Asymptotic Large-ZAtomic Wave Functions

Abstract

A new technique for calculating wave functions for atoms and ions is developed. Formal asymptotic expansions with largeness parameter $Z$ (nuclear charge) of the form $\psi =e^{-Zh}\Sigma a_n{Z}^{-n}$, where $h$ and the set of $a_n\text{'}\mathrm{s}$ are functions of the electron coordinates, are determined through first order in $\frac{1}{Z}$. In so doing, the Schrödinger 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, $\frac{\left(\ln Z\right)}{Z}$ 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.