Bipolar spherical harmonics

Starting with the parametrization for two-electron emission given in equ. (4.68), and using equ. (4.67) which relates the bipolar spherical harmonics to spherical harmonics attached to the actual directions ka = a, electron emission, the triple differential cross section for two-electron emission can be rewritten as... 

Here, v, unlike n, Z m, is not a quantum number but a dummy non-negative integer. We will now derive a two-range ADT for the two types of orbitals defined in Eq. 4.5 by further breaking them up into convenient components. Two-range ADTs are more naturally expressed in terms of bipolar spherical harmonics, X, defined as [3] ... 

Because of the conservation of the total angular momentum (J = j -f-1), we have to search for a proper combination of spherical harmonics. Introducing then the coupling coefficients of quantum mechanical angular momenta (the Clebsch-Gordan coefficients), we obtain a bipolar harmonic... 

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