Perturbation theories degenerate problems

The convergence of the iterative determination of eigenvalues and eigenvectors is accelerated appreciably if spin-orbit Cl and quasi-degenerate perturbation theory procedures are combined. To this end, the perturbation matrix is set up in the basis of the most important LS contracted Cl vectors Em 1 1). The solutions of this small eigenvalue problem... 

Let Ho be the Hamiltonian of the independent electron atom. We use the formalism of time-independent, degenerate perturbation theory to describe the problem, the variation being that, in the present case, the states which are degenerate in energy belong to the continuum on one hand and to the discrete spectrum on the other. This is a very interesting complication it is fundamental to quantum mechanics that discrete energy levels appear in what would otherwise be a fully continuous spectrum. Autoionisation is a mechanism which couples bound states of one channel to continuous states of another. 

In common with all problems of the same type in degenerate perturbation theory (see [259]), since the coefficient has factored out of the equations, it must be determined by normalising the perturbed function thus ... 

An example will show the application of some of the ideas introduced above. Let us start with the simple two-ccntcr-two-orbital problem described exhaustively in Chapter 2. In the language of perturbation theory these two orbitals experience a degenerate interaction for the case of H2 where the energies of each atomic orbital are the same. The result is an in-phase (bonding) combination and an out-of-phase (antibonding) combination, between the centers A and B. A more complicated example arises when there are two orbitals on A and one on B as when the orbitals of linear H3 are constructed from those of II2 + H (3.10). This is shown in Figure 3.1, where the relative phases of the orbitals have been chosen so that Sij and are positive. 

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