Combining the $\Delta$-Self-Consistent-Field and GW Methods for Predicting Core Electron Binding Energies in Periodic Solids

For the computational prediction of core electron binding energies in solids, two distinct kinds of modelling strategies have been pursued: the $\Delta$-Self-Consistent-Field method based on density functional theory (DFT), and the GW method. In this study, we examine the formal relationship between these two approaches, and establish a link between them. The link arises from the equivalence, in DFT, between the total energy difference result for the first ionization energy, and the eigenvalue of the highest occupied state, in the limit of infinite supercell size. This link allows us to introduce a new formalism, which highlights how in DFT - even if the total energy difference method is used to calculate core electron binding energies - the accuracy of the results still implicitly depends on the accuracy of the eigenvalue at the valence band maximum in insulators, or at the Fermi level in metals. We examine, whether incorporating a quasiparticle correction for this eigenvalue from GW theory improves the accuracy of the calculated core electron binding energies, and find that the inclusion of vertex corrections is required for achieving quantitative agreement with experiment.