Basis-set methods for the Dirac equation

Abstract

The pathologies associated with finite basis-set approximations to the Dirac Hamiltonian HDirac are avoided by applying the variational principle to the bounded operator 1 / (H Dirac – W) where W is a real number that is not in the spectrum of HDirac. Methods of calculating upper and lower bounds to eigenvalues, and bounds to the wave-function error as measured by the L2 norm, are described. Convergence is proven. The rate of convergence is analyzed. Boundary conditions are discussed. Benchmark energies and expectation values for the Yukawa potential, and for the Coulomb plus Yukawa potential, are tabulated. The convergence behavior of the energy-weighted dipole sum rules, which have traditionally been used to assess the quality of basis sets, and the convergence behavior of the solutions to the inhomogeneous problem, are analyzed analytically and explored numerically. It is shown that a basis set that exhibits rapid convergence when used to evaluate energy-weighted dipole sum rules can nevertheless exhibit slow convergence when used to solve the inhomogeneous problem and calculate a polarizability. A numerically stable method for constructing projection operators, and projections of the Hamiltonian, onto positive and negative energy states is given. PACS Nos.: 31.15Pf, 31.30Jv, 31.15-p