Optical matrix elements in tight-binding calculations

Calculations of the optical properties of solids require knowledge of momentum matrix elements. Within the tight-binding method, these are not readily available since wave functions are expanded in an unknown basis. A popular solution to this problem is based on taking the k-space gradient of the tight-binding Hamiltonian. This technique was previously claimed to be exact, but is in fact only approximate since intra-atomic contributions are emitted. We highlight the reason for this misconception. By comparison to exact results from a simple model it is demonstrated, however, that the k-gradient term provides a good approximation to the interatomic contribution. Subsequently, we demonstrate that adding a k-independent intra-atomic term to the momentum matrix leads to excellent agreement with exact results for momentum matrix elements and dielectric spectra.