Singularity subtraction methods to reduce finite-size effects in periodic quantum chemistry calculations

Quantum chemistry calculations with periodic boundary conditions suffer from finite-size errors (FSEs), a third fundamental source of error alongside basis set incompleteness and correlation errors. In the context of density functional theory (DFT), Hartree-Fock (HF), and higher-order wavefunction methods, FSEs arise primarily from the long-range Coulomb interaction and can manifest as a slowly converging quadrature error when approximating an integral in the reciprocal space by a finite k-point summation. Thus, converging to the thermodynamic limit (TDL) for these theories remains a challenge for routine calculations.

The singularity subtraction (SS) method offers a systematic approach for reducing this quadrature error by adding and subtracting an integrable auxiliary function that captures the leading orders of the singularity. We first investigate the performance of the SS method in the simplest setting, aiming at reducing the FSE in exact exchange calculations used in HF and hybrid DFT calculations. For a range of semiconductors and insulators, the SS method with the proposed auxiliary functions achieves robust, millihartree-level accuracy, including cases with sparse k-meshes and large basis sets. Furthermore, we demonstrate that SS can also be used to correct MP2 correlation energies, showcasing its promise in reducing FSE contributions beyond the HF level.