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.
Best Answer
Why don't you change integration variables from $\vec Q$ to $\vec q = \vec Q+\vec k$? Your $\vec S \cdot (\vec k \times \vec Q) = \vec S\cdot (\vec k\times \vec q)$, and your expansion in components will now be with $Y(\hat q)$ and $Y(\hat k)$.