Direct perturbation theory of relativistic effects for explicitly correlated wave functions: The He isoelectronic series

Abstract

Direct perturbation theory (DPT) of relativistic effects is formulated for two-electron states described by a Dirac-Coulomb or a Dirac-Gaunt Hamiltonian. The relativistic wave function, a 16-component spinor, is—after a change of the metric—expanded in powers of c−2. An expression for the leading relativistic correction E2 to the energy is derived, that reduces to the Breit-Pauli form if the nonrelativistic problem is solved exactly; otherwise a correction term appears. The method is applied in two ways to the ground state of He-like ions including electron correlation. In the first way, via a conventional configuration interaction (partial wave expansion) in a Slater-type orbital (STO) basis, the nonrelativistic partial wave increments to the energy go as (l+1/2)−4, and those of the leading relativistic correction as (l+1/2)−2. Knowing the exact analytic behaviour of the leading terms in the partial wave expansions, an extrapolation to l→∞ is possible. More accurate results, with a rather rapidly converging partial wave expansion, are obtained in the second way, where the nonrelativistic wave function contains terms linear in the interelectronic coordinate r12 (R12-method). For the He ground state both the nonrelativistic energy and the relativistic correction are obtained with an error of a few nanohartrees (nEh). The importance of various contributions to the energy for different values of the atomic charge Z is discussed.