Scaling rules for resonance dynamics near a saddle point: The pendulum as a zero-order model

The pendulum is the simplest zero-order model for an isomerizing vibrationa l mode (one which passes through a saddle point). We utilize the classical action/angle theory of the pendulum, for which new results are given in the appendix, to determine generic scaling laws between the quantum mechanical pendulum eigenvalue distribution and the coupling matrix elements. These s caling rules are more appropriate for isomerizing vibrational modes than ar e the usual harmonic oscillator scaling rules, encoded in traditional spect roscopic effective Hamiltonians, which break down catastrophically at a sad dle point. As a simple example of resonant quantum dynamics in the vicinity of a saddle point, we analyze a system consisting of a pendulum model for bend/internal rotor motion, anharmonically coupled to a stretching harmonic oscillator, in qualitative agreement with the known dynamics of HCP. The d ominance of just two of the infinite number of resonances, 2:1 and 4:1, at all energies including that of the saddle point, is related to the scaling properties of the zero-order pendulum model.