We need that the coordinate vector \( {\bf r} \) can be written in terms of spherical components as $$ {\bf r}_\alpha = r\sqrt{\frac{4\pi}{3}} Y_{1\alpha} $$ Using this expression we get $$ \begin{eqnarray*} \left[{\bf r} \otimes {\bf r} \right]^{(2)}_\mu &=& \frac{4\pi}{3}r^2 \sum_{\alpha ,\beta}\langle 1\alpha 1\beta\vert 2\mu \rangle Y_{1\alpha} Y_{1\beta} \end{eqnarray*} $$