Lower Bound to the Ground-State Energy and Mass Polarization in Helium-Like Atoms

Abstract

Using a method which involves finding the expectation value of the square of the Hamiltonian, lower bounds to the ground state energies of He, ${\mathrm{Li}}^{+}$, and ${\mathrm{O}}^{6+}$ have been obtained from existing Hylleraas-type wave functions. The limit of error in the Chandrasekhar-Herzberg 18-parameter helium upper bound, -2.9037063 atomic units (a.u.), is found to be in absolute value less than 0.00179 a.u. and "probably" less than 0.00009 a.u. (\ensuremath{\sim}20 ${\mathrm{cm}}^{\ensuremath{-}1}$). These results would seem to indicate that the Hylleraas series is not as rapidly convergent as previously thought.Again using existing Hylleraas-type wave functions, values have been obtained for the mass polarization (specific mass effect) in the same helium-like atoms. The value obtained for helium is somewhat different from that given by Bethe.