Quotients of group algebrae in the calculation of intermediate ligand field matrix elements

The structure of the classes of symmetry elements excluded during the subduction of the representations of SU(2) onto the finite group 0* is shown to quantitatively define the relationship of the coupling algebrae of these two groups. This relationship is formalized as a quotient algebra. This quotient algebra is realized as 3-like symbols which exist whether or not the quotient can be defined as a group. These symbols distribute the value of a reduced matrix element of SU(2) onto the subduced reduced matrix elements of O* and are termed Partition Coefficients. Since the structure of the excluded symmetry classes is independent of the quantization of O*, these Partition Coefficients can be used to define the values of the matrix elements of O* without reference to the form of its basis set. Thus, the choice of physical interpretation of the ligand field is unimportant. The strong field, weak field, Russell-Saunders and j-j coupling models are all unified in terms of the Partition Coefficients and the 3 symbols which are appropriate to the quantization.