Quantum defect theory I. General formulation

Quantum defects provide a measure of the differences between energy levels of many-electron atoms and energy levels of hydrogenic systems. The theory for central potentials has been developed by Hartree in 1927 and 1929, Ham in 1955, and Seaton in 1955 and 1958. The present paper is the first in a series concerned with a many-channel generalization of quantum defect theory. Wave functions ψ( N +1) for an atomic system containing N +1 electrons are expanded in terms of products ψ α ( N ) F α where the ψ α ( N ) are functions for the corresponding N -electron system and the F α are functions for the added electron. The F α satisfy coupled differential equations with potentials V αβ ( r ) = -1/ r ( Z-N )δ αβ + ½ U αβ ( r ), Z being the nuclear charge. If ψ α ( N ) or ψ β ( N ) is a bound state, U αβ goes to zero faster than 1/ r as r → ∞ . The present discussion is restricted to systems for which Z-N > 0 and the simplifying assumption is made that U αβ = 0, for r > r 0 where r 0 is finite. For all values of the total energy E the asymptotic forms of the functions F α can be expressed in terms of a scattering matrix S . It is shown that S can be expressed in terms of known functions and of a matrix IJ -1 where the matrices I and J are analytic functions of E . Once the matrix IJ -1 is known one may calculate the bound-state energy eigenvalues, allowing for configuration interaction and series perturbations, the asymptotic forms of the normalized bound-state eigenfunctions, elastic and inelastic collision cross sections, and the positions and shapes of resonances in cross sections.