Classical transition states in quantum theory

A prescription is given for the calculation of multidimensional transmission probabilities below and just above a threshold energy. In a semiclassical approximation of quantum wave propagation, the method uses complexified dynamics to provide a smooth transition in the probability of transmission in regions of phase space where classical transmission switches off sharply. The calculation is presented using the language of bimolecular reactions and can be viewed as a semiclassical extension of classical transition state theories based on periodic orbit dividing surfaces and normally hyperbolic invariant manifolds. The approach is similar to that used in Miller's classical S-matrix formalism except that probabilities instead of amplitudes are considered. The calculation makes no use of action-angle variables and applies equally to integrable and nonintegrable systems. The main result is an approximation for an operator which acts on the internal degrees of freedom of the reacting molecules and which has a simple classical limit above a threshold energy as a measure of the classical flux through a phase space bottleneck. This operator is formally like the density-matrix for a Fermi gas of harmonic oscillators and uniform asymptotic approximations are given for its coherent-state and Wigner-Weyl representations.