Perturbational Approaches to Spin-Orbit Coupling

As in all perturbational approaches, the Hamiltonian is divided into an unperturbed part and a perturbation V. The operator is a spin-free, one-component Hamiltonian and the spin-orbit coupling operator takes the role of the perturbation. There is no natural perturbation parameter X in this particular case. Instead, J4 so is assumed to represent a first-order perturbation The perturbational treatment of fine structure is an inherent two-step approach. It starts with the computation of correlated wave functions and energies for pure spin states—mostly at the Cl level. In a second step, spin-orbit perturbed energies and wavefunctions are determined. 

Only one of the matrix elements needs to be evaluated explicitly. All others can be obtained from the reduced matrix element by means of the Wigner-Eckart theorem, Eq. [166]. The eigenvectors 

If the degeneracy is lifted completely in first-order or if at least second-order effects do not introduce an additional splitting of degenerate levels, the second-order energy can be expressed as 

The corresponding first-order perturbed wave function reads 

Moreover, first and second orders of perturbation theory are not defined in a stringent way in a molecule. Consider, for instance, all potential energy curves of a diatomic molecule that correspond to a specific dissociation 

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