Trigonally quantized ligand field potentials, d-orbitals and d-orbital energies

Abstract

Abstract The most generalized ligand field potentials for all possible geometries of coordination numbers one to twelve containing a trigonal axis have been derived in trigonal quantization. These potentials have been specialized for idealized geometries and the corresponding trigonally quantized d-orbitals and their energies have been derived. The applicability of the potentials and the d-orbitals in the study of the systems of idealized as well as slightly deviated trigonal geometries has been described. The ligand field potentials for a number of geometries containing a three-fold axis of symmetry have been evaluated along the trigonal axis and are categorized into three kinds, cylindrical potentials (no spherical harmonics of non-zero order), cubic potentials (related to each other by a constant factor) and non-cubic potentials (not related to cubic potentials by a constant factor). The energies of the d-orbitals defined along the trigonal axis have been calculated. The original spherical harmonics of the d-orbitals are pure wave functions for geometries of cylindrical potentials. The trigonally quantized d-orbitals which are the eigenfunctions of cubic geometries have been derived. The d-orbital wave functions for regular cubic based and for some trigonally distorted cubic and non-cubic geometries have also been obtained. The original spherical harmonics are the appropriate zero order functions for geometries of cylindrical potentials with small trigonal distortions. Similarly the cubic d-orbitals can be used as the appropriate zero order functions for regular cubic based geometries of non-cubic potentials and for trigonally distorted cubic and cubic-based geometries with slight deviations of the trigonal angles.