Integrals over products of rotation matrices

The expressions are particularly useftd in the evaluation of integrals over products of rotational matrices, as we shall see. They are widely used in many branches of physics and chemistry from multipole expansions through to statistical mechanical averaging. 

If we recall the definition of the rotation matrix 0, x) from equation (5.45) and 

We can then use this result in the integration of equation (5.96) to give 

The evaluation of this integral by conventional methods (even for the special case m = m2 = m2 = 0 when the rotation matrices reduce to spherical harmonics) is extremely laborious. We shall use this result in the derivation of the Wigner-Eckart theorem and other angular momentum relationships later in this chapter. 

If we recall the definition ofthe rotation matrix 9) , ((/ , 9, x) from equation (5.45) and integrate over the volume element dry = sin0 d d0 dx, we obtain 

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We now make use of a standard result for integrating over the product of three rotational matrices (see chapter 5, equation (5.100)),... 

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