Understanding the Empirical Shifts Required for Quantitative Computation of X-ray Spectroscopy

Theoretical modeling has played a crucial role in the emergence of X-ray spectroscopic techniques as essential characterization methods. Most electronic structure methods require empirical shifts to obtain quantitative agreement with the experimental data. Modeling X-ray spectroscopy of open-shell systems is particularly challenging for electronic structure methods due to the presence of singly occupied orbitals, which can further contribute to the empirical shift. The requirement of these empirical shifts has been attributed to different inadequacies of electronic structure methods, e.g., orbital relaxation, static and dynamic electron correlation, self-interaction error, spin adaptation of the wave function, and relativistic effects. While few studies in the literature have studied the contribution of individual factors, systematic studies have not been performed to understand the relative importance of different factors. In this article, we have tested a wide range of electronic structure methods, both within single- and multireference frameworks, to compute core-excited and core-ionized states of both light elements and transition metal-based systems and qualitatively analyzed the effect of different factors on the empirical shift. We have found that orbital relaxation is the most essential factor for the accurate computation of the core excitation and ionization energies. Two density functional-based approaches, excited-state density functional theory (eDFT) and multiconfiguration pair-density functional theory (MC-PDFT), performed the best among all of the methods tested in this study. While both of these methods can provide sub-eV accuracy for organic systems, eDFT and MC-PDFT show a much larger error for transition metal systems, which can possibly be attributed to the inadequacy of conventional relativistic treatment.