Consider the many-body state \( \Psi_{\lambda} \) expressed as linear combinations of Slater determinants (\( SD \)) of orthonormal single-particle states \( \phi({\bf r}) \): $$ \begin{equation} \Psi_{\lambda} = \sum_i C_{\lambda i} SD_i \tag{2} \end{equation} $$ Using the Slater-Condon rules the matrix elements of any one-body (\( {\cal O}_1 \)) or two-body (\( {\cal O}_2 \)) operator expressed in the determinant space have simple expressions involving one- and two-fermion integrals in our given single-particle basis. The diagonal elements are given by: $$ \begin{eqnarray} \langle SD | {\cal O}_1 | SD \rangle & = & \sum_{i \in SD} \langle \phi_i | {\cal O}_1 | \phi_i \rangle \\ \langle SD | {\cal O}_2 | SD \rangle & = & \frac{1}{2} \sum_{(i,j) \in SD} \langle \phi_i \phi_j | {\cal O}_2 | \phi_i \phi_j \rangle - \nonumber \\ & & \langle \phi_i \phi_j | {\cal O}_2 | \phi_j \phi_i \rangle \nonumber \end{eqnarray} $$