Four component regular relativistic Hamiltonians and the perturbational treatment of Dirac’s equation

By combining the ideas of the direct perturbation theory approach to the solution of the Dirac equation with those underlying the regular expansion as used to obtain the two‐component Chang–Pélissier–Durand Hamiltonian, a four‐component form of the regular expansion is proposed. This formulation lends itself naturally to systematic improvement by a nonsingular form of perturbation theory. Alternatively it can be viewed as a double perturbation version of direct perturbation theory, where relativistic effects on the Hamiltonian and the metric are considered separately and the Hamiltonian perturbation is summed to infinite order. The scaling procedure that was earlier shown to be exact in the case of a hydrogenic potential and that greatly improved the core orbital energies, is found to follow naturally from the current formulation. The accuracy of the various approximations to the wave functions is assessed with respect to several radial expectation values weighing different regions in the uranium atom as a test case