Efficient all-electron periodic Fourier-transformed Coulomb method

We present an efficient algorithm for constructing an all-electron periodic Coulomb matrix based on Ewald summation combined with the Fourier-transformed Coulomb method. The short-range contributions involving compact densities are evaluated in real space using standard Gaussian density fitting. For the long-range contributions, we introduce an integral-direct plane wave density fitting scheme that is applicable to both compact and diffuse densities. The resulting method achieves orders-of-magnitude speedups for prototypical solid-state systems compared to a closely related approach, the range-separated density fitting method. Using the dispersion-corrected PBE functional with all-electron Dunning and Karlsruhe basis sets, we apply our method to compute the cohesive energy of the benzene crystal and the adsorption energy of CO on the MgO(001) surface. These results are in good agreement with existing literature. Our approach enables efficient Gaussian-based semi-local density functional calculations using dense k-point meshes and traditional molecular Gaussian basis sets.