Rayleigh-Ritz approximation of the Dirac operator in atomic and molecular physics - art. no. 022508

Four-component (spinor) solutions of the Dirac equation may be approximated by L-spinor expansions. We discuss their orthogonality and completeness an d relate L-spinor properties to those of the Coulomb Sturmians. The mathema tics of Rayleigh-Ritz approximations for one-electron Schrodinger and Dirac operators provides a rigorous setting for applying finite L-spinor matrix approximations to the relativistic hydrogenic atom. Convergence of eigenval ues and eigenvectors with respect to the size of the L-spinor set, of expec tation values of quantum-mechanical operators, sum rules, and perturbation expansions is examined. The contribution to perturbation sums over states f rom solutions with eigenvalues in the continuum range (mc(2),infinity) (ele ctronic scattering states) and (- infinity, - mc(2)) (positronic scattering states) is shown to be essential to get accurate results.