Transition Path Sampling - Rate Constant Computation

Rate Constant Computation

In the Bennett–Chandler procedure the rate constant kAB for the transition from A to B is derived from the correlation function

C(t) = \frac{\langle h_A(0) h_B(t) \rangle}{\langle h_A \rangle}

where hA(B) is the characteristic function of state A(B), and hA(B)(t) is either 1 if the system at time t is in state A(B) or 0 if not. The time-derivative C'(t) starts at time 0 at the transition state theory (TST) value kABTST and reaches a plateau kAB ≤ kABTST for times of the order of the transition time. Hence once the function is known up to these times, the rate constant is also available.

In the TPS framework C(t) can be rewritten as an average in the path ensemble

k_{AB}^{TPS}(t) = \frac{d}{dt}C(t) = \frac{\langle \dot{h_B(t)} \rangle_{AB}}{\langle h_B(t') \rangle_{AB}} C(t')

where the subscript AB denotes an average in the ensemble of paths that start in A and visit B at least once. Time t' is an arbitrary time in the plateau region of C(t). The factor C(t') at this specific time can be computed with a combination of path sampling and umbrella sampling.

Read more about this topic:  Transition Path Sampling

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