Density fitting approach to Coulomb integrals for infinite systems with translational periodicity is reformulated in direct space. Despite of the Coulomb infinite decay of some integrals, direct-space calculation is shown to be feasible. Moreover, we show that the direct-space ansatz is completely equivalent to our previous formulation in reciprocal space. Computational demands scale linearly with the number of unit cells. In addition, direct-space treatment has some practical advantages over the reciprocal-space formulation. The efficiency of our scheme is demonstrated on systems with translational periodicity in one dimension. Computation time takes only a small fraction of the conventional calculation with exact integrals. We show that for infinite systems auxiliary basis sets of equally good quality as for molecules can be constructed in a systematic way.
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