Random-phase approximation excitation energies from approximate equation-of-motion ring coupled-cluster doubles

The ground-state correlation energy calculated in the random-phase approximation (RPA) is known to be identical to that calculated using a subset of terms appearing in coupled-cluster theory with double excitations. In particular, this equivalence requires keeping only those terms that generate time-independent ring diagrams, in the Goldstone sense. Here I show that this equivalence extends to neutral excitation energies, for which those calculated in the RPA are identical to those calculated using an approximation to equation-of-motion coupled-cluster theory with double excitations (EOM-CCD). The equivalence requires three approximations to EOM-CCD: first, the ground-state double-excitation amplitudes are obtained from the ring-CCD equations (the same as for the correlation energy); second, the EOM eigenvalue problem is truncated to the single-excitation (one particle + one hole) subspace; third, the similarity transformation of the Fock operator must be neglected, as it corresponds to a dressing of the single-particle propagator, which is not present in the conventional RPA.