Properties of Wannier functions

Wannier functions provide us with a means for the development of rigorous theorems which are analogous to the crude theorems we might derive with a tight binding approximation. However, a good and accurate Wannier function would be extremely difficult to calculate and its structure would be quite complicated. Excluding calculations obtained using the O.A.O. approximation few works exist on the mathematical properties of Wannier functions and only very few have been explicitly determined. The present study attempts to provide a partial solution for this situation and seeks to extend the use and knowledge of these functions. First, a theoretical investigation is made of the mathematical properties of Wannier functions and then some calculations are carried out to obtain some of these functions for various forms of potential fields. The original results obtained concern; I) Free electron Wannier functions for many of the familiar crystal types. II) A method for obtaining Wannier functions from relatively simple functions (unperturbed Wannier functions). The method has been appli ed to several particular cases (potential field with inversional symmetry etc.) and has been modified for other applications. III) Nearly free electron Wannier functions calculated both using analytical methods and the method outlined in section II. IV) The extension of Kohn's one dimensional treatment to three dimensions using a new approach.