Fig. 5: Illustrations and analyses of the four systems taken from ref. ¹² with different charge states and charge transfer.

Atom colors are as follows: H (white), C (gray), O (red), Na (purple), Mg (dark green), Cl (light green), Ag (silver), and Au (gold). a The \({{{\rm{C}}}}_{10}{{{\rm{H}}}}_{2}/{{{\rm{C}}}}_{10}{{{\rm{H}}}}_{3}^{+}\) set. b The \({{{\rm{Ag}}}}_{3}^{+/-}\) set has Ag trimers in positive or negative charge states. c The \({{{\rm{Na}}}}_{8/9}{{{\rm{Cl}}}}_{8}^{+}\) set. d The Au₂-MgO(001) set has a wetting (left) or unwetting (right) Au₂ on the doped (left) or undoped (right) MgO(001) surface. e Potential energies for the Au₂ cluster adsorbed at the MgO(001) substrate for the non-wetting geometry for the Al-doped and undoped cases. SR and LR stand for short-range and long-range models. The equilibrium density functional theory (DFT) bond lengths, DFT energy and the associated minimum energies are denoted in black symbols. The Au-O bond length is the minimum distance between Au and O atoms. f The atomic charges (q) from the underlying DFT data (left), and the predicted atomic charge from Cartesian atomic cluster expansion long-range (CACE-LR, right) for the nonwetting Au₂ cluster adsorbed on the doped MgO(001) substrate. g The change of atomic charges (Δq) due to doping, from the DFT data (left), and the predicted atomic charge from CACE-LR (right). The color bars are in linear scale.