Fourier transform
A Fourier
transform to momentum space of the radial part
$$
V_{\alpha}\left(r\right)
$$
is rather simple since
the radial depends only on the magnitude of the relative distance and
thereby the relative momentum
$$
\mathbf{q}=\frac{1}{2}\left(\mathbf{k}_{p}-\mathbf{k}_{q}-\mathbf{k}_{r}+\mathbf{k}_{s}\right)
$$
Omitting spin and isospin dependencies, the momentum space version of the
interaction reads
$$
\begin{equation}
\langle \mathbf{k}_p \mathbf{k}_q \vert V_{\alpha}\vert\mathbf{k}_r\mathbf{k}_s\rangle=
\frac{V_{\alpha}}{L^{3}}\left(\frac{\pi}{\alpha}\right)^{3/2}\exp{(\frac{-q^{2}}{4\alpha})}\delta_{\mathbf{k}_{p}+\mathbf{k}_{q},\mathbf{k}_{r}+\mathbf{k}_{s}}
\tag{8}
\end{equation}
$$