Wavefunctions in LSJ-, intermediate, and .-coupling

If spin-orbit effects are negligible, LSJ-coupling can be applied, and the two wavefunctions for J = 1 and M = 1 are given by 

In the limiting case of strong spin-orbit interactions, /s)-coupled single-particle orbitals have to be used. With the help of equs. (7.42)-(7.44) they can be traced back to the common spin orbitals, giving 

The relation between the wavefunctions (p j and q w of the two extreme coupling cases can be worked out using the representations directly in the Slater determin-antal wavefunctions, equs. (7.95) and (7.96), or alternatively, using the coupling transformation in equ. (7.52). In both cases one gets 

For intermediate coupling, however, the wavefunction has to be derived from a 

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