I know that dipole transition rate is

[itex]\frac{32}{3}\frac{\pi^3 \alpha c}{\lambda^3}\left<\psi_1|\mathbf{d}|\psi_2\right>[/itex]

The matrix elements on the right can be separated into a radial integral and an angular part. The radial part is

[itex]\int R_{nl} R_{n'l'} r^3\,\mathrm{d}r[/itex]

If we pretend that spin doesn't exit, the angular part goes something like

[itex]\iint {Y_l^m}^{*} Y_1^{0,\pm 1} Y_{l'}^{m'} \sin(\theta) \,d\theta\,d\phi[/itex]

I can solve this using Clebsch Gordan coefficients, and it seems to give the right answer. But, I have no idea what to do when I add in spin angular momentum. It seems this integral doesn't make sense if I replace l with with 1/2 integer j.

I tried some various things, and I ended up with something that gave the right answer in some cases but not in others. I'm always off by some multiple of some rational. I know it has to do with degeneracy and angular momentum addition, but I can't figure it out. Any resources?