Energy operators as idempotents in the angular overlap model of the ligand field

Abstract

In the angular overlap model, the ligand field is parametrized on the assumption that it is proportional to the square of overlap integrals between the orbitals of the open d or f shell of the central ion and those orbitals of the ligands which participate in the bonding. For linearly ligating ligands the orbital energies within the open shell is expressed by the semi-empirical parameters eσ, eπ, … eλ …. el. It is shown that the solid harmonic form of the functions, describing the open shell, makes it possible to calculate the coefficients to the e parameters by taking matrix elements of certain projection operators, belonging to each ligand, with respect to a standard central ion function basis. These idempotents project out of the standard set, those functions which have a maximum overlap with the ligand orbitals. The formalism also makes it possible to define shift operators, rotation operators, and those operators which make possible the parametrization of central ion l mixing by the ligand field.