Clebsch summation

Secondly, due to the smallness of the rotational temperature for the majority of molecules (only hydrogen and some of its derivatives being out of consideration), under temperatures higher than, say, 100 K, we replace further on the corresponding summation over rotational quantum numbers by an integration. We also exploit the asymptotic expansion for the Clebsch-Gordan coefficients and 6j symbol [23] (JJ1J2, L > v,<0... 

Fig. 8.5. Summation of two Clebsch-Gordan coefficients over one projection parameter.

Performing the summation of the products of Clebsch-Gordan coefficients over all projection parameters, we obtain the invariants, called 3n/-coefficients. Their trivial cases were presented in Figs 8.6b-e (Kro-necker and triangular deltas). The first non-trivial case is represented by formula (6.16) or the transformation matrix of three angular momenta, connected with 6/-coefficient (see formula (6.42) and Fig. 8.7)). 

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... 

The summation limits over parameters K and K in (19.14) are defined by the conditions of non-zero values of the Clebsch-Gordan and 67-coefficients. The spin-spin operator H (the second sum in (1.22)) may be... 

The double summation in equation (3.54) therefore reduces to a single summation since m2=m-mj is fixed once m and m are specified. This condition can be incorporated in the Clebsch-Gordan coefficient and the third projection quantum... 

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