Large-ZExpansion Theory for the Ground State of a One-Electron Ion Perturbed byrN

Abstract

Large-$Z$ (nuclear charge) expansion theory is applied to the nonrelativistic ground state of a one-electron ion which is perturbed by a repulsive central field potential which goes as the $N\mathrm{th}$ power of the radial distance of the electron from the nucleus where $N$ is an integer ($N=-1\mathrm{}$; $N\geqq 1$). The energy-expansion parameters for these problems are determined self-consistently within the asymptotic method itself. General recursive relations for the wave-function expansion coefficients in the basic asymptotic expansion and the stretched solution are obtained. For the case of $N=1\mathrm{}$, the wave function and the energy expansion are obtained explicitly through third order. For $N=-1\mathrm{}$, the basic asymptotic solution is the exact solution. For $r$ large, a few terms of the conventional large-$Z$ perturbation-theory expansion for the $N=-1\mathrm{}$ case give an inaccurate representation for the exact wave function. The relative error associated with first-order perturbation-theory expectation-value results is shown to increase as the operator probes the outer part of the atom to a greater extent.