Fig. 1. Main features of SpookyNet and problems addressed in this work.

a Optimized geometries of Ag${}_3^{+}$/Ag${}_3^{-}$ (left) and singlet/triplet CH₂ (right). Without information about the electronic state (charge/spin), machine learning models are unable to distinguish between the different structures. b Au₂ dimer on a MgO(001) surface doped with Al atoms (Au: yellow, Mg: gray, O: red, Al: pink). The presence of Al atoms in the crystal influences the electronic structure and affects Au₂ binding to the surface, an effect which cannot be adequately described by only local interactions. c Potential energy E_pot (solid black) for O–H bond dissociation in water. The asymptotic behavior of E_pot for very small and very large bond lengths can be well-approximated by analytical short-ranged E_sr (dotted red) and long-ranged E_lr (dotted orange) energy contributions, which follow known physical laws. When they are subtracted from E_pot, the remaining energy (solid blue) covers a smaller range of values and decays to zero quicker, which simplifies the learning problem. d Visualization of a random selection of learned interaction functions for SpookyNet trained on the QM7-X⁷¹ dataset. They are designed to closely resemble atomic orbitals, facilitating SpookyNet’s ability to extract chemical insight from data.