Extremal path approach to rate constant calculations by the linearized semiclassical initial value representation

To extend the applicability of the linearized initial value representation (LIVR) method to lower temperatures and realistic potentials, a generalization to barriers other than the inverted parabola is proposed. The LIVR method calculates rate constants of chemical reactions involving quantum effects by weighting classical trajectories by the Wigner distribution function (WDF) of the Boltzmann-averaged flux operator. These calculations can be performed efficiently if the WDF is available in analytical form, which is the case for harmonic barriers only. The proposed generalization to anharmonic barriers is based on the recognition that above a critical temperature T*=ℏω/πkB, where ω is the curvature at the top of the barrier and kB is the Boltzmann constant, the WDF is dominated by an extremal trajectory. The evaluation of WDFs and thus of thermal rate constants is thereby reduced to the search for the extremal path defined by a steepest-descent condition for the WDF. This extremal trajectory is the high-temperature analogue of the instanton (bounce path), which exists for temperatures lower than T*/2. Explicit formulas are derived for the generation of realistic WDFs and barrier crossing rate constants for symmetric barriers of arbitrary shape. Approximations are introduced that will reduce the extra computational effort required for these anharmonic barriers. They are based on the fact that above the critical temperature the WDF of any anharmonic potential can be represented with good approximation in an analytical form analogous to that of the parabolic barrier by the introduction of effective parameters. Results obtained for two standard model potentials, the quartic potential and the symmetric Eckart barrier, are compared with the well-known parabolic barrier results. The formal and actual temperature limits for calculating tunneling rate constants and the extension of the method to asymmetric barriers are briefly discussed.

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