Perturbation theory of relativistic effects: relative inaccuracies of approximate results

The first-order relativistic corrections to energies can be determined either from the direct perturbation expansion of the Dirac equation or by using the so-called Pauli operator. The two formulae are equivalent for exact non-relativistic reference functions but lead to different results for approximate solutions. The analysis of a model problem with controlled deviation from the exact non-relativistic solution clearly shows a preference for the first-order relativistic energy formula derived from direct perturbation theory. The use of the Pauli approximation with inaccurate non-relativistic reference functions may introduce significant inaccuracies in the calculated relativistic corrections for low-lying s-states.