A general shell-model many-body state $$\vert Psi_{\gamma}\rangle = \sum_{i} C_i \vert \Phi_i\rangle,$$ can be expanded as $$|\Psi_{\gamma}\rangle=C_0|\Phi_0\rangle+\sum_{ai}C_i^a|\Phi_i^a\rangle+\sum_{abij}C_{ij}^{ab}|\Phi_{ij}^{ab}\rangle+\dots.$$