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We describe an all-electron $G_0W_0$ implementation for periodic systems with$k$-point sampling implemented in a crystalline Gaussian basis. Ourfull-frequency $G_0W_0$ method relies on efficient Gaussian density fittingintegrals and includes both analytic continuation and contour deformationschemes. Due to the compactness of Gaussian bases, no virtual state truncationis required as is seen in many plane-wave formulations. Finite size correctionsare included by taking the $q \to 0$ limit of the Coulomb divergence. Using ourimplementation, we study quasiparticle energies and band structures across arange of systems including molecules, semiconductors, rare gas solids, andmetals. We find that the $G_0W_0$ band gaps of traditional semiconductorsconverge rapidly with respect to the basis size, even for the conventionallychallenging case of ZnO. Using correlation-consistent bases of polarizedtriple-zeta quality, we find the mean absolute relative error of theextrapolated $G_0W_0$@PBE band gaps to be only 5.2% when compared toexperimental values. For core excitation binding energies (CEBEs), we find that$G_0W_0$ predictions improve significantly over those from DFT if the $G_0W_0$calculations are started from hybrid functionals with a high percentage ofexact exchange.

All-electron Gaussian-based $G_0W_0$ for Valence and Core Excitation Energies of Periodic Systems