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We present an implementation of the Bethe-Salpeter equation (BSE) forcore-conduction band pairs within the framework of the projector augmented-wavemethod. For validation, the method is applied to the $K$-edges of diamond,graphite, hexagonal boron-nitride, as well as four lithium-halides (LiF, LiCl,LiI, LiBr). We compare our results with experiment, previous theoretical BSEresults, and the density functional theory-based supercell core-hole method. Inall considered cases, the agreement with experiment is excellent, in particularfor the position of the peaks as well as the fine structure. Comparing BSE tosupercell core-hole spectra we find that the latter often qualitativelyreproduces the experimental spectrum, however, it sometimes lacks importantdetails. This is shown for the $K$-edges of diamond and nitrogen in hexagonalboron-nitride, where we are capable to resolve within the BSE experimentalfeatures that are lacking in the core-hole method. Additionally, we show thatin certain systems the supercell core-hole method performs better if theexcited electron is added to the background charge. We attribute this improvedperformance to a reduced self-interaction.

Core-hole excitations using the projector augmented-wave method and the Bethe-Salpeter equation