Development of analytical approach for tunneling calculations in double well potentials

A new strategy for the analytical solution of the Schrödinger Equation (SE) for double well potentials using angular prolate spheroidal (APS) functions was developed. The APS functions have a suitable shape and solving it is a simple and accurate way of obtaining the energy levels and their splittings, and the tunneling frequency. The strategy starts with the modeling of the potential energy surface (PES) using the curve generated by the intrinsic reaction coordinates (IRC), which is fitted using a symmetrical trigonometric potential, or generating the PES from the height and width of the barrier fitted to the symmetrical trigonometric potential. The trigonometric potential is then used in the resolution of the SE. In transforming the variables to unitless ones, a parameter L must be defined, which cannot be obtained via any experimental values. The issue was solved defining it as a fitting parameter. The strategy accuracy was compared to the WKB method and to the instanton-soliton approach in calculating the tunneling splitting and the tunneling frequency of the molecules NH3 , ND3 , CH3NH2 , (CH3 )2NH and (CH3 )3N. The calculated tunneling frequencies for the NH3 molecule were 2,56, 3,67 and 1,12 ×1010 s−1 , for the developed method, the WKB method and for the instanton-soliton approach. The experimental tunneling frequency is 2, 38 × 1010 s−1 . The results show that the developed strategy was the best in generating accurate values with simple and straightforward calculations. It also has the advantage that it can be applied in regions where the semiclassical fails.