Transition path ensemble defined

In principle, we could compute C (t) from an ordinary path ensemble simulation. This would imply that we generate an ensemble of paths of length t that start at A and we would count all the paths that are at time t in B. However, since the transition from A to B is a rare event, the number of paths that end in B is so small that such an approach would require very long simulations. Therefore, we need to help the system explore the regions of interest. Suppose that region B can be defined by the value of an order parameter A xt e B if Amin < A (xt) < Amax-In principle, one could use more order parameters to characterize region B. For equation 6.1, we can write... 

In order to elucidate a mechanism, one must first consider the nature of the states initially formed by photoexcitation as well as the natures of other expected states eventually populated by internal conversion/intersystem crossing. Although it is by no means universally true, many transition metal complexes, when excited, undergo efficient relaxation to a bound, lowest energy excited state (LEES) or an ensemble of thermally equilibrated LEESs from which the various chemical processes lead to photoproducts. In such systems, the simplest model of which is illustrated by Figure 9, one can comfortably apply transition state theory to the rates and consider pressure effects in terms of the mechanisms of the individual decay LEES processes. In this case, the quantum yield of product formation would be defined by the ratio of rate constants by which the various chemical and photophysical paths for ES decay are partitioned. For Figure 9, in the absence of a bimolecular quencher Q, this would be... 

In an ergodic system, every possible trajectory of a particular duration occurs with a unique probability. This fact may be used to define a distribution functional for dynamical paths, upon which the statistical mechanics of trajectories is based. For example, with this functional one can construct partition functions for ensembles of trajectories satisfying specific constraints, and compute the reversible work to convert between these ensembles. In later sections, we will show that such manipulations may be used to compute transition rate constants. In this section, we derive the appropriate path distribution functionals for several types of microscopic dynamics, focusing on the constraint that paths are reactive, that is, that they begin in a particular stable state. A, and end in a different stable state, B. 

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