  Recently, we introduced a class of molecular representations for kernel-based
regression methods -- the spectrum of approximated Hamiltonian matrices
(SPA$^\mathrm{H}$M) -- that takes advantage of lightweight one-electron
Hamiltonians traditionally used as an SCF initial guess. The original
SPA$^\mathrm{H}$M variant is built from occupied-orbital energies (ie,
eigenvalues) and naturally contains all the information about nuclear charges,
atomic positions, and symmetry requirements. Its advantages were demonstrated
on datasets featuring a wide variation of charge and spin, for which
traditional structure-based representations commonly fail.
SPA$^\mathrm{H}$M(a,b), as introduced here, expand the eigenvalue
SPA$^\mathrm{H}$M into local and transferable representations. They rely upon
one-electron density matrices to build fingerprints from atomic and bond
density overlap contributions inspired from preceding state-of-the-art
representations. The performance and efficiency of SPA$^\mathrm{H}$M(a,b) is
assessed on the predictions for datasets of prototypical organic molecules
(QM7) of different charges and azoheteroarene dyes in an excited state.
Overall, both SPA$^\mathrm{H}$M(a) and SPA$^\mathrm{H}$M(b) outperform
state-of-the-art representations on difficult prediction tasks such as the
atomic properties of charged open-shell species and of $\pi$-conjugated
systems.
