Calculation of Tanabe-Sugano Diagrams by Matrix Diagonalization

The chemistry and spectroscopy of transition elements depend on complex interactions found for partially filled d subshells. There are complex electron-electron repulsion effects within a partially filled subshell and additional complications due to participation of some of the d orbitals in chemical bonding. The number of states involved can be quite large, and the mathematical treatment involves matrices as large as 28 by 28. The direct solution for eigenvalues and eigenvectors of matrices this large was not even attempted in the 1930s, but is well within the capabilities of current undergraduates and computers. Full matrices in the form of spreadsheets are provided in this paper for octahedral and tetrahedral symmetry for all cases from d2 through d8. Diagonalization of such matrices with MACSYMA is illustrated. Construction of full Tanabe-Sugano diagrams is possible for students for any choice of input parameters.