For a more quantitative assessment, the 2-mer test set was used to calculate the root-mean-square-deviations (RMSDs) of machine-learned electrostatic potentials compared to reference ab initio quantum calculations at varying isovalues of the density (Fig. ). During this assessment, the averages of the electrostatic potentials were aligned for better comparison. By changing the isovalue, the distance of the potential surface from the molecule changes as well, which is quantified here as the average distance of the potential from the nearest atom. Smaller or larger isovalues of the density result in a potential surface farther away from or closer to the molecule, respectively. For comparison, the RMSDs for the electrostatic portion of the Amber BSC1 classical force field, which consists of parameterized partial atomic charges, are shown alongside the machine-learned potentials. At longer distances from the molecule, the machine learning and classical potentials agree closely. Conversely, at shorter distances the model performs much better as the error in the classical potential increases sharply. This behavior is expected since at short distances, the classical potential enters the region of the electron cloud, which is approximated here using the van der Waals radius of hy- G/C base pairs. Units for the potential are given in a.u. The dark red portions at the ends of the structures are the negatively charged phosphate groups. (c) RMSDs against quantum reference calculations on the 2-mer test set for the machine learning electrostatic potential and the classical BSC1 force field. To get a sense of the range of interaction, the van der Waals radius for hydrogen is plotted at r = 1.2 Å (dotted line), and one half of the distance for DNA base stacking is plotted at r = 1.7 Å (dashed line). (d) Machine-learned electrostatic potentials on the major and minor groove sides of a DNA structure with an A-tract (PDB code: 264d ). The isovalue of the density is set such that the potential is calculated at an average distance of 1.7 Å from the nearest atom. Electrostatic potential plots were made using Plotly. drogen at r = 1.2 Å. Classical force fields do not contain direct information about the electron density distribution and only approximate it with partial atomic charges, so they tend to show deficiencies at short-ranges. Since machine-learned electrostatic potentials are calculated directly from electron densities, they should produce accurate short-and long-range interactions, as reflected by the stable RMSD curve in Fig. . Moving beyond the test set, we demonstrate that the model can produce a machine-learned electrostatic potential for a full size DNA structure. Figure shows a machine-learned electrostatic potential for a DNA structure with an A-tract (PDB code: 264d ). A-tracts have received considerable attention due to their unique structural properties, including an unusually narrow minor groove with an enhanced negative electrostatic potential. Both the shape and the negativity of the potential are thought to enable site-specific binding of certain proteins. Although most computational studies on DNA A-tracts were carried out in solvent environments, the machine-learned electrostatic potential for DNA in a vacuum environment was also able to capture the enhanced negativity in the narrow minor groove (Fig. ).