In the case that the interaction is central, \( V(\mathbf{r}) = V(r) \), then $$ \begin{equation} V_{lm, l'm'}(r) = V(r) \int d{\hat{r}}Y_{lm}^*(\hat{r})Y_{l'm'}(\hat{r}) = V(r) \delta_{l,l'}\delta_{m,m'}, \tag{11} \end{equation} $$ and $$ \begin{equation} V_{lm, l'm'}(k,k') = \frac{2}{\pi} \int_0^\infty drr^2j_l(kr) V(r) j_{l'}(k'r)\delta_{l,l'}\delta_{m,m'} = V_l(k,k') \delta_{l,l'}\delta_{m,m'} \tag{12} \end{equation} $$ where the momentum space representation of the interaction finally reads, $$ \begin{equation} V_{l}(k,k') = {2 \over \pi} \int_0^\infty dr\: r^2 \: j_l(kr) V(r) j_{l}(k'r). \tag{13} \end{equation} $$