  Persistent homology is a relatively new tool often used for
\emph{qualitative} analysis of intrinsic topological features in images and
data originated from scientific and engineering applications. In this paper, we
report novel \emph{quantitative} predictions of the energy and stability of
fullerene molecules, the very first attempt in employing persistent homology in
this context. The ground-state structures of a series of small fullerene
molecules are first investigated with the standard Vietoris-Rips complex. We
decipher all the barcodes, including both short-lived local bars and long-lived
global bars arising from topological invariants, and associate them with
fullerene structural details. By using accumulated bar lengths, we build
quantitative models to correlate local and global Betti-2 bars respectively
with the heat of formation and total curvature energies of fullerenes. It is
found that the heat of formation energy is related to the local hexagonal
cavities of small fullerenes, while the total curvature energies of fullerene
isomers are associated with their sphericities, which are measured by the
lengths of their long-lived Betti-2 bars. Excellent correlation coefficients
($>0.94$) between persistent homology predictions and those of quantum or
curvature analysis have been observed. A correlation matrix based filtration is
introduced to further verify our findings.
