The synthesis of metallic nanoparticle assemblies is nowadays well-controlled, such that these systems offer the possibility of controlling light at a sub-wavelength scale, thanks, in parts to surface plasmons. Determining the energy dispersion of plasmons likely to couple to light in these nanostructures is, therefore, a necessary preliminary task on the way to understanding both their photonic properties and their physical nature, in particular the role of the quadrupole contribution. Starting with a general model that takes account of all energy modes, we show that its low-lying energy dispersion, gained numerically, can be compared to that of a minimal model that treats dipoles and quadrupoles on the same footing. The main advantage of the latter relies on the fact that its formulation is tractable, such that a semi-analytical Bogoliubov transformation allows one to access the experimentally relevant energy bands. Based on this semi-analytical derivation, we determine quantitatively the limit of validity of both the dipole-only model and the presently proposed dipole and quadrupole model, compared to a full-plasmon-mode Hamiltonian. The results show that the dispersion relation, which accounts for dipoles and quadrupoles, is sufficient to capture the low-energy physics in most experimental situations. Besides, we show that at small lattice spacing, the contribution of quadrupoles is dominant around the Brillouin zone center.
Masset et al. (Mon,) studied this question.