Can One Hear the Shape of a Molecule (from its Coulomb Matrix Eigenvalues)?

Coulomb matrix eigenvalues
(CMEs) are global 3D representations
of molecular structure, which have been previously used to predict
atomization energies, prioritize geometry searches, and interpret
rotational spectra. The properties of the CME representation and its
relationship to molecular structure are established using the Gershgorin
circle theorem. Numerical bounds are studied using a data set of 309 000
conformational samples of all constitutional isomers of acyclic alkanes,
CnH2n+2, from
methane (n = 1) to undecane (n =
11), to establish the extent to which the CME preserves chemical intuitions
about isomer and conformer similarity and its ability to distinguish
constitutional isomers. Neither supervised nor unsupervised machine-learning
algorithms can perfectly distinguish constitutional isomers as the
molecular size increases, but the misclassification rate can be kept
below 1%.