Matrix formulation of direct perturbation theory of relativistic effects in a kinetically balanced basis

Abstract

Direct perturbation theory (DPT) and its quasi-degenerate version (QD-DPT) in a matrix formulation, i.e. DPT-mat and QD-DPT-mat are derived from the matrix representation of the Dirac operator in a kinetically balanced basis, both in the intermediate and the unitary normalization. The results are compared with those of an earlier formulation in terms of operators and wave functions. In the wave function formulation it is imperative to describe the weak singularities of the wave function at the position of a point nucleus correctly and to satisfy the key relation between large and small components locally. This formulation is incompatible with an expansion in a regular basis. In a matrix formulation in a kinetically balanced basis both the large and the small component are expanded in regular basis sets and the key relation is only satisfied in the mean. DPT is essentially a theory at bispinor level. Although it is possible to eliminate the small component to arrive at a quasi-relativistic theory, this requires some care. A both compact and numerically stable formulation is in terms of the large and the small component. The generalization from a theory for one state to a quasi-degenerate formulation for a set of states, is very simple in the matrix formulation, but rather complicated and somewhat indirect at wave function level, where an intermediate quasi-relativistic step is needed. The advantage of the matrix formulation is particularly pronounced in the unitary normalization.