Explicit core-hole single-particle methods for <i>L</i>- and <i>M</i>-edge x-ray absorption and electron energy-loss spectra

Single-particle methods based on Kohn–Sham unoccupied states to describe near-edge X-ray absorption (XAS) spectra are routinely applied for the description of K-edge spectra, as there is no complication due to spin–orbit (SO) coupling. L- and M-edge spectra are often addressed via variants of time-dependent density functional theory (TDDFT) based on SO calculations. Here, we present a computationally efficient implementation based on single-particle calculations with core holes within the frozen-core approximation. Combined with a semiempirical energy shift and a fixed SO splitting for each core level, this allows for a computationally cheap, while overall accurate, prediction of experimental spectra on the absolute energy scale. The spectra are compared to about 40 times slower linear-response TDDFT calculations for molecules and show similar or even better matches with experiment. An exception is multiplet effects that we analyze in detail and show that they cannot be covered by a single-particle approximation. A similar picture emerges for solids, where good qualitative and sometimes even quantitative agreement to experimental XAS and electron energy-loss spectra is achieved.