Exact representation ofexp(iq⋅r)in the empirical tight-binding method and its application to electromagnetic interactions

Abstract

We show how the exact representation of the operator $\mathrm{exp}(i\mathbf{q}\mathbf{\ensuremath{\cdot}}\mathbf{r})$ in the Bloch sum basis used in the empirical tight-binding method follows from the transformation law obeyed by cell-periodic operators, such as the Hamiltonian. From this representation, we derive matrix elements of its product with cell-periodic operators, since product operators of this type arise when the crystal is subject to various perturbations. Using these results, we calculate the expression for the linear transverse dielectric tensor.