I'm trying to calculate:

$$<\Phi_{n'l'm'}| e^{-ib\cdot r}|\Phi_{nlm}>,$$

where the hydrogen wave functions are $\Phi_{nlm}$ and $\Phi_{n'l'm'}$. If I use the Rayleigh plane-wave expansion:

$$e^{ik\cdot r} = 4 \pi \sum_{l=0}^\infty \sum_{m=-l}^{m=l} i^l j_l(kr) {Y_l^{m}}^*(\hat{k})Y_l^m(\hat{r})$$

I can reduce the integral to something of this form:

$$\int dr^3 j_l(\hat{b} \hat{r}) R_{n'l'}^*(r) R_{nl}(r) Y_{l'm'}(\hat{r}) Y_{lm}(\hat{r})Y_{l''m''}(\hat{r}),$$

where $j_l$ is the spherical Bessel function. I know that integrals of similar type can be solved with the help of the 3-j symbol (http://en.wikipedia.org/wiki/3-j_symbol). But I don't see anyway in which I could bring it to this form. How do I calculate this integral?

In the end this is the overlap matrix for Wannier functions.


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