Electronic tunneling in H +2 evaluated from the large-dimension limit

Abstract

We derive a simple, analytic expression for the energy splitting Δ E between the lowest pair of H + 2 states (1sσ g and 2pσ u) that results from electron exchange between two protons. The calculation employs the semiclassical instanton method, with two unorthodox features which markedly simplify the treatment: (1) The double-minimum potential and corresponding wavefunctions that govern the electronic tunneling are evaluated in the large-dimension limit. (2) The time variable is rescaled to cure divergent behavior of fluctuations about the instanton path that otherwise appears because of the potential develops sharp cusps as the internuclear distance increases. By virtue of exact interdimensional degeneracies, the large- D limit yields valid results for a 3D molecule. Indeed, a simple dimensional scaling law gives Δ E for all pairs of g, u states that stem from separated atom states with m= l= n−1, for n=1→∞. For a wide range of internuclear distances, our analytic expression for Δ E, which pertains to the leading order in ł>, gives for such states good agreement with comparable semiclassical methods as well as with exact numerical calculations. It is remarkable that use of the effective potential for the large-dimension limit, which is exactly calculable from classical electrostatics, yields quantitatively results for electronic tunneling, an intrinsically quantal phenomenon.