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The $GW$ approximation has been recently gaining popularity among the methodfor simulating molecular core-level X-ray photoemission spectra. Traditionally,$GW$ core-level binding energies have been computed using either the cc-pV$n$Zor def2-$n$ZVP basis set families, extrapolating the obtained results to thecomplete basis set limit, followed by a an element-specific relativisticcorrection. Despite of achieving good accuracy, these binding energies arechronically underestimated. By using first-row elements and standard techniquesknown to offer good cost-accuracy ratio in other theories, we show that thecc-pV$n$Z and def2-$n$ZVP families show large contraction errors and lead tounreliable complete basis set extrapolations. On the other hand, we demonstratethat uncontracted versions of these basis sets offer vastly improvedconvergence. Even faster convergence can be obtained using core-rich,property-optimized, basis sets families like pcSseg-$n$, pcJ-$n$ and ccX-$n$Z.Finally, we also show that the improvement over the core properties does notdegrade the calculation of the valence excitations, and thus offer a balanceddescription of both core and valence regions.

On the basis set selection for molecular core-level $GW$ calculations