A method for determining "good" action-angle variables and semiclassical eigenvalues in nonseparable systems

A method for determining semiclassical eigenvalues and the canonical transformation relating "good" and harmonic action-angle variables for nonseparable molecular vibrations is developed. The method makes use of Fourier expansions of the harmonic action and good angle variables in terms of the harmonic angle variables (for fixed good actions). The coefficients in these expansions are determined by requiring that each expansion, when truncated at N terms, be exactly satisfied at N appropriately chosen times during the integration of a trajectory for the system of interest. Applications of this method are made to the determination of semi-classical eigenvalues for several model two mode systems which have been extensively studied using other semiclassical methods. Essentially exact agreement with these earlier calculations is obtained. We then study the dependence of cartesian coordinates and harmonic actions on good angle variables for two of the models. Weak correlation between motions in different modes is found, and this leads to a simple but reasonably accurate method for decoupling the two modes based on motional time scale separations.