Quantum defect theory for bound states

Quantum defect theory (QDT) was developed by Seaton [111] and his collaborators, from ideas which can be traced to the origins of quantum mechanics, through the work of Hartree and others. They relate to early attempts to extend the Bohr theory to many-electron systems (see e.g. [114]). 

In chapter 2, we saw how the quantum defect is defined from a slight modification of the Rydberg formula for H. It is found experimentally to be nearly constant for different series members, especially for unperturbed series in atoms with a compact core. The first task of QDT is to explain this fact, and to extract from this empirical observation an appropriate wavefunction, consistent with an effective one-electron Schrodinger equation, such that the quantum defect would turn out to be be nearly constant as the principal quantum number n is changed. 

QDT is not an ab initio theory, i.e. it is not an attempt to solve the many-body problem from first principles. Rather, it is a theoretically-based parametrisation. One seeks a form for the wavefunctions and for their dependence on n this in turn leads to precise rules for the variation of many other quantities with n because, in quantum mechanics, once the wavefunctions are known, many observable properties of the system become calculable. 

The organisation of the present chapter is as follows. First, we give the basic principles of single channel QDT. Next, we move on to series perturbations, and to simple graphical methods which can be used to analyse them by MQDT. Finally, we give examples of how such methods can be extended to more complex situations, and we deal with extrapolations beyond the series limit which allow autoionisation profiles to be predicted from the data for interacting bound states. 

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