Fig. 1: Illustration of density-based machine learning for water conformer energies.

For all panels, DFT energies (orange) are shown alongside CC energies (blue) for the same molecular conformers, with optimized geometries indicated by open diamonds. a The nuclear potential, represented by an approximate Gaussians potential, is the input to a set of ML models that return the electron density⁵³. This learned density is the input for independent ML predictions of molecular energies based on DFT or CC electronic structure calculations, or the difference between these energies, in order to correct the DFT energy (final term in Eq. (3)). b Calculated energies for CC (dark blue) and DFT (dark orange) for 102 sample geometries relative to the lowest training energy (top), along with the relative energy errors for DFT compared to CC for each conformer (bottom). Note that the DFT energy errors are not a simple function of the energy relative to the minimum energy geometry (see Supplementary Fig. 2), as short O–H bond lengths tend to be too high in energy and stretched bonds are overstabilized. c Average out-of-sample prediction errors for the different ML functionals compared to the reference E^CC energies. The MAE of the E^DFT energies w.r.t. E^CC is also shown as a dashed line. d The energy surface (in kcal mol⁻¹) of symmetric water geometries for \({E}_{{\mathrm{ML}}}^{{\mathrm{DFT}}}\) (orange) and \({E}_{\Delta \,\text{-}\,{\mathrm{DFT}}}^{{\mathrm{CC}}}\) (blue) after applying the Δ-DFT  correction (bottom). For this figure, DFT calculations use the PBE functional, and CC calculations use CCSD(T) (see “Methods” for more details).