This paper describes program LEVEL, which can solve the radial or one-dimensional Schrodinger equation and automatically locate either all of, or a selected number of, the bound and/or quasibound levels of any smooth single- or double-minimum potential, and calculate inertial rotation and centrifugal distortion constants and various expectation values for those levels. It can also calculate Franck–Condon factors and other off-diagonal matrix elements, either between levels of a single potential or between levels of two different potentials. The potential energy function may be defined by any one of a number of analytic functions, or by a set of input potential function values which the code will interpolate over and extrapolate beyond to span the desired range.

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LEVEL: A computer program for solving the radial Schrödinger equation for bound and quasibound levels

Various methods for calculating the energies and widths of quasibound levels (orbiting or shape resonances) for spherical potentials are critically compared. A derivation for the previously‐proposed Airy function boundary condition method is presented, and a Weber function boundary condition method for locating resonances which lie above the potential barrier maximum is derived, tested, and found wanting. It is shown that the Weyl m‐function method of Hehenberger et al. [J. Chem. Phys. 65, 4559 (1976)] yields results in essentially exact agreement with the time‐delay maximum method of Le Roy and Bernstein [J. Chem. Phys. 54, 5114 (1971)]. An improved semiclassical method of calculating these resonance widths, suggested by M.S.Child, is presented and shown to be reliable even for levels lying right at a potential barrier maximum.

Energies and widths of quasibound levels (orbiting resonances) for spherical potentials

For pt.I see ibid., vol.14, p.4213 (1980). The combination of Milne's theory for calculating bound-state energies and wavefunctions with the complex rotation method yields an appealingly simple and powerful tool for the computation of complex-valued resonance (Siegert) energies and wavefunctions. The method provides an unambiguous assignment of a quantum number n=0,1,... to a resonance state and permits a unified picture of bound-state and resonance properties. The numerical technique is sufficiently fast and stable to enable reliable calculations of higher resonances. This is illustrated for some model potentials. Furthermore the complex energy WKB quantisation is discussed and tested numerically.

Milne's differential equation and numerical solutions of the Schrodinger equation. II. Complex energy resonance states

The spectral properties of a singular second order differential operator L is analysed in terms of Weyl's theory A spectral function is derived and the results are briefly compared with scattering theory The analytic properties of L, its associated Green's function and related quantities of modern scattering theory are analysed It is shown that the analytic continuation of an appropriately dilated operator L allows for an analytic extension of the singular case to a higher order Riemann sheet The present theory is thus applicable to the study of the entire bound and continuous spectrum of L A numerical realization of the theory is presented and applied to the scattering potential of Moiseyev et al (1978, Molec Phys, 36, 1613) New resonance structures are thereby found Their asymptotic behaviour is discussed and an attempt to classify this type of structure is made Implications of the present theory for predissociations in diatomic molecules, analysis of scattering experiments and ab initio stud

Weyl's theory and the method of complex rotation

The radial equation (or set of equations) derived in scattering theory is analyzed by means of Titchmarsh–Weyl theory for singular second‐order differential equations. In particular we have focused on the spectral density concept and the corresponding relation to the scattering cross section. The method of complex deformations is brought in as a necessary ingredient in the evaluation of the underlying pole strings, which together with the background build up the actual dispersion relation data. The analysis is supported by numerical applications to a centrifugal family of simple potentials.

Titchmarsh–Weyl theory and its relations to scattering theory: Spectral densities and cross sections; Theory and applications

Weyl’s theory for singular ordinary second order differential equations is presented as a numerical method for analyzing the spectral properties of the Schrodinger 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.

B. Laskowski

Papers

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