Barrier sampling, classical trajectory

The key idea that supplements RRK theory is the transition state assumption. The transition state is assumed to be a point of no return. In other words, any trajectory that passes through the transition state in the forward direction will proceed to products without recrossing in the reverse direction. This assumption permits the identification of the reaction rate with the rate at which classical trajectories pass through the transition state. In combination with the ergodic approximation this means that the reaction rate coefficient can be calculated from the rate at which trajectories, sampled from a microcanonical ensemble in the reactants, cross the barrier, divided by the total number of states in the ensemble at the required energy. This quantity is conveniently formulated using the idea of phase space. 

Figure 6 An illustration of the problems that can arise when parts of parameter space are nonphysical. Assume that one is interested in sampling the potential for energies less than the maximum of the spline fit function. Classical trajectories coming from the right cannot surmount the barrier and will behave almost tihe same on the spline fit function as on the true potential. However, the GA can sample anywhere and has the possibility of accessing the nonphysical region to the left of the maximum in the spline function. One solution is to add a penalty function whose value is zero except for regions of small distance. The sum of die spline fit function and the penalty function will be at least as large as the true potential in this classically forbidden. region and will thereby push solutions to larger distances.

We tested whether sampling efficiency could be further enhanced by using swapping [44] and jump-walking [45], two improvements on the classical MMC method that combine sampling at two different temperatures to generate search trajectories that can more easily cross barriers. 

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