Improved Multidimensional Semiclassical Tunneling Theory

We show that the analytic multidimensional
semiclassical tunneling
formula of Miller et al. [Miller, W. H.; Hernandez, R.; Handy, N.
C.; Jayatilaka, D.; Willets, A. Chem. Phys. Lett. 1990, 172, 62] is qualitatively incorrect
for deep tunneling at energies well below the top of the barrier.
The origin of this deficiency is that the formula uses an effective
barrier weakly related to the true energetics but correctly adjusted
to reproduce the harmonic description and anharmonic corrections of
the reaction path at the saddle point as determined by second order
vibrational perturbation theory. We present an analytic improved semiclassical
formula that correctly includes energetic information and allows a
qualitatively correct representation of deep tunneling. This is done
by constructing a three segment composite Eckart potential that is
continuous everywhere in both value and derivative. This composite
potential has an analytic barrier penetration integral from which
the semiclassical action can be derived and then used to define the
semiclassical tunneling probability. The middle segment of the composite
potential by itself is superior to the original formula of Miller
et al. because it incorporates the asymmetry of the reaction barrier
produced by the known reaction exoergicity. Comparison of the semiclassical
and exact quantum tunneling probability for the pure Eckart potential
suggests a simple threshold multiplicative factor to the improved
formula to account for quantum effects very near threshold not represented
by semiclassical theory. The deep tunneling limitations of the original
formula are echoed in semiclassical high-energy descriptions of bound
vibrational states perpendicular to the reaction path at the saddle
point. However, typically ab initio energetic information is not available
to correct it. The Supporting Information contains a Fortran code, test input, and test output that implements
the improved semiclassical tunneling formula.