Perturbative second-order optical susceptibility of bulk materials: a symmetry-enforced return to non-orthogonal localized basis sets

The second-order optical susceptibility of semiconductorsχijk(2)(-2ω;ω,ω)finds application in metrology, spectroscopy, telecommunications, material characterization, and quantum information. Pioneering calculations ofχijk(2)(-2ω;ω,ω)utilized non-orthogonal Gaussian orbitals centered at atoms. That formulation transitioned into plane-wave-based algorithms as time went by. As of late, nevertheless, multiple tools for calculating optical susceptibilities have recast the problem using Wannier (i.e.<i>localized</i>) orbitals, making a comeback onto frameworks based on localized basis sets. Here, we present an approach for calculatingχijk(2)(-2ω;ω,ω)reliant on numerical pseudo-atomic orbitals (PAOs) within perturbation theory in the velocity gauge. Its salient feature is a calculation of 'Slater-Koster-like' two-center integrals of the momentum operator in between PAOs identified by symmetry. The approach was successfully tested on paradigmatic cubic silicon carbide (3C-SiC) and gallium arsenide, for which linear responses are contributed as well.