All-electron periodic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>G</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msub><mml:mi>W</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math> implementation with numerical atomic orbital basis functions: Algorithm and benchmarks

We present an all-electron, periodic ${G}_{\text{0}}{W}_{\text{0}}$ implementation within the numerical atomic orbital (NAO) basis framework. A localized variant of resolution-of-the-identity (RI) approximation is employed to significantly reduce computational cost evaluating and storing two-electron Coulomb repulsion integrals. demonstrate that error arising from RI can be reduced insignificant level by enhancing set auxiliary functions, used expand products two single-particle NAOs. An efficient algorithm introduced deal with singularity in Brillouin zone sampling suitable for NAO perform systematic convergence tests identify a parameters, which serve as default choice most practical purposes. Benchmark calculations are carried out prototypical semiconductors insulators, compared independent reference values obtained based on linearized augmented plane waves (LAPWs) plus high-energy orbitals (HLOs) set, well experimental results. With moderate (FHI-aims tier 2) our produce band gaps typically lie between standard LAPW $\mathrm{LAPW}+\mathrm{HLO}$ Complementing 2 highly Slater-type (STOs), we find show overall towards The algorithms techniques developed this work pave way implementations correlated methods