used to probe changes in absorption or other properties caused by the excitation. At short time scales the delay is readily adjusted by varying the path length travelled by the beams, letting the speed of light set the delay. 

We recently received a preprint from Dellago et al. [9] that proposed an algorithm for path sampling, which is based on the Langevin equation (and is therefore in the spirit of approach (A) [8]). They further derive formulas to compute rate constants that are based on correlation functions. Their method of computing rate constants is an alternative approach to the formula for the state conditional probability derived in the present manuscript. 

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