Flux autocorrelation function

Day P N and Truhlar D G 1991 Benchmark calculations of thermal reaction rates. II. Direct calculation of the flux autocorrelation function for a canonical ensemble J. Chem. Phys. 94 2045-56... 

Constants Recursive Development of the Flux Autocorrelation Function. 

W. H. Miller The expression for the reaction rate (in terms of a flux-flux autocorrelation function) obtained by myself, Schwartz, and Tromp in 1983 is very similar (though not identical) to the one given earlier by Yamamoto. It is also an example of Green-Kubo relations. 

We see that the rate constant may be determined as the time integral of the canonical averaged flux autocorrelation function for the flux across the dividing surface between reactants and products. It is also clear that we only need to calculate the flux correlation function for trajectories starting on the dividing surface, for otherwise F(p(0), q(0)) = 0 and there will be no contributions to the product formation. 

To give an idea of the form of the flux autocorrelation function, we consider the dynamics of a free particle with a constant potential energy of Eo, H = P2 /(2m) + Eo, which to a first approximation can describe the dynamics along a relevant reaction coordinate in the barrier region of the potential surface. The flux correlation function (5.115) can, in the coordinate representation, be written in the form [2] (see Appendix F)... 

Qr is the partition function of the reactants, and v = 2 and 1 for a bimolecular and a unimolecular reaction, respectively. Note that in the flux autocorrelation function, the position of the Boltzmann operator differs from the form given in Eq. (5.115). 

Here C is the gas phase (uncoupled) flux autocorrelation function, Zbath is the bath partition function, J(co) is the bath spectral density (computed as described above from a classical molecular dynamics computation), Bi and B2 are combinations of trigonometric functions of the frequency a> and the inverse barrier frequency, and Anally ... 

As in other flux correlation function computations, f is the complex time t ——. Thus, given the Quantum Kramers model for the reaction in the complex system, and the re-summed operator expansion as a practical way to evaluate the necessary evolution operators needed for the flux autocorrelation function, the quantum rate in the complex system is reduced to a simple combination of gas phase correlation functions with simple algebraic functions. 

In this section we discuss the flux-flux autocorrelation function approach to computation of reaction rates in both classical and quantum theories. 

As we have shown above the classical flux-flux autocorrelation function Cp can be explicitly analyzed within the framework of the CNF theory. The following natural question arises Can one obtain the time dependence of the quantum flux-flux autocorrelation function Cp using the methods of the QNF theory Of course, the QNF technique provides one with an approximation of the original Hamiltonian operator. This approximation is only accurate in the vicinity of the equilibrium saddle point in phase space, so one should not expect a perfect agreement between the QNF flux-flux autocorrelation function and the exact one to hold up to infinitely long times. Instead, the QNF theory will provide an approximation of the exact flux-flux autocorrelation function in a certain time interval whose length will depend on the effective Planck s constant among other parameters. 

A spectral representation of the flux-flux autocorrelation function takes the form... 

The reaction path Hamiltonian also provides a very useful framework for the rigorous calculation of the Boltzmann (i.e., thermally averaged) rate constant for a chemical reaction using the path integral methods described by Miller, Schwartz, and Tromp. In that paper it is shown that the rate constant can be expressed as the time integral of a flux-flux autocorrelation function... 

The Kubo approach has also proven to be extremely fruitful for the theory of chemical rate processes. In fact, chemical rate constants can also be expressed as an integral of a flux autocorrelation function. As was shown by Chandler [5], however, the brute-force approach does not work in this case. In fact, in the context... 

EXACT QUANTUM RATE CONSTANTS FOR H+H2 REACTION VIA FLUX-FLUX AUTOCORRELATION FUNCTION ... 

Some time ago Miller derived an exact quantum formulation of the thermal rate constant, and more recently Miller et al. re-cast it in terms of the time integral of the flux-flux autocorrelation function. Although, as might be expected for an exact formulation, the time... 

Tlie tqiproaches were illustra by three exan les the photodissociaiton of Nal via a two surface wave packet calculation the J=0 vibrational states of all symmetries for H3- and the exact quantum thermal rate constant for die H+H2 reaction via the 3D DVR formulation of the flux-flux autocorrelation function. The split time exponential propagator was used for the Bist problem, and die sequential diagonalization-truncation approach for the latter two problems. Each of diese cases illustrates the substantial advances represented by die DVR, pomitting the efficient solution of problems on a much larger scale than heretofore possible. 

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