Thermal rate of transmission through a barrier: Exact expansion of up to and including terms of order h over line 4

Nine decades after Wigner's formulation of quantum rate theory, his celebrated result was recently generalized to the asymmetric barrier using an exact first-order expansion of the transmission probability in terms of h over bar 2. This paper extends the first-order quantum correction to second-order correction of order h over bar 4 for the thermally averaged transmission probability through an arbitrary barrier. The derivation employs a systematic expansion of the projection operator onto products and the thermal distribution which involves a Taylor expansion of the potential about the barrier up to eighth order. The resulting exact analytical expression is calibrated with numerical calculations of several model potentials and shows excellent agreement when the h over bar 4 term is included. In comparison, the semiclassical transition state theory cannot reproduce the correct h over bar 4 terms when the anharmonicity is treated on the level of VPT-4 (vibrational perturbation theory-fourth order) and will potentially need a VPT-6 expansion. Further analysis of the quartic barrier reveals suppressed transmission due to the dominant role of quantum reflection above the barrier. These results not only provide a conceptual framework but can also be applied to heavy atom tunneling and machine learning.