Starting with self-consistent fields (SCFs), localized orbitals should facilitate the calculation of the correlation energy in extended, and in particular periodic, systems. This idea is exploited on model ring systems (H4n+2). It is shown that for insulating materials [(H-2)(2n+1), presenting a large gap in the band structure], most of the energy lowering brought by the orders larger than 2 in the canonical many-body perturbation expansion are due to the local-hole-local-particle interaction and that the localized Epstein-Nesbet second-order energies are close to the best correlation-energy estimates. The situation is completely different for small-gap (metalliclike) systems, such as cyclic H4n+2, where the localized second-order approach misses a large fraction of the correlation energy, involving the propagation of the holes and of the particles and implying specific higher-order diagrams.