Quasi-Spin and Pseudo-Cylindrical Selection Rules

1 Quasi-Spin and Pseudo-Cylindrical Selection Rules 

Low-symmetry LF operators are time-even one-electron operators that are non-totally symmetric in orbit space. They thus have quasi-spin K = 1, implying that the only allowed matrix elements are between 2P and 2D (Cf. Eq. 28). Interestingly in complexes with a trigonal or tetragonal symmetry axis a further selection rule based on the angular momentum theory of the shell is retained. Indeed in such complexes two -orbitals will remain degenerate. This indicates that the intra-t2g part of the LF hamiltonian has pseudo-cylindrical D h symmetry. As a result the 2S+1L terms are resolved into pseudo-cylindrical 2S+1 A levels (/l = 0,1. L ). It is convenient to orient the z axis of quantization along the principal axis of revolution. In this way each A level comprises the ML = A components of the L manifold. In a pseudo-cylindrical field only levels with equal A are allowed to interact, in accordance with the pseudo-cylindrical selection rule  

In reality the cylindrical LF has to be added to the octahedral interaction which lifts the degeneracy of the 2D term as described in Sect. 3.1. When combining these two perturbations it matters whether the cylindrical field coincides with the trigonal or with the tetragonal directions of the octahedron, because the remaining D3i or DAh subgroups are not equivalent. Two different spectral patterns thus will result from this superposition of octahedral and cylindrical ligand fields. 

---