Wavevector-dependent optical properties from wavevector-independent conductivity tensor

We discuss the standard ab initio calculation of the refractive index by means of the scalar dielectric function and show its inherent limitations. To overcome these, we start from the general, microscopic wave equation in materials in terms of the frequency- and wavevector-dependent dielectric tensor, and we investigate under which conditions the standard treatment can be justified. We then provide a more general method of calculating the frequency- and direction-dependent refractive indices by means of a $(2 \times 2)$ complex-valued "optical tensor", which can be calculated from a purely frequency-dependent conductivity tensor. Finally, we illustrate the meaning of this optical tensor for the prediction of optical material properties such as birefringence and optical activity.