Fitting of local densities in periodic systems

We present a formalism for density fitting, a widely used method in molecular context, to approximate the electron repulsion integrals between local densities in periodic systems. An efficient treatment is achieved through a separation of the fitting into a reciprocal space part, done with an extended, but momentless fitting basis set, and a direct space part with a small, local set of usual Gaussian-type orbitals. This approach can be applied to periodic Hartree-Fock, Kohn-Sham, post-Hartree-Fock correlation, etc. methods. We demonstrate the efficiency of the method in the periodic local M{}ller-Plesset perturbation theory of the second order, which in the conventional formulation is for crystals very expensive.