Rates of convergence of the partial-wave expansion beyond the Kato cusp condition

Abstract

The rates of convergence for the partial-wave expansion with odd-power ${r}_{12}$ terms for the ground-state energy of the helium atom are derived. For both the second-order $1/Z$ expansion and the Rayleigh-Ritz variational method, the energy increments of the partial-wave expansion converge as $O({L}^{\ensuremath{-}N\ensuremath{-}7})$, where $N$ is the highest odd-power ${r}_{12}$ function. The derivations require assumptions of the regularities for the ground-state helium wave function, which have not been established. Numerical results are presented for supporting the theoretical rates of convergence.