Perturbation theory is a useful tool for evaluating small corrections to a system, and as we noted in the preceding chapter, relativity is a small correction for much of the periodic table. If we can use perturbation theory based on an expansion in 1/c we assign much of the work associated with a more complete relativistic treatment to the end of an otherwise nonrelativistic calculation. The problems with the Pauli Hamiltonian—the singular operators and the questionable validity of an expansion in powers of p/mc—are essentially circumvented by the use of direct perturbation theory. For systems containing heavier atoms it is necessary to go to higher order in 1/c perturbation theory, and possibly even abandon perturbation theory altogether. If we could perform an expansion that yielded a zeroth-order Hamiltonian incorporating relativistic effects to some degree and that was manifestly convergent, it might be possible to use perturbation theory to low order for heavy elements. If we wish to incorporate some level of relativistic effects into the zeroth-order Hamiltonian, we cannot start from Pauli perturbation theory or direct perturbation theory. But can we find an alternative expansion that contains relativistic corrections and is valid for all r: that is, can we derive a regular expansion that is convergent for all reasonable values of the parameters? The expansion we consider in this chapter has roots in the work by Chang, Pélissier, and Durand (1986) and Heully et al. (1986), which was developed further by van Lenthe et al. (1993, 1994). These last authors coined the term “regular approximation” because of the properties of the expansion. The Pauli expansion results from taking 2mc2 out of the denominator of the equation for the elimination of the small component (ESC). The problem with this is that both E and V can potentially be larger in magnitude than 2mc2 and so the expansion is not valid in some region of space. In particular, there is always a region close to the nucleus where |V − E|/2mc2 > 1. An alternative operator to extract from the denominator is the operator 2mc2 − V , which is always positive definite for the nuclear potential and is always greater than 2mc2.
Geometry optimizations in the zero order regular approximation for relativistic effects
