The terms we need to evalute then are just the elements $$ \langle SD_i |a_p^{\dagger} a_q | SD_j \rangle, $$ which can be rewritten in terms of spectroscopic factors by inserting a complete set of Slater determinats as $$ \langle SD_i |a_p^{\dagger} a_q | SD_j \rangle = \sum_{l}\langle SD_i \vert a_p^{\dagger}\vert SD_l\rangle \langle SD_l \vert a_q \vert SD_j \rangle, $$ where \( \langle SD_l\vert a_q(a_p^{\dagger})\vert SD_j\rangle \) are the spectroscopic factors. These can be easily evaluated in \( m \)-scheme. Using the Wigner-Eckart theorem we can transform these to a \( J \)-coupled scheme through so-called reduced matrix elements.