Spherical harmonic addition theorem ($P_{\ell }(\hat{\mathbf{n}}\cdot \hat{\mathbf{m}})=\frac{4\pi }{2\ell +1}{\sum }_m{Y}_{\ell m}(\hat{\mathbf{n}})Y_{\ell m}^{\ast }(\hat{\mathbf{m}})$)

For orthonormal complex spherical harmonics, $P_{\ell }(\hat{\mathbf{n}}\cdot \hat{\mathbf{m}})=4\pi {\sum }_{m=-\ell }^{\ell }{Y}_{\ell m}(\hat{\mathbf{n}})Y_{\ell m}^{\ast }(\hat{\mathbf{m}})/(2\ell +1)$. This converts an angle between two directions into a separable sum. Together with Orthogonality of Legendre polynomials, it projects rotationally invariant scattering kernels onto angular multipoles.