3j symbol

The problem is not simplified by Eq. (15), since there exists a closed-form expression for the multi-scattering matrix for n spheres in terms of spherical Bessel and Hankel functions, spherical harmonics and 3j-symbols, where l, l and to, m are total angular momentum and z-projection quantum numbers, respectively (Henseler, Wirzba and Guhr, 1997) ... 

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

A general matrix element is first reduced with the 3j-symbol (or alternatively with a Clebsch-Gordan coefficient)... 

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... 

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

It is important here to note that in Eq. (D.14) it is assumed that the given matrix element is employed in the form of (D.8), and in Eq. (D.15) it is employed in the form of (D.ll). Then, applying the explicit form for the 3j-symbols, we obtain directly for zero rank elements... 

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. 

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. 

From the fact that / = 0 and from the triangular conditions in the 3j symbol of Eq. (250) 0 < 1 <2, sothatA = 2. The selection rules on k with respect to 1 = 2 are k = 1 or 3. The q values for k = 1 and 3 are found in fhe odd crysfal field pofential terms (see Gorller-Walrand and Binnemans, 1996 Gorller-Walrand and Birmemans, 1998 Prather, 1961 Wyboume, 1965). For the four symmetries considered here the even and odd terms are... 

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 ... 

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