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

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-N-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.