  We introduce a generalized Dicke-like model to describe two-level systems
coupled with a single bosonic mode. In addition, the two-level systems mutually
interact via direct dipole-dipole interaction. We apply the model to an
ensemble of dye molecules coupled to a plasmonic excitation in a metallic
nanoparticle and study how the dipole-dipole interaction and configurational
randomness introduced to the system affect the energy spectra. Comparing the
system eigenenergies obtained by our model with the light spectra from a
multiple-scattering simulation, we suggest a way to identify dark modes in our
model. Finally, we perform a parameter sweep in order to determine the scaling
properties of the system and to classify the regions of the parameter space
where the dipole-dipole interactions can have significant effects.
