GENERALIZED WANNIER FUNCTIONS AND EFFECTIVE HAMILTONIANS

An effective Hamiltonian for an electron in a periodic lattice in a uniform magnetic field can be formulated rigorously in terms of an orthonormal set of basis functions. These functions are obtained from one another by application of magnetic translation operators. The Wannier functions turn out to be a special case of this more general set, applicable in the absence of magnetic fields. Not only are these functions not uniquely defined, but the form of the effective Hamiltonian will depend on their manner of selection, with the exception that at zero field the effective Hamiltonian is unique. A multiband effective Hamiltonian suitable for breakdown calculations is described. © 1968 The American Physical Society.