Application of discrete-basis-set methods to the Dirac equation

Variational solutions to the Dirac equation in a discrete ${L}^{2}$ basis set are investigated. Numerical calculations indicate that for a Coulomb potential, the basis set can be chosen in such a way that the variational eigenvalues satisfy a generalized Hylleraas-Undheim theorem. A number of relativistic sum rules are calculated to demonstrate that the variational solutions form a discrete representation of the complete Dirac spectrum including both positive-and negative-energy states. The results suggest that widely used methods for constructing ${L}^{2}$ representations of the nonrelativistic electron Green's function can be extended to the Dirac equation. As an example, the relativistic basis sets are used to calculate electric dipole oscillator strength sums from the ground state, and dipole polarizabilities.