Probability flux correlation function

For a dielectric continuum environment, the probability flux correlation function is given by ... 

The rate expression in Eq. (16.15) can be simplifled signiflcantly under a series of well-deflned approximations. In the short-time high-temperature approximation for a Debye solvent, the solvent dynamics is negligible on the time scale of the probability flux correlation function, leading to the following simpliflcation ... 

Moreover, when the probability flux correlation function decays on a time scale shorter than the time scale of the solvent effects on the R motion, the solvent effects on the R motion can be neglected. In this case, Cf,(t) can be approximated by the standard analytical expression for the time correlation function of an undamped quantum mechanical harmonic oscillator [53] ... 

We have used this theoretical formulation to analyze the dynamical aspects of model PCET reactions in solution with molecular dynamics simulations [56, 57]. For these model systems, the time dependence of the probability flux correlation function is dominated by the solvent damping term, and only the short-time equilibrium fluctuations of the solvent impact the rate. The proton donor-acceptor motion does not impact the dynamical behavior of the reaction but does influence the magnitude of the rate. 

For the H-rCH4 H2+CH3PO] and O+CH4 OH+CHspI] reactions, accurate Multi Conhgurational Time Dependent Hartree (MCTDH) calculations have been performed. These calculations include all vibrational modes and are for a total angular niomcntum J = 0. The flux correlation function formalism[68. 69, 70] was employed to calculate cumulative reaction probabilities for J — 0. 

To deal with the ET rate in such a case, our strategy is to combine the generalized nonadiabatic transition state theory (NA-TST) and the Zhu-Nakamura (ZN) nonadiabatic transition probability.The generalized NA-TST is formulated based on Miller s reactive flux-flux correlation function approach. The ZN theory, on the other hand, is practically free from the drawbacks of the LZ theory mentioned above. Numerical tests have also confirmed that it is essential for accurate evaluation of the thermal rate constant to take into account the multi-dimensional topography of the seam surface and to treat the nonadiabatic electronic transition and nuclear tunneling effects properly. 

In energy domain, the cumulative reaction probability can also be calculated employing flux correlation functions [3, 4] ... 

We begin the calculation of the cumulative reaction probability with an expression derived from the analysis of reactive flux correlation functions [4], given by... 

Time correlation functions are our main tools for conveying this information in stationary systems. These are systems at thermodynamic equilibrium or at steady state with steady fluxes present. In such systems macroscopic observables do not evolve in time and there is no time origin that specifies the beginning of a process. However, it is meaningful to consider conditional probabilities such as P(B, t2 I A, Ix dB—the probability that a dynamical variable B will have a value in the range (B,... .B + dB) at time Z2 if another dynamical variable A had the value A at time Zi, and the joint probability P B, t2, A, tfjdBdA that A will be in the range (A,..., A + dA) at time t = ti and B will be in (5,..., 5 + d/jf) at time Z2 These two probabilities are connected by the usual relation (cf. Eq. (1.188))... 

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