opt-DDAP: Optimisable density-derived atomic point charges via automatic differentiation

Interatomic potentials which accurately describe long-range electrostatics require atom-centred charges. One such method to determine these atom-centred charges from density functional theory (DFT) calculations is the density-derived atomic point (DDAP) charge method. DDAP fits atom-centred 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 behaviour, 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 optimisation 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 pseudo-inverse solution followed by charge renormalisation, maintaining stability even in the presence of ill-conditioned matrices. We validate the framework on NaCl vacancy supercells and on MoS$_2$, demonstrating faithful reconstruction of both absolute and difference charge densities. The optimised charges are intended to serve as inputs to effective electrostatic models in machine-learning and empirical interatomic potentials that incorporate long-range interactions.