Direct Perturbation Theory of Properties

The calculation of properties using direct perturbation theory follows exactly the same lines as we used for Breit-Pauli theory. As we noted above, stationary direct perturbation theory leads to precisely the same equations we would have obtained by simply expanding the perturbed wave functions in the set of eigenfunctions of the zeroth-order Hamiltonian, and on this basis we proceed with the development of multiple direct perturbation theory for properties. 

The main difference between the direct and the Breit-Pauli perturbation schemes is that in the former we have to expand the metric as well as the Hamiltonian. To illustrate 

As in section 17.4, we use tlie parameters /u and r] for relativity and the external field, with fi,= l/c and r] the field strength. The Hamiltonian we write as 

The two perturbations will have a different form for direct and Breit-Pauli perturbation theory. The wave function also will be different in form two components in Breit-Pauli theory and four components in direct perturbation theory. In addition, the metric must be expanded in the direet scheme, 

The energy and wave function are expanded in both perturbation parameters, 

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