Rayleigh-Ritz approximation of the Dirac operator in atomic and molecular physics

Abstract

Four-component (spinor) solutions of the Dirac equation may be approximated by L-spinor expansions. We discuss their orthogonality and completeness and relate L-spinor properties to those of the Coulomb Sturmians. The mathematics of Rayleigh-Ritz approximations for one-electron Schr\"odinger and Dirac operators provides a rigorous setting for applying finite L-spinor matrix approximations to the relativistic hydrogenic atom. Convergence of eigenvalues and eigenvectors with respect to the size of the L-spinor set, of expectation values of quantum-mechanical operators, sum rules, and perturbation expansions is examined. The contribution to perturbation sums over states from solutions with eigenvalues in the continuum range ${(mc}^{2},\ensuremath{\infty})$ (electronic scattering states) and $(\ensuremath{-}\ensuremath{\infty},\ensuremath{-}{\mathrm{mc}}^{2})$ (positronic scattering states) is shown to be essential to get accurate results.