Tunneling through the one-dimensional Eckart barrier is treated by applying a higher order semiclassical approximation which adds corrections proportional to powers of ℏ to the Herman–Kluk (HK) initial value approximation. Although the usual, zero-order HK treatment is very poor in this case, the first- and second-order corrections substantially improve the accuracy of the computed tunneling probabilities. To investigate how this works, the HK expression is shown to be equivalent, for the present purposes, to a formula involving a single integral over the initial momentum, with an integrand that has a simple analytical form. Similarly, the most important part of the first-order correction term is shown to be expressible in a very simple form. For a particular range of energies, the integral can be analyzed in terms of a steepest descent treatment along a path through a caustic. In this way, it is verified that the zero-order approximation does not approach the correct classical limit as ℏ → 0, but the first-order term, which does not vanish in this limit, improves the accuracy of the result. More generally, the corrections of each order contain terms that are of all orders in ℏ1/3, including those that are O(ℏ0) and which survive in the classical limit. The infinite sum over such terms is performed analytically and shown to yield the correct classical limit for the tunneling amplitude.

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A principal weakness of the Herman–Kluk (HK) semiclassical approximation is its failure to provide a reliably accurate description of tunneling between different classically allowed regions. It was previously shown that semiclassical corrections significantly improve the HK treatment of tunneling for the particular case of the one-dimensional Eckart system. Calculations presented here demonstrate that the lowest-order correction also substantially improves the HK description of tunneling across barriers in two-dimensional systems. Numerical convergence issues either do not arise or are easily overcome, so that the calculations require only a moderate number of ordinary, real, classical trajectories.