Mixed-space formalism for the dielectric response in periodic systems

We present a useful formalism for the calculation of the polarizability and dielectric response of periodic systems. Our approach is to introduce intermediate ``mixed-space'' functions with the full translational periodicity of the lattice. This is a considerable advantage over existing real-space methods since the decay length of a response function [such as {ε}(r,r'{}{ω})] can be significantly larger than the Wigner-Seitz cell radius. Further, we show that, in supercell calculations, these mixed-space functions decay as fast as the corresponding real-space quantities within one supercell, so that the present scheme can be combined with usual real-space cutoff techniques. The advantage of the present method compared to a standard reciprocal space approach is exemplified for the case of bulk silicon and for the case of a Si surface in a slab geometry.