Generalized Clebsch-Gordan coefficients

Clebsch-Gordan coefficients have already occurred several times in our considerations in the Introduction (formula (2)) while generalizing the quasispin concept for complex electronic configurations, while defining a relativistic wave function (formulas (2.15) and (2.16)), in the addition theorem of spherical functions (5.5) and in the definition of tensorial product of two tensors (5.12). Let us discuss briefly their definition and properties. There are a number of algebraic expressions for the Clebsch-Gordan coefficients [9, 11], but here we shall present only one ... 

Many special cases of the Clebsch-Gordan coefficients have very simple algebraic expressions, usually without sums. Thus, a general formula for the Clebsch-Gordan coefficient with all projection parameters equal to zero may be easily found from Eqs. (5.8) and (5.6). If one of the parameters ji = 0, then... 

Let us now put (14.59) into the general formula (13.23) and rearrange the creation and annihilation operators so that when the second-quantization operators are placed side by side the rank projections enter into the same Clebsch-Gordan coefficient. A summation over the projections gives... 

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

This set of operators is in general reducible but a proper linear combination of them can be taken irreducible. It can be used in constructing an irreducible tensor product of order K with (2K + 1) elements indexed as -K < M < K by using Clebsch-Gordan coefficients... 

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