I have an integral involving spherical harmonics and a cross product. It reads
$$ \int d^3k d^3Q \phi^{*}_{L'}(k+Q)Y^{*}_{L',M'_{L'}}(\widehat{k+Q})C^{J'M'_{J'}}_{L'S'}\mathbf{S}\cdot \mathbf{(k\times Q)} \phi_{L}(k)Y_{L,M_{L}}(\widehat{k})C^{JM_{J}}_{LS} $$
with
$$ \phi_{L}(k)=e^{-k^2} $$
and
$$ C^{JM_{J}}_{LS}=<sm,\bar{s}\bar{m}|S M_S><S M_S,L M_L|J M_J> $$
are the Clebsch-Gordan coefficients associated with the addition of angular momentum, with sums over magnetic quantum numbers are assumed. Also, $L=L'=1$ but the $M_L$ and $M'_{L'}$ are unspecified. I want to perform the angular integrals first, so that the radial integrals over Gaussians can be easily performed after. My first attempt was to write the triple product as
$$ \mathbf{S}\cdot \mathbf{(k\times Q)} =\epsilon_{ijk}S_{i}k_{j}Q_{k} $$
and then write the cartesian components of $k$ and $Q$ in terms of spherical harmonics via
$$ k_{i}=\sqrt{\frac{4\pi}{3}}k\sum_{\alpha}\epsilon^{i}_{\alpha}(\hat{z})Y_{1 \alpha}(\hat{k}) $$
where the $\epsilon$ are the usual polarization vectors. This gives me something like
$$ \int d^3k d^3Q kQ\phi^{*}_{L'}(k+Q)Y^{*}_{L',M'_{L'}}(\widehat{k+Q})C^{J'M'_{J'}}_{L'S'}\frac{4\pi}{3}\sum_{\alpha,\alpha'}\mathbf{S}\cdot \mathbf{(\epsilon_{\alpha}\times \epsilon_{\alpha'})} Y_{1 \alpha}(\hat{k})Y_{1 \alpha'}(\hat{Q})\phi_{L}(k)Y_{L,M_{L}}(\widehat{k})C^{JM_{J}}_{LS}. $$
From here, I was going to integrate the spherical harmonics of the same variable, like
$$ \int d\Omega_{Q} Y^{*}_{1,a}(\hat{Q})Y_{1,b}(\hat{Q})=\delta_{a,b} $$
but I have spherical harmonics of $k$, $Q$ and $k+Q$, so I am stumped. Any help would be appreciated. Mostly, I was wondering if my approach is completely off.
Thanks.
