Wigner, Eckhart theorem

Spin projection coefficients as given in (4.84) can be obtained in a general and mathematically proper manner by using the Wigner-Eckhart theorem. A detailed description of this topic is found in the textbook by Bencini and Gatteschi [106]. 

The quadrupolar term is more complicated, but after judicious application of the Wigner-Eckhart theorem,37 the electric Hamiltonian, for cylindrical symmetry, may be written as... 

The crystal field potential W cf is developed in Legendre polynomials, and with the help of the Wigner-Eckhart theorem (Edmonds, 1957), can be expressed in terms of operator equivalents O which are functions of angular momentum operators J J+, J- This method of Stevens (1952) is described in detail by Hutchings (1966), whose nomenclature is adopted here. The hamiltonian as a function of angular momentum operators is... 

From symmetry considerations it can be shown that only terms with 6 contribute to the crystal field potential. In the above equation = A"(r") 6 where (r") = expectation value of r" for 4f electrons (Freeman and Watson, 1962). 6n depends upon the reduced matrix elements in the Wigner-Eckhart theorem and upon the electron configuration. For 4f electrons all terms 62 =... 

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