The Coulomb singularity corrections in plane-wave electronic structure calculations: Implementation and benchmarking

The Kohn-Sham density functional theory (DFT) is a fundamental tool for investigating the electronic structures of molecules and solids. For periodic systems, the singularity problem does not arise in pure DFT calculations. However, when DFT is combined with the Hartree-Fock method and other wavefunction methods to improve accuracy, the long-range nature of the Coulomb potential $1/r$ leads to cumbersome interactions and introduces a $1/G^2$ singularity at the $\Gamma$ point in reciprocal space.Handling these divergence singularities is crucial for enhancing both the accuracy and efficiency of calculations in periodic systems, particularly in various computational methods such as hybrid functionals, random phase approximation (RPA), second-order Møller-Plesset perturbation theory (MP2), GW approximation, and time-dependent density functional theory (TDDFT).In this work, we conduct detailed comprehensive calculations and comparisons of various approaches, including spherical truncation, Wigner-Seitz truncation, screened Coulomb potentials, auxiliary function, and Probe charge method, to addressing this singularity problem in plane-wave calculations within periodic boundary conditions.In the context of plane-wave-based RPA and GW calculations, we pioneeringly implement the Wigner-Seitz truncation scheme and systematically investigate its convergence and accuracy through both supercell expansion and k-point sampling. Our results reveal optimal correction strategies tailored to different systems and computational methods, providing valuable guidance for future developments in the field.