Above we argued that a similarity transformation preserves all eigenvalues. Nevertheless, the CCD correlation energy is not the exact correlation energy. Explain!
The CCD approximation does not make \( \vert\Phi_0\rangle \) an exact eigenstate of \( \overline{H_N} \); it is only an eigenstate when the similarity-transformed Hamiltonian is truncated to at most 2p-2h states. The full \( \overline{H_N} \), with \( T=T_2 \), would involve six-body terms (do you understand this?), and this full Hamiltonian would reproduce the exact correlation energy. Thus CCD is a similarity transformation plus a truncation, which decouples the ground state only from 2p-2h states.