Spatial distribution of photoelectrons participating in formation of x-ray absorption spectra

Interpretation of x-ray absorption near-edge structure (XANES) experiments is often done via analyzing the role of particular atoms in the formation of specific peaks in the calculated spectrum. Typically, this is achieved by calculating the spectrum for a series of trial structures where various atoms are moved and/or removed. A more quantitative approach is presented here, based on comparing the probabilities that a XANES photoelectron of a given energy can be found near particular atoms. Such a photoelectron probability density can be consistently defined as a sum over squares of wave functions which describe participating photoelectron diffraction processes, weighted by their normalized cross sections. A fine structure in the energy dependence of these probabilities can be extracted and compared to XANES spectrum. As an illustration of this technique, we analyze the photoelectron probability density at the Ti K pre-edge of ${\mathrm{TiS}}_2$ and at the Ti K-edge of rutile ${\mathrm{TiO}}_2.$