Efficient All-Electron Periodic Fourier-Transformed Coulomb Method

We present an efficient algorithm for the all-electron periodic Coulomb matrix based on the Ewald summation combined with the Fourier-transformed Coulomb method. The short-range contributions involving compact densities are evaluated in real space using the standard Gaussian density fitting method. For the long-range contributions, we introduce an integral-direct planewave density fitting, 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 dispersion-corrected PBE functional and 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 $\mathbf {k}$-point meshes and traditional molecular Gaussian basis sets.