Core–hole delocalization for modeling x-ray spectroscopies: A cautionary tale

In recent years, the field of X-ray spectroscopy has progressed rapidly, in particular thanks to the development of X-ray free electron lasers and the continuous improvement in time and energy resolution of time-resolved measurements. The interpretation of data collected with these advanced techniques is directly linked to theory and modeling and their ability to correctly describe the interactions between X-rays and matter. It is, therefore, important to understand the benefits and pitfalls of various approaches to compute X-ray spectra. I will discuss several common methods used to calculate steady-state X-ray absorption, emission, and photoelectron spectroscopy of molecules using first principles density- and wave function-based methods. After discussing their accuracy and precision, I will focus on the question of core-hole (CH) delocalization, i.e. the extent to which core-hole orbitals may delocalize over several atoms of symmetric and extended molecular systems, and how CH delocalization can affect the quality of calculated spectra. Capturing electronic relaxation is crucial for X-ray properties, since the strong attractive polarization of the charge density related to core-ionization or excitation is one of the main effects at play. Relaxation errors can be introduced in calculations that consider explicit CHs, if care is not given to localize the CH, or to describe electron correlation at a high level of theory. Besides discussing the effect of the delocalization-induced relaxation error (DIRE) for different X-ray spectroscopies, I will show that DIRE directly relates to the level at which exchange and correlation are included in the calculation. While a localized CH can describe orbital relaxation well, a delocalized CH with the proper symmetry requires higher levels of theory to capture the correlation effects responsible for relaxation. With the emergence of new X-ray spectroscopy software that use explicit CHs, it becomes significant to be aware of how DIRE can affect different types of X-ray spectra and mitigate it by using localized CHs, or an appropriately high theory level.