We need to determine the unknown cluster amplitudes that enter in CCSD. Let
$$ \begin{align} \vert\Phi_i^a\rangle &= a^\dagger_a a_i \vert \Phi_0\rangle , \tag{18}\\ \vert\Phi_{ij}^{ab}\rangle &= a^\dagger_a a^\dagger_b a_j a_i \vert \Phi_0\rangle \tag{19} \end{align} $$be 1p-1h and 2p-2h excitations of the reference. Computing matrix elements of the Schroedinger Equation (12) yields
$$ \begin{align} \tag{20} \langle \Phi_0\vert \overline{H_N}\vert \Phi_0\rangle &= E_c , \\ \langle \Phi_i^a\vert \overline{H_N}\vert \Phi_0\rangle &= 0 , \tag{21}\\ \langle \Phi_{ij}^{ab}\vert \overline{H_N}\vert \Phi_0\rangle &= 0 . \tag{22} \end{align} $$