A note on quantum thermodynamic rate theories

Abstract

The relationship between quantum transition state theory, the mixed quantum classical rate theory and the Hansen–Andersen analytic continuation methods is analyzed. We prove that the first three time derivatives of a coordinate dependent operator are the same in quantum and classical mechanics. As a result, the mixed quantum classical theory, in which the quantum projection operator is replaced by the classical, may be considered as a specific case of the Hansen–Andersen methodology. The same holds true for quantum transition state theory for one dimensional systems, where the exact quantum propagator is replaced by its parabolic barrier approximation. In the multidimensional case, quantum transition state theory errs somewhat in the second nonzero time derivative, however it may be reformulated to assure that it too remains exact for the first two nonzero initial time derivatives. Further systematic improvement of the mixed quantum classical theory may be obtained by including higher order terms in the ℏ2 expansion of the Wigner–Liouville equation. An iterative solution of the integral form of the Wigner–Liouville equation is suggested, which is based on propagation of classical trajectories only.