Uniform semiclassical theory of avoided crossings

A voided crossings influence spectra and intramolecular redistribution of energy. A semiclassical theory of these avoided crossings shows that when primitive semiclassical eigenvalues are plotted vs a parameter in the Hamiltonian they cross instead of avoiding each other. The trajectories for each are connected by a classically forbidden path. To obtain the avoided crossing behavior, a uniform semiclassical theory of avoided crossings is presented in this article for the case where that behavior is generated by a classical resonance. A low order perturbation theory expression is used as the basis for a functional form for the treatment. The parameters in the expression are evaluated from canonical invariants (phase integrals) obtained from classical trajectory data. The results are compared with quantum mechanical results for the splitting, and reasonable agreement is obtained. Other advantages of the uniform method are described

Semiclassical calculation of bound states in a multidimensional system. Use of Poincaré's surface of section

A method utilizing integration along invariant curves on Poincaré's surfaces of section is described for semiclassical calculation of eigenvalues. The systems treated are dynamically nonseparable and are quasiperiodic. Use is also made of procedures developed in the previous paper of this series. The calculated eigenvalues for an anharmonically coupled pair of oscillators agree well with the exact quantum values. They also agree with the previous semiclassical calculations in this laboratory, which instead used integrations along the caustics. The present paper increases the number of systems capable of being treated. Using numerical counter examples for nondegenerate systems, it is also shown that an alternative view in the literature, which assumes that periodic trajectories alone suffice, leads to wrong results for the individual eigenvalues

Semiclassical calculation of bound states in a multidimensional system for nearly 1:1 degenerate systems

The method is devised to calculate eigenvalues semiclassically for an anharmonic system whose two unperturbed modes are 1:1 degenerate, by introducing a curvilinear Poincaré surface of section. The results are in reasonable agreement with the quantum ones. The classical trajectories also frequently show a large energy exchange among the two unperturbed normal modes. Implications for Slater's theory of unimolecular reactions, which neglects this effect, and for "quantum ergodicity" are described