Relative intensities for multiplet and crystal-field-split transition-metal-ion photoemission spectra

Previously, the photoemission spectra of the $d$ levels of ionic transition-metal systems have been analyzed assuming localized-ion levels and taking into account the multiplet and crystal-field splitting of the final states. However, theoretical analyses of the relative intensities of these levels given in several recent papers are inconsistent and usually incorrect. The analysis required to obtain geometrical intensity rules for systems with several open shells (i.e., $t_{2g}$ and $e_g$ shells in transition-metal systems) is quite involved and difficult. In this paper, we review and physically interpret the analysis. We explicitly state the main assumptions used and show where they enter the problem. Finally, we present tables of our results for $d^n$ ions ($n=1\mathrm{}-9$) in octahedral and in tetrahedral coordination.