The first equation states that the coupled-cluster correlation energy is an expectation value of the similarity-transformed Hamiltonian. The second and third equations state that the similarity-transformed Hamiltonian exhibits no 1p-1h and no 2p-2h excitations. These equations have to be solved to find the unknown amplitudes \( t_i^a \) and \( t_{ij}^{ab} \). Then one can use these amplitudes and compute the correlation energy from the first line of Eq. (20).
We note that in the CCSD approximation the reference state is not an exact eigenstates. Rather, it is decoupled from simple states but \( \overline{H} \) still connects this state to 3p-3h, and 4p-4h states etc.