  Molecular representations are of fundamental importance for the modeling and
analysis of molecular systems. Representation models and in general approaches
based on topological data analysis (TDA) have demonstrated great success in
various steps of drug design and materials discovery. Here we develop a
mathematically rigorous computational framework for molecular representation
based on the persistent Dirac operator. The properties of the spectrum of the
discrete weighted and unweighted Dirac matrices are systemically discussed and
used to demonstrate the geometric and topological properties of both
non-homology and homology eigenvectors of real molecular structures. This
allows us to asses the influence of weighting schemes on the information
encoded in the Dirac eigenspectrum. A series of physical persistent attributes,
which characterize the spectrum of the Dirac matrices across a filtration, are
proposed and used as efficient molecular fingerprints. Finally, our persistent
Dirac-based model is used for clustering molecular configurations from nine
types of organic-inorganic halide perovskites. We found that our model can
cluster the structures very well, demonstrating the representation and
featurization power of the current approach.
