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

The second-order optical susceptibility of semiconductors ${\chi }_{ijk}^{(2)}(-2\omega ;\omega ,\omega )$ finds application in metrology, spectroscopy, telecommunications, material characterization, and quantum information. Pioneering calculations of ${\chi }_{ijk}^{(2)}(-2\omega ;\omega ,\omega )$ 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 ({\em i.e.}, {\em localized}) orbitals, making a comeback onto frameworks based on localized basis sets. Here, we present an approach for calculating ${\chi }_{ijk}^{(2)}(-2\omega ;\omega ,\omega )$ reliant on numerical pseudoatomic 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.