Smoothed truncated Coulomb potential for periodic Gaussian-basis Hartree--Fock exchange
Truncated Coulomb (TC) potentials reduce finite-size errors and accelerate thermodynamic-limit convergence in periodic Hartree--Fock (HF) calculations, but their use with Gaussian basis sets is complicated by the evaluation of electron repulsion integrals (ERIs), particularly for nonspherical truncation domains and all-electron calculations. We introduce the smoothed truncated Coulomb (sTC) potential as a systematically improvable approximation to a parent TC potential. A real-space Gaussian convolution smooths the sharp truncation boundary, and a single dimensionless parameter, $\eta$, controls the width of the smoothing window, which can be tightened to systematically approach the TC reference. The smoothing by sTC enables a dual-space algorithm for evaluating periodic Gaussian-basis ERIs that requires neither a large plane-wave basis nor new molecular integral kernels and is applicable to both pseudopotential and all-electron calculations, including the important Wigner--Seitz-cell truncation boundaries. Benchmarks spanning insulating, semiconducting, layered, metallic, and molecular-crystal systems show that sTC-based HF closely reproduces TC results in pseudopotential calculations and extends TC-quality calculations to all-electron settings. Across these systems, sTC substantially improves thermodynamic-limit convergence relative to the probe-charge Ewald method while remaining practical for all-electron calculations in which direct TC-based calculations are computationally challenging.