A multichannel quantum defect analysis of diatomic predissociation and inelastic atomic scattering

Abstract

Given an NT×NT interaction matrix W∞(R) which describes the dissociation of a diatomic molecule into NT asymptotic atomic channel states, we can generate exact numerical solutions to the close-coupled scattering equations. At total energies E above the highest dissociation threshold we obtain an NT×NT scattering matrix S(E) which defines the asymptotic structure of the NT- fold degenerate multichannel scattering or continuum wave functions. This matrix varies rapidly with energy and is nonanalytic at thresholds. However, based on a multichannel quantum defect analysis (MCQDA) of the coupled equations we find that the numerical S(E) matrix can be made to yield a real, symmetric matrix Y(E) which is analytic in E. This matrix can then be analytically continued across threshold to provide rigorous analytic descriptions of the multichannel diatomic wave functions in the predissociating and bound-state regions of the energy spectrum. Since the extraction of Y(E) is predicated on assigning a reference potential 𝒱γ(R) to each channel, the detailed energy variation of Y(E) is dependent on the choice of potentials. Fortunately, the physics contained in W∞(R) generally dictates an obvious set of reference potentials which usually make Y(E) slowly varying. As E is reduced below threshold we can use Y(E), often well represented as a constant over a wide range of energies, to provide a description of the predissociating molecule, including such observable properties as linewidths, level shifts, and branching ratios. At still lower energies, when all channels are closed, Y(E) offers a complete, nonperturbative description of the configuration interaction between bound electronic-rotational states of the molecule. Although many insights in this paper are provided by a WKB analysis of the reference solutions associated with each reference potential, the MCQDA yields a complete quantum mechanical description of the exact close-coupled wave functions. The quality of these wave functions is limited only by the accuracy of the molecular potentials and interaction matrix elements that are used to construct W∞(R), and by the number NT and specific set of channels we choose to include in the close-coupled theory. The analysis is equally valid when applied to either adiabatic avoided crossings or diabatic curve crossings. More importantly, the formal structure of the close-coupled wave functions dictated by MCQDA yields a rigorous framework for the analysis of strongly interacting continuum, predissociating, and bound state channels without recourse to perturbation theory.