Quantum defect theory with long-range multipole potentials

The formula K = K-oo - K-oc (K-cc + tau)(-1) K-co is familiar in quantum defect theory: K is the reactance matrix for electron-ion collisions; K is a matrix with partitioning for open-open, open-closed, closed-open and closed-closed elements; and tau = tan(pi nu) where nu is a diagonal matrix of effective quantum numbers in the closed channels. In the simplest case it may be assumed that K is a slowly varying function of the energy. This formula has usually been obtained by assuming only a pure Coulomb potential at radial distances r greater than or equal to r(0), with r(0) finite. This paper discusses the derivation of a similar formula for the case in which long-range non-Coulomb potentials are included, behaving like r(-lambda-1) for r greater than or equal to r(0), with lambda greater than or equal to 1.