Atoms degenerate perturbation theory

Excited States of the Helium Atom. Degenerate Perturbation Theory... 

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. 

Theory can now provide much valuable guidance and interpretive assistance to the mechanistic photochemist, and the evaluation of spin-orbit coupling matrix elements has become relatively routine. For the fairly large molecules of common interest, the level of calculation cannot be very high. In molecides composed of light atoms, the use of effective charges is, however, probably best avoided, and a case is pointed out in which its results are incorrect. It seems that the mean-field approximation is a superior way to simplify the computational effort. The use of at least a double zeta basis set with a method of wave function computation that includes electron correlation, such as CASSCF, appears to be imperative even for calculations that are meant to provide only semiquantitative results. The once-prevalent degenerate perturbation theory is now obsolete for quantitative work but will presumably remain in use for qualitative interpretations. 

In quantum mechanics the splitting of electronic terms is described by using the degenerate version of the perturbation theory. The Hamiltonian for electrons in the atom in the crystal environment acquires the form ... 

In the perturbation theory for degenerate states the resonant hyperpolarizability is determined by the tensor part of polarizability [9] and may be extracted out of the fourth-order terms self-consistently in the case of nondegenerate perturbation theory the resonant part appears for separate sublevels of an atomic multiplet. The numerical results are listed in Table 2. 

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