Two-body matrix elements for particles moving in central potentials are important in many areas of physics. These typically involve the matrix elements of Legendre polynomials, which can be written using the spherical harmonic addition theorem as

\[P_{k}\left(\cos \theta_{12}\right)=\frac{4 \pi}{2 k+1} \sum_{q} Y_{k q}^{*}\left(\theta_{1}, \phi_{1}\right) Y_{k q}\left(\theta_{2}, \phi_{2}\right) \equiv \boldsymbol{C}_{k}^{(1)} \cdot \boldsymbol{C}_{k}^{(2)}\]

Calculate the reduced matrix element $\left\langle l_{1} l_{2} L\left|\boldsymbol{C}{k}(1) \cdot \boldsymbol{C}{k}(2)\right| l_{1}^{\prime} l_{2}^{\prime} L^{\prime}\rightangle$.