We rewrite $$ |\Psi_0\rangle=C_0|\Phi_0\rangle+\sum_{ai}C_i^a|\Phi_i^a\rangle+\sum_{abij}C_{ij}^{ab}|\Phi_{ij}^{ab}\rangle+\dots, $$ in a more compact form as $$ |\Psi_0\rangle=\sum_{PH}C_H^P\Phi_H^P=\left(\sum_{PH}C_H^P\hat{A}_H^P\right)|\Phi_0\rangle, $$ where \( H \) stands for \( 0,1,\dots,n \) hole states and \( P \) for \( 0,1,\dots,n \) particle states. Our requirement of unit normalization gives $$ \langle \Psi_0 | \Phi_0 \rangle = \sum_{PH}|C_H^P|^2= 1, $$ and the energy can be written as $$ E= \langle \Psi_0 | \hat{H} |\Phi_0 \rangle= \sum_{PP'HH'}C_H^{*P}\langle \Phi_H^P | \hat{H} |\Phi_{H'}^{P'} \rangle C_{H'}^{P'}. $$