Higher powers of spin operators

The origin of zero-field terms in the spin Hamiltonian is rooted in crystal-field theory, in which coordination complexes are represented as geometric structures of point charges (Stevens 1997). The crystal-field potential of these point charges is 

Rzero = DOQO + D2Q2. DOQO + D2Q2 + D4Q4 nrhombic D2KJ2 D2KJ2 D4KJ4 D4 4 n4 4 

For higher integer spins the number of allowed zero-field interaction terms further increases, and so does the convolution of comparable effects, except once more for a unique term that directly splits the highest non-Kramer s doublet. For S 3 we have the addition, valid in cubic (and, therefore, in tetragonal, rhombic, and triclinic) symmetry  

As an illustration consider then a zero-field Hamiltonian for S = 4 in which we have retained only the familiar axial D- and rhombic E-term plus the cubic terms that split the non-Kramer s doublets in first order  

Note the characteristic band structure of these energy matrices. When we first con-  

---