  The validity of the few-level approximation in dipole-dipole interacting
collective systems is discussed. As example system, we study the archetype case
of two dipole-dipole interacting atoms, each modelled by two complete sets of
angular momentum multiplets. We establish the breakdown of the few-level
approximation by first proving the intuitive result that the dipole-dipole
induced energy shifts between collective two-atom states depend on the length
of the vector connecting the atoms, but not on its orientation, if complete and
degenerate multiplets are considered. A careful analysis of our findings
reveals that the simplification of the atomic level scheme by artificially
omitting Zeeman sublevels in a few-level approximation generally leads to
incorrect predictions. We find that this breakdown can be traced back to the
dipole-dipole coupling of transitions with orthogonal dipole moments. Our
interpretation enables us to identify special geometries in which partial
few-level approximations to two- or three-level systems are valid.
