We write the similarity-transformed Hamiltonian as
$$ \begin{align} \overline{H_N}=\sum_{pq} \overline{H}^p_q a^\dagger_q a_p + {1\over 4} \sum_{pqrs} \overline{H}^{pq}_{rs} a^\dagger_p a^\dagger_q a_s a_r + \ldots \tag{25} \end{align} $$with
$$ \begin{align} \overline{H}^p_q &\equiv \langle p\vert \overline{H_N}\vert q\rangle , \tag{26}\\ \overline{H}^{pq}_{rs} &\equiv \langle pq\vert \overline{H_N}\vert rs\rangle . \tag{27} \end{align} $$Thus, the CCSD Eqs. (20) for the amplitudes can be written as \( \overline{H}_i^a = 0 \) and \( \overline{H}_{ij}^{ab}=0 \).