opt-DDAP: Optimizable Density-Derived Atomic Point Charges via Automatic Differentiation

Interatomic potentials which accurately describe long-range electrostatics require atom-centered charges. One such method to determine these atom-centered charges from density functional theory (DFT) calculations is the density-derived atomic point (DDAP) charge method. DDAP fits atom-centered Gaussians to the ground-state DFT charge density and preserves the multipole moments that govern long-range electrostatics. While these charges accurately predict long-range behavior, in practice, they are limited by their reliance on fixed, heuristic parameters and a constrained solver that becomes numerically unstable for complex or covalent systems. In this work, we present opt-DDAP, which solves this limitation by reformulating the algorithm as a differentiable computational graph. This reformulation allows for the optimization of Gaussian basis parameters and the reciprocal-space cutoff using automatic differentiation. To ensure numerical robustness through this automatic differentiation process, we replace the conventional Lagrange-multiplier approach with a pseudoinverse solution followed by charge renormalization, maintaining stability even in the presence of ill-conditioned matrices. We validate the framework on NaCl vacancy supercells and on MoS2, demonstrating faithful reconstruction of both absolute and difference charge densities. The optimized charges are intended to serve as inputs to effective electrostatic models in machine learning and empirical interatomic potentials that incorporate long-range interactions.