Reactive flux correlation function

Using these distribution functions, we can write the reactive flux correlation function in the compact form... 

The Hamiltonian in Eq. (39) has bear used to calculate the adiahatic free energy as a function of the solvent coordinate using the umbrella sampling method, and reactive flux correlation function calculations have been used to determine the adiabatic rate constant. The results were qualitatively similar to the results based on the two-state model. 

The rate coefficient of a reactive process is a transport coefficient of interest in chemical physics. It has been shown from linear response theory that this coefficient can be obtained from the reactive flux correlation function of the system of interest. This quantity has been computed extensively in the literature for systems such as proton and electron transfer in solvents as well as clusters [29,32,33,56,71-76], where the use of the QCL formalism has allowed one to consider quantum phenomena such as the kinetic isotope effect in proton transfer [31], Here, we will consider the problem of formulating an expression for a reactive rate coefficient in the framework of the QCL theory. Results from a model calculation will be presented including a comparison to the approximate methods described in Sec. 4. 

For a quantum mechanical system in thermal equilibrium undergoing a transformation A B, rate constant kAB may be calculated from the time-dependent reactive-flux correlation function [28],... 

Fig. 14.11 The reactive flux correlation function, Eq. (14.99) plotted against time. After the initial transient period this function becomes essentially constant on the time scale shown.

The transmission coefficient k was calculated from a series of transition-state trajectories by monitoring the recrossings ( = St) that occur as a function of time until each trajectory is finally trapped in the product or reactant well. The normalized reactive flux-correlation function x(t) defined in Eq. 19 was constructed from this set of trajectories 131 the result is shown in Fig. 29. From its initial value, equal to the. transition-state result, ic(<) decreases rapidly until it becomes approximately constant for an extended period. The ra-... 

Figure 29. Value of the reactive flux-correlation function (/) versus time the function is normalized to the transition-state-theory value (see the text).

The exact quantum expression for the activated rate constant was first derived by Yamamoto [6]. The resulting quantum reactive flux correlation function expression is given by... 

Starting from its quantum mechanical origins, we showed how one can arrive at a computationally tractable expression for the rate constant of a condensed phase quantum process. The calculation of this reactive-flux correlation function expression involves QCL dynamics of the species variables with the initial conditions sampled from the quantum equilibrium distribution. This approach was demonstrated for the calculation of the rate constant of a proton transfer reaction in a hydrogen-bonded complex dissolved in a polar solvent. 

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

The activation free energy AA can be used to compute the TST approximation of the rate constant = Ce, where C is the preexponential factor. Because not every trajectory that reaches the transition state ends up as products, the actual rate is reduced by a factor k (the transmission coefficient) as described earlier. The transmission coefficient can be calculated using the reactive flux correlation function method. " " " Starting from an equilibrated ensemble of the solute molecules constrained to the transition state ( = 0), random velocities in the direction of the reaction coordinate are assigned from a flux-weighted Maxwell-Boltzmann distribution, and the constraint is released. The value of the reaction coordinate is followed dynamically until the solvent-induced recrossings of the transition state cease (in less than 0.1 ps). The normalized flux correlation function can be calculated using " ... 

Yamamoto, T., Wang, H.B., and Miller, W.H. (2002) Combining semidassical time evolution and quantum Boltzmann operator to evaluate reactive flux correlation function for thermal rate constants of complex systems. J. Chem. Phys., 116 (17), 7335-7349. 

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