  We consider plasmonic metasurfaces constituted by an arbitrary periodic
arrangement of spherical metallic nanoparticles. Each nanoparticle supports
three degenerate dipolar localized surface plasmon (LSP) resonances. In the
regime where the interparticle distance is much smaller than the optical or
near-infrared wavelength associated with the LSPs, the latter couple through
the dipole-dipole interaction and form collective plasmonic modes which extend
over the whole metasurface. Within a Hamiltonian model which we solve exactly,
we derive general expressions which enable us to extract analytically the
quasistatic plasmonic dispersion for collective modes polarized within the
plane and perpendicular to the plane of the metasurface. Importantly, our
approach allows us not only to consider arbitrary Bravais lattices, but also
non-Bravais two-dimensional metacrystals featuring nontrivial topological
properties, such as, e.g., the honeycomb or Lieb lattices. Additionally, using
an open quantum system approach, we consider perturbatively the coupling of the
collective plasmons to both photonic and particle-hole environments, which
lead, respectively, to radiative and nonradiative frequency shifts and damping
rates, for which we provide closed-form expressions. The radiative frequency
shift, when added to the quasistatic dispersion relation, provides an
approximate analytical description of the fully retarded band structure of the
collective plasmons. While it is tempting to make a direct analogy between the
various systems which we consider and their electronic tight-binding
equivalents, we critically examine how the long-range retarded and anisotropic
nature of the dipole--dipole interaction may quantitatively and qualitatively
modify the underlying band structures and discuss their experimental
observability.
