Orbital Quenching and the Spin-Only Formula

Consider the orbital angular momentum of a free-ion term. Here L = 3 and the orbital degeneracy is 7. Application of Van Vleck s formula (5.8) predicts an effective magnetic moment. 

In octahedral symmetry, the F term splits into Aig + T2g + Tig crystal-field terms. Suppose we take the case for an octahedral nickel(ii) complex. The ground term is 2g. The total degeneracy of this term is 3 from the spin-multiplicity. Since an A term is orbitally (spatially) non-degenerate, we can assign a fictitious Leff value for this of 0 because 2Leff+l = 1. We might employ Van Vleck s formula now in the form 

Strictly, L is defined only as a quantum number for a spherical environment - the free ion. The use of L ff = 0 for A terms or Leff = 1 for L terms on the grounds that (2Leff + 1) equals the degeneracy of these terms is, however, legitimate as used here. There is a close parallel between the quantum mechanics of T terms in octahedral or tetrahedral symmetry on the one hand, and of P terms in spherical symmetry on the other. 

Now take the case for an octahedral vanadium(iii) ion. For d, the ground term is Tig. The spatial degeneracy of a 7 term is three-fold and we describe this with Leff = 1. Using (5.10) we find eff = VlO. So for this Tig term, the crystal field has quenched some, but not all, of the angular momentum of the parent free ion F term. 

Analogous arguments apply to the various ground terms of octahedral or tetrahedral d, d, d andd complexes. 

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