Electronic tunneling and exchange energy in the D-dimensional hydrogen-molecule ion

Abstract

Dimensional scaling generates an effective potential for the electronic structure of atoms and molecules, but this potential may acquire multiple minima for certain ranges of nuclear charges or geometries that produce symmetry breaking. Tunneling among such minima is akin to resonance among valence bond structures. Here we treat the D-dimensional H2+ molecule ion as a prototype test case. In spheroidal coordinates it offers a separable double-minimum potential and tunneling occurs in only one coordinate; in cylindrical coordinates the potential is nonseparable and tunneling occurs in two coordinates. We determine for both cases the ground state energy splitting ΔED as a function of the internuclear distance R. By virtue of exact interdimensional degeneracies, this yields the exchange energy for all pairs of g, u states of the D = 3 molecule that stem from separated atom states with m = l = n - 1, for n = 1 → ∞. We evaluate ΔED by two semiclassical techniques, the asymptotic and instanton methods, and obtain good agreement with exact numerical calculations over a wide range of R. We find that for cylindrical coordinates the instanton path for the tunneling trajectory differs substantially from either a straightline or adiabatic path, but is nearly parabolic. Path integral techniques provide relatively simple means to determine the exact instanton path and contributions from fluctuations around it. Generalizing this approach to treat multielectron tunneling in several degrees of freedom will be feasible if the fluctuation calculations can be made tractable. © 1991 American Institute of Physics.