  The study of nanostructures' vibrational properties is at the core of
nanoscience research, they are known to represent a fingerprint of the system
as well as to hint the underlying nature of chemical bonds. In this work we
focus on addressing how does the vibrational density of states (VDOS) of the
carbon fullerene family ($C_n:~n=20\to720$ atoms) evolves from the molecular to
the bulk material (graphene) behavior using density functional theory. We found
that the fullerene's VDOS smoothly converges to the graphene characteristic
shape-line with the only noticeable discrepancy in the frequency range of the
out-of-plane optic (ZO) phonon band in graphene. From a comparison of both
systems we obtain as main results that: 1)The pentagonal faces in the
fullerenes impede the existence of the analog of the high frequency graphene's
ZO phonons, 2)which in the context of phonons this could be interpreted as a
compression (by 43\%) of the ZO phonon band by decreasing its maximum allowed
radial-optic vibration frequency. 3)As a result, the deviation of fullerene's
VDOS relative to graphene should result on important thermodynamical
implications. The obtained insights can be extrapolated to other structures
containing pentagonal rings such as nanostructure or as pentagonal defects in
graphene.
