Electronic states degenerate perturbation theory

In this section, we shall use the degenerate perturbation theory approach to derive the form of the effective Hamiltonian for a diatomic molecule in a given electronic state. Exactly the same result can be obtained by use of the Van Vleck or contact transformations [12, 13]. The general expression for the operator up to fourth order in perturbation theory is given in equation (7.43). Fourth order can be considered as the practical limit to this type of approach. Indeed, even its implementation is very laborious and has only been used to investigate the form of certain special terms in the effective Hamiltonian. We shall consider some of these terms later in this chapter. For the moment we confine our attention to first- and second-order effects only. 

Thus far, we have investigated the various contributions to the effective Hamiltonian for a diatomic molecule in a particular electronic state which arise from the spin-orbit and rotational kinetic energy terms treated up to second order in degenerate perturbation theory. Higher-order effects of such mixing will also contribute and we now consider some of their characteristics. 

In two previous papers [8,9] we have calculated the static polarizabilities and hyperpolarizabilities for ls3p Pj (J = 0, 2)-states of helium. The method was based on degenerate perturbation-theory expressions for these quantities. The necessary dipole matrix elements were found by using the high-precision wave function on framework of the configuration-interaction (Cl) method [10]. The perturbed wave functions are also expanded in a basis of accurate variational eigenstates [11]. These basis sets of the wave functions explicitly take account of electron correlation. To control the result we have also carried out similar calculations with Fues model potential method. 

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. 

S. Wilson, K. Jankowski, and J. Paldus, Int.]. Quantum Chem., 28,525 (1985). Applicability of Non-Degenerate Many-Body Perturbation Theory to Quasi-Degenerate Electronic States. II. A Two-State Model. 

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