On the choice of d-orbital coordinate system in charge-density studies of low-symmetry transition-metal complexes

The distribution of the d electrons over the corresponding orbitals in transition-metal complexes is a central concept in the theory of metal-ligand bonding. The description requires the assignment of an axis of quantization, which is unambiguous in symmetric environments but not clear-cut in the now commonly encountered case of a low-symmetry coordination environment. As the d-electron population can be derived from accurate diffraction data using the methods of charge-density analysis [Holladay et al. (1983). Acta Cryst. A39, 377-387], the need for an appropriate procedure is relevant in this area of crystallography. Several criteria for the choice of coordinate system based on the resulting orbital populations are discussed. They are tested on a cobalt atom in a trigonal bipyramidal site and applied to transition-metal sites in Cu^II-alanyl-valine, and an open zirconocene. The population of the d-orbital cross terms for the different coordinate-system orientations is used to judge the results. In the cases examined, the intuitively most reasonable coordinate system corresponds to the one with smaller value of the sum of the populations of the d-orbital cross terms.