Asymptotic Large-ZAtomic Wave Functions

Abstract

A new for calculating wave functions for atoms and ions is developed. Formal asymptotic expansions with largeness parameter $Z$ (nuclear charge) of the form $\ensuremath{\psi}={e}^{\ensuremath{-}Zh}\ensuremath{\Sigma}{a}_{n}{Z}^{\ensuremath{-}n}$, where $h$ and the set of ${a}_{n}'\mathrm{s}$ are functions of the electron coordinates, are determined through first order in $\frac{1}{Z}$. In so doing, the Schr\odinger equation for a general atomic system with $N$ electrons is reduced to a set of first-order partial differential equations for successive ${a}_{n}$. These equations are then solved recursively. Screening and correlation are exhibited explicitly in the resulting asymptotic atomic wave functions. Applications made to the ground state of 2-electron systems show that the asymptotic wave function obeys the virial theorem through first order. Magnetic susceptibilities within 5% of the accepted values are obtained for helium and singly ionized lithium. Other expectation values, $\frac{1}{2}({s}_{1}+{s}_{2})$, ${{s}_{12}}^{2}$, and ${s}_{12}$, are found to be in excellent agreement with Pekeris's variational calculations utilizing many parameters. In the neighborhood of certain singular points the large-$Z$ asymptotic solutions obtained by a matching technique are shown to satisfy the correct cusp conditions. In certain of these regions, $\frac{(\mathrm{ln}Z)}{Z}$ terms enter. The omission of such terms in ordinary variational-perturbation wave functions may result in a loss of accuracy when computing expectation values other than the energy.