  The earliest known examples of quantum superposition were found in organic
chemistry -- in molecules that `resonate' among multiple arrangements of
$\pi$-bonds. Small molecules such as benzene resonate among a few bond
arrangements. In contrast, a large system that stretches over a macroscopic
lattice may resonate among an extensive number of configurations. This is
analogous to a quantum spin liquid as described by Anderson's resonating
valence bond (RVB) picture. In this article, we study the intermediate case of
somewhat large molecules, the C$_{20}$ and C$_{60}$ fullerenes. We build a
minimal description in terms of quantum dimer models, allowing for local
resonance processes and repulsion between proximate $\pi$-bonds. This allows us
to characterize ground-states, e.g., with C$_{60}$ forming a superposition of
5828 dimer covers. Despite the large number of contributing dimer covers, the
ground state shows strong dimer-dimer correlations. Going beyond the ground
state, the full spectrum of C$_{60}$ shows many interesting features. Its
Hilbert space is bipartite, leading to spectral reflection symmetry. It has a
large number of protected zero-energy states. In addition, the spectrum
contains many scar-like states, corresponding to localized
dimer-rearrangement-dynamics. Resonance dynamics in QDMs can manifest in the
behaviour of defects, potentially binding defects via an effective attractive
interaction. To test this notion, we introduce pairs of vacancies at all
possible separations. Resonance energy reaches its lowest value when the
vacancies are closest to one another. This suggests confinement of monomers,
albeit within a finite cluster. We discuss qualitative pictures for
understanding bonding in fullerenes and draw connections with results from
quantum chemistry.
