  A method for incorporating electromagnetic fields into empirical
tight-binding theory is derived from the principle of local gauge symmetry.
Gauge invariance is shown to be incompatible with empirical tight-binding
theory unless a representation exists in which the coordinate operator is
diagonal. The present approach takes this basis as fundamental and uses group
theory to construct symmetrized linear combinations of discrete coordinate
eigenkets. This produces orthogonal atomic-like "orbitals" that may be used as
a tight-binding basis. The coordinate matrix in the latter basis includes
intra-atomic matrix elements between different orbitals on the same atom.
Lattice gauge theory is then used to define discrete electromagnetic fields and
their interaction with electrons. Local gauge symmetry is shown to impose
strong restrictions limiting the range of the Hamiltonian in the coordinate
basis. The theory is applied to the semiconductors Ge and Si, for which it is
shown that a basis of 15 orbitals per atom provides a satisfactory description
of the valence bands and the lowest conduction bands. Calculations of the
dielectric function demonstrate that this model yields an accurate joint
density of states, but underestimates the oscillator strength by about 20% in
comparison to a nonlocal empirical pseudopotential calculation.
