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Recently, we introduced a class of molecular representations for kernel-basedregression methods -- the spectrum of approximated Hamiltonian matrices(SPA$^\mathrm{H}$M) -- that takes advantage of lightweight one-electronHamiltonians traditionally used as an SCF initial guess. The originalSPA$^\mathrm{H}$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 demonstratedon datasets featuring a wide variation of charge and spin, for whichtraditional structure-based representations commonly fail.SPA$^\mathrm{H}$M(a,b), as introduced here, expands eigenvalueSPA$^\mathrm{H}$M into local and transferable representations. It relies uponone-electron density matrices to build fingerprints from atomic or bond densityoverlap contributions inspired from preceding state-of-the-art representations.The performance and efficiency of SPA$^\mathrm{H}$M(a,b) is assessed on thepredictions for datasets of prototypical organic molecules (QM7) of differentcharges and azoheteroarene dyes in an excited state. Overall, bothSPA$^\mathrm{H}$M(a) and SPA$^\mathrm{H}$M(b) outperform state-of-the-artrepresentations on difficult prediction tasks such as the atomic properties ofcharged open-shell species and of $\pi$-conjugated systems.

SPA$^\mathrm{H}$M(a,b): encoding the density information from guess Hamiltonian in quantum machine learning representations