PERFECT LOCALIZED BASIS FUNCTIONS FOR SOLIDS - CHEMICAL PSEUDOPOTENTIALS AND THE KRONIG-PENNEY MODEL

Anderson's chemical pseudopotential scheme is supposed to define a set of highly localized basis functions in terms of which the eigenstates of a solid can be expanded exactly. This sounds good in principle but little is known about the method in practice and it has not even been established that the basis functions always exist. This paper discuss es some of the general properties of localized basis sets spanning a b and of Bloch eigenstates and then looks at the Kronig-Penney model as an example. It is shown that the chemical pseudopotential equation has no solutions when the strength P of the attractive delta function pot entials is less than about 1.4018, and that the basis functions genera ted are never unique when they do exist. Fortunately, it turns out tha t there is a generalized version of the chemical pseudopotential equat ion that can be solved no matter how weak the potentials, although the non-uniqueness of the basis functions remains. The solutions resemble atomic orbitals in the extreme tight-binding limit (although even the n there are alternatives) but this is not true in general and so chemi cal pseudopotential theory should not be taken as a justification for using atomic orbitals as basis functions.