  We derive the explicit expression of the three self-energies that one
encounters in many-body perturbation theory: the well-known $GW$ self-energy,
as well as the particle-particle and electron-hole $T$-matrix self-energies.
Each of these can be easily computed via the eigenvalues and eigenvectors of a
different random-phase approximation (RPA) linear eigenvalue problem that
completely defines their corresponding response function. For illustrative and
comparative purposes, we report the principal ionization potentials of a set of
small molecules computed at each level of theory. The performance of these
schemes on strongly correlated systems (\ce{B2} and \ce{C2}) is also discussed.

Chemical Physics;Materials Science;Strongly Correlated Electrons;Nuclear Theory