Optimized Auxiliary Functions for Robust Mitigation of Finite-Size Errors in Periodic Hybrid Density Functional Theory

When
calculating properties of periodic systems at the thermodynamic
limit (TDL), the dominant source of finite size error (FSE) arises
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 sum. The singularity subtraction (SS)
method offers a systematic approach for reducing this quadrature error
and thus the FSE. In this work, we first investigate the performance
of the SS method in the simplest setting, aiming at reducing the FSE
in exact exchange calculations by subtracting the Coulomb contribution
with a single, adjustable Gaussian auxiliary function. We demonstrate
that a simple fitting method can robustly estimate the optimal Gaussian
width and leads to rapid convergence toward the TDL. Furthermore,
we suggest new forms of the auxiliary function, whose optimal parameters
could also be determined through least-squares fitting. For a range
of semiconductors and insulators, the proposed auxiliary functions
achieve robust, millihartree-level accuracy in hybrid density functional
theory calculations, including cases with sparse k-meshes
and large basis sets.