Relativistic all-order equations for helium

The ground-state energy of helium is calculated to all orders in perturbation theory starting from a relativistic ``no-pair'' many-body Hamiltonian in which contributions of virtual electron-positron pairs are ignored. In this calculation only Coulomb interactions between the electrons are considered. Two all-order calculations are presented: one calculation is based on nuclear Coulomb-field orbitals and the second is based on Hartree-Fock orbitals. For each calculation the all-order equations of many-body perturbation theory are solved iteratively. The resulting numerical value for the ground-state energy from the Coulomb-field calculation is -2.903 856(1) a.u., while the Hartree-Fock calculation gives -2.903 855(2) a.u. Both of these values agree with that obtained by correcting the well-established nonrelativistic energy for lowest-order relativistic effects.