This paper is a contribution to the theory of Wannier functions. The main emphasis is on practical methods for the ab initio construction of Wannier functions for simple and composite bands from appropriate variational principles. The cases of a simple $s$ band, a hybridized $s-d$ band, and the valence and conduction bands of the diamond structure are treated in detail. The calculations of Bloch waves, energy bands, and densities of states from Wannier functions is described. Questions of uniqueness and nonuniqueness and problems due to attachment to other bands are also discussed.

• Received 6 April 1972

DOI:https://doi.org/10.1103/PhysRevB.7.4388