Wave Function of Ionized Lithium

Abstract

A nonrelativistic wave function for the $1s2{s}^{3}S$ state of singly ionized lithium has been found by using an expansion of the function in Legendre polynomials of the cosine of the angle between the lines joining the electrons to the nucleus. Previous work had indicated that the distance dependent coefficient of the angle-independent term of the expansion predominates over the others. This made it possible to obtain an accurate function by computing to high precision an approximate value of the coefficient as a function of the distances of the electrons from the nucleus. The calculation of this function of the two distances was facilitated by the separability of the wave equation satisfied by the function. An additive correction to this function and also the distance dependent coefficients of higher order Legendre polynomials, all of which were comparatively small, could then be calculated by relaxation and numerical variational methods of less accuracy. Application of tests based on the virial theorem and Green's theorem to the wave function were used as criteria of accuracy. The Ritz integral led to the energy value $\ensuremath{-}1.135724 \mathrm{Rhc}{Z}^{2}$, where $R$ is Rydberg's constant for lithium, $Z$ is the atomic number, and $h$ and $c$ are Planck's constant and the velocity of light, respectively. This result compares favorably with the experimental value, corrected for the relativistic effect and nuclear motion, of $\ensuremath{-}1.135722\ifmmode\pm\else\textpm\fi{}0.000025 \mathrm{Rhc}{Z}^{2}$. The hyperfine structure integral $1+\ensuremath{\epsilon}$ of Breit and Doermann was found to have the value 1.06191\ifmmode\pm\else\textpm\fi{}0.00003.