Relativistic all-order pair functions from a discretized single-particle Dirac Hamiltonian

Relativistic all-order pair functions are obtained by summation over a complete set of eigenvectors to a discretized single-particle Dirac Hamiltonian. The discretization of the Dirac equation, by substituting finite-difference formulas for derivatives, is discussed in detail. It is shown how to obtain a symmetric eigenvalue problem, and a way to avoid spurious states in the spectrum is presented. The number of operations required to solve for a radial pair function is proportional to $N^3$, where N is the number of radial lattice points used. The method is applied to the ground state of helium using the Dirac-Coulomb Hamiltonian and the no-virtual-pair approximation. An accuracy of a few parts in ${10}^8$ is achieved for the total energy. This accuracy allows a determination of the leading term in the partial-wave expansion of the relativistic corrections to approximately 0.075${\alpha }^2$(l${+(1/2)\mathrm{}}^{\mathrm{-}2}$, which implies a slow convergence compared to the partial-wave expansion of the nonrelativistic energy.