When we diagonalize the Hamiltonian matrix, the eigenvectors are the coefficients \( C_{\lambda i} \) used to express the many-body state \( \Psi_{\lambda} \) in terms of a linear combinations of Slater determinants (\( SD \)) of orthonormal single-particle states \( \phi({\bf r}) \).

With these eigenvectors we can compute say the transition likelyhood of a one-body operator as $$ \langle \Psi_{\lambda} \vert {\cal O}_1 \vert \Psi_{\sigma} \rangle = \sum_{ij}C_{\lambda i}^*C_{\sigma j} \langle SD_i | {\cal O}_1 | SD_j \rangle . $$ Writing the one-body operator in second quantization as $$ {\cal O}_1 = \sum_{pq} \langle p \vert {\cal o}_1 \vert q\rangle a_p^{\dagger} a_q, $$ we have $$ \langle \Psi_{\lambda} \vert {\cal O}_1 \vert \Psi_{\sigma} \rangle = \sum_{pq}\langle p \vert {\cal o}_1 \vert q\rangle \sum_{ij}C_{\lambda i}^*C_{\sigma j} \langle SD_i |a_p^{\dagger} a_q | SD_j \rangle . $$