Quantitative Comparison of the Symmetry Components of a Ligand Field: Illustrations of the Orthonormal Operators Formalism

Abstract

A recently proposed formulation of ligand field models in terms of orthonormal operator sets allows a quantitative comparison of the various symmetry components of a ligand field operator in terms of percentages. This is illustrated here by the detailed analysis of an orthorhombic system using the hierarchy O ⊃ D₄ ⊃ D₂ of cubic, tetragonal and orthorhombic symmetries. The first part of the analysis pertains to a set of general octahedral e functions. The formalism introduced is later applied to an entire set of d orbitals, and in this connection we further discuss the concepts of atomic and molecular ligand field theory. In the present treatment all ligand field parameters and splittings are defined with respect to sign as well as magnitude. In the example cited the tetragonal splitting in the e space may be positive as well as negative, whereas the total splitting and the quantity called the orthorhombic splitting are non-negative. It is demonstrated that for any ligand field component of octahedral E symmetry it is possible to choose a coordinate frame in such a way that the resulting decomposition of the field component into tetragonal and orthorhombic parts will describe the system as being at least 75% tetragonal.