Summary: A simple method has been developed for calculating the d-orbital energy levels of transition-metal ions in coordination polyhedra with both orthogonal and non-orthogonal distortions, using equations based on those derived by Ballhausen (1954). The input data are atomic coordinates, a standard value of the crystal field splitting parameter Δ at known metal-ligand distance, and the ratio of radial integrals B₂/B₄, which is approximately constant for a given ion. The method can be applied to polyhedra containing different ligands.

Application of the equations to the Mn³⁺ (M3) site in piemontite and the Fe²⁺ (M2) site in orthopyroxene gives calculated transition energies in good agreement with the observed band energies.

The calculations permit definite assignment of the great majority of d-d absorption bands even in multi-site phases, and enable discrimination of crystal-field and charge-transfer bands in mineral spectra. They also throw light on the fine structures of both oxygen → metal and metal → metal charge-transfer bands, and allow the calculation of crystal-field stabilization enthalpy and electronic entropy. The latter is a previously neglected energy term that contributes significantly to the energetics of reactions within and between phases containing transition-metal ions.