Hartree-Fock and Correlation Energies for 1s2s3S and1S States of Helium-like Ions

The energies of the 1s2s 3S and 1S excited states of an atom of nuclear charge Z are calculated in the form of a series in ascending powers of 1/Z. The first four terms in the series for these non-relativistic, but otherwise exact, energies are obtained numerically; the corresponding zero-order and first-order wave functions are calculated - the zero-order function exactly, and the first-order function by a variational method, up to twelve variable parameters being taken. The energies of these same states are also calculated within the Hartree-Fock scheme. A common misunderstanding in the Hartree-Fock equations for the excited singlet state is corrected, and then the zero-order and first-order wave functions are calculated exactly. Formulae are obtained from which, by numerical integration, these Hartree-Fock energies may be computed in a power series similar to that for the exact energy. Electron correlation is very small for the triplet, but larger for the singlet. But both values are much less than for the 1s2 ground state.