Hylleraas Variational Calculation of Auto-Ionization States

Abstract

Auto-ionization states of two-electron systems (${\mathrm{H}}^{-}$ and He) have been calculated using the Feshbach $Q$-projection-operator formalism with a Hylleraas-type variational wave function. Results for most of the $S$ states in He and the first two in ${\mathrm{H}}^{-}$ are lower than any previously calculated. The energy of the lowest ${}_{}{}^1{}_{}{}^{}S$ state in He is found to be -1.5575487 Ry using 50 terms, and the convergence indicates that the first 6 figures are significant. Except for the lowest ${}_{}{}^3{}_{}{}^{}P$ state in He, calculations of odd-parity $P$ states do not yield as low results as previous relative partial-wave calculations. Comparison with accurate resonant-energy calculations of Burke and Taylor in $S$ states allows us to deduce energy shifts which are inherent in the projection-operator technique.