Weyl’s theory applied to predissociation by rotation. I. Mercury hydride

Abstract

Weyl’s theory for singular ordinary second order differential equations is presented as a numerical method for analyzing the spectral properties of the Schrödinger operator associated with predissociation by rotation in diatomic molecules. It is pointed out that the poles of Weyl–Titchmarsh’s m function characterize both the discrete eigenvalues and the resonances in the continuous spectrum, thus allowing a unified treatment of the entire spectrum. The conventional hard core treatment of the origin is compared to two alternative approaches, both applicable to more general singular behavior of the potential. A detailed discussion of the other singularity, infinity, is given and Kodaira’s theorem is generalized to an expression involving Wronskian’s between the regular and the asymptotic solutions. The calculation of the Weyl–Titchmarsh m function is based on numerical Runge–Kutta integrations of both independent solutions, combined with asymptotic information derived from a Riccati expansion. Im(m) is shown to have its physical significance as flux of predissociating particles. The resonances in the continuous spectrum are obtained by numerical analytical continuation of Im(m) across the real energy axis. The method is applied to Stwalley’s isotopically combined (IC1) potential for HgH. All resonance energies and lifetimes are reported and compared to Stwalley’s phase shift results. Excellent agreement is found for sharp resonances whereas considerable discrepancies occur for the broader ones. This latter fact is attributed to the failure of the Breit–Wigner fit to account for asymmetric non-Lorentzian shapes.