I am trying to calculate the electric quadrupole transition matrix elements for an alkali for transitions between hyperfine Zeeman levels. The electric quadrupole Hamiltonian consists of terms of the form $r^2 Y_{2,q}$.
I decomposed the hyperfine Zeeman states in terms of the decoupled nuclear angular momentum and total electronic angular momentum basis, and further in terms of the orbital angular momentum ($l, m$) and electronic spin basis using Clebsch-Gordan coefficients. This allowed me to evaluate the angular part of the quadrupole matrix element using the known integral, $$ \begin{equation} \sqrt{\frac{4\pi}{3}} \int \sin(\theta) \; d\theta \; d\phi \; Y_{l', m'} Y_{2,q} Y_{l,m} = (-1)^{l'-m'} \sqrt{\max(l,l')} \; W_{3j}(l',2,l,-m',q,m) \end{equation} $$ where $W_{3j}$ is the Wigner-3j function.
The integral that remains to be evaluated is $\int R_{n', l'}^* r^2R_{n,l} \;r^2 \;dr$, which I don't know how to do. Particularly, I am not sure how to write the radial wavefunction for an alkali, and if there is a closed-form solution of the integral.
