Numerical solution of the relativistic pair equation

Abstract

Correlation has been treated relativistically by solutions of the pair equation, an inhomogeneous differential equation in two dimensions. The equation is solved within the no–(virtual-)pair approximation and is formally correct to order ${\alpha }^2$ Ry. Continuum dissolution is rigorously avoided, by the use of projection operators. Correlation is treated starting from a suitable choice of a fully relativistic one-particle description of the atom. The solution of the equation involves additional difficulties as compared with the nonrelativistic case, which are discussed. Second-order energies arising from electrostatic and magnetic correlation have been calculated for the 1$s^2$ ground state for a series of heliumlike systems (Z=2–50). The starting point has been either hydrogenlike wave functions or Dirac-Fock orbitals. Comparison has been made with other methods, showing good, sometimes excellent, agreement, and trends in the isoelectronic series are discussed. By iterative solutions of the exact pair equation a nonperturbative treatment of the two-body part of the static Coulomb interaction is obtained. This procedure has been used for s excitations.