Semiclassical tunneling splittings for arbitrary vibrational states in multidimensional double wells

A semiclassical theory developed in a previous paper [K. G. Kay, Phys. Rev. A 96, 042116 (2017)] is applied to calculate tunneling splittings for arbitrary vibrational states of model two-dimensional double-well systems. Cases in which the classical dynamics for the wells is chaotic, mixed, and regular are considered. A perturbative treatment, based on the condition of small tunneling amplitudes, is found to be sufficiently accurate for the cases studied and is applied for most of the calculations. Treatments that approximate certain imaginary-time trajectories in the classically forbidden region by linearization about a variety of judiciously selected reference trajectories yield good results for all systems treated. These calculations can be greatly simplified by approximating all imaginary-time trajectories as linearizations about a single reference trajectory. A simple way to determine optimal reference trajectories for this purpose is presented. It is found that their use yields splittings of satisfactory accuracy for the cases studied.