Wigner 3j-symbol

Sometimes, instead of Clebsch-Gordan coefficients, one uses the Wigner 3j-symbols which possess simpler symmetry properties. These are con-... 

The Wigner 3j-symbol is often defmed as the coefficient coupling a product of three irreducible tensors (of the same variance) to an invariant Invoking this definition, it immediately follows that the function c, Q) is an invariant. 

Weisstein EW (2010) Wigner 3j-Symbol (ftitothWorld A Wolfram Web Resource), http //mathworld.wolffam. com/Wigner3j-SymboLhtml. Accessed 2nd October, 2010... 

Because of the renormalization, the same result is obtained using the Racah V-coefficients, Clebsh-Gordan coefficients or Wigner 3j-symbols. The coupling coefficients have been tabulated in specialized monographs ... 

Here the factor in parentheses is a Wigner 3j symbol[3] In terms of these S functions the energy can be expressed in the form... 

The 3j-symbols occurring in the Wigner-Eckart theorem are expressed analytically, giving rise to the final formulae for the individual components of the... 

If this condition is not fulfilled, the Clebsch-Gordan coefficients vanish, otherwise they have certain numerical values (see Table 7.1). The Clebsch-Gordan coefficients are related to the Wigner coefficients [Wig51], also called 3j symbols jt = a, h = b, j3 = c,m1 = a, m2 = P,m3 = y), defined by... 

The tensorial structure of the spin-orbit operators can be exploited to reduce the number of matrix elements that have to be evaluated explicitly. According to the Wigner-Eckart theorem, it is sufficient to determine a single (nonzero) matrix element for each pair of multiplet wave functions the matrix element for any other pair of multiplet components can then be obtained by multiplying the reduced matrix element with a constant. These vector coupling coefficients, products of 3j symbols and a phase factor, depend solely on the symmetry of the problem, not on the particular molecule. Furthermore, selection rules can be derived from the tensorial structure for example, within an LS coupling scheme, electronic states may interact via spin-orbit coupling only if their spin quantum numbers S and S are equal or differ by 1, i.e., S = S or S = S 1. 

Some of the consequences of the spherical symmetry of the hamiltonian of an atomic system for the Green s function are explored in this section. A number of results concerning tensor operators are invoked and the reader are assumed to be familiar with the Wigner 3j and 6j symbols. 

The Wigner 3j and 6j symbols, their basic properties uid their fundamental relations are discussed in a great many different sources. 

After arranging our notation to accommodate spherical tensors we turn to a powerful tool for the calculation of matrix elements. The Wigner-Eckart theorem allows the factorisation of a matrix element into a part containing the dependence of the magnetic quantum numbers, essentially a 3j symbol, and a part independent of these, called the reduced matrix element and already employed in sect. 2 ... 

Related to the Clebsch-Gordan vector coupling coefficients are Wigner s 3j-symbols... 

The following relations used above can be readily proved by using the properties of 3j Wigner symbols (19),... 

---