Determination of tunneling rates in bound systems using the complex coordinate method

Abstract

Up to now tunneling rates in bound systems have been obtained primarily by semiclassical or wave packet calculations. A new accurate quantum time-independent method is presented. Those irregular eigenfunctions of bound systems which diverge asymptotically, but upon complex scaling of coordinates X→X exp(iΘ) become square integrable functions and are associated with complex eigenvalues are found to describe barrier penetration processes. The imaginary part of each of the complex eigenvalues of the complex scaled Hamiltonian contains the tunneling decay rate provided that the Balslev–Combes rotation angle is large enough. The appearance of a critical value Θc as the rotational angle Θ is varied, at which a sharp transition from a real energy spectrum of the bound system to a complex eigenvalue spectrum is an indication of an exponential decay through the potential barrier. Tunneling in multiple barrier problems is important in several areas of physics and chemistry, including isomerization reactions, Josephson junction superconductors, electron tunneling from a 1D metallic lattice under the influence of a uniform electric field (field emission), and tunneling in the EF 1Σg state of molecular hydrogen. Several representative numerical examples are presented.