Variational methods for Dirac wave equations

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Abstract
Several difficulties associated with the use of the Dirac operator in atomic and molecular structure calculations are analysed. The physical stability of the ground states of atoms and molecules requires the use of Dirac's 'hole' (positron) theory to prevent the decay of bound electrons into the lower continuum. Thus relativistic bound-state calculations for atoms or molecules can, in practice, more or less ignore the negative energy states. A rigorous theorem on the eigenvalue distribution in finite-basis-set variational calculations with one-body Dirac operators is presented, and its implications for many-body atomic or molecular systems are described. The theory shows that the basis sets must satisfy suitable boundary conditions to ensure acceptable results. The theory is illustrated with numerical studies of convergence to hydrogenic solutions using finite basis sets, and to closed-shell Dirac-Fock problems. In all cases, the bound-state variational energies are rigorous upper bounds. Finally, the author presents the first results of a calculation in which the Breit interaction has been included in the self-consistent field instead of being treated as a first-order perturbation.

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