LAMP-QDF: Local Attention Map for Potential in Quantum Deep Field

The Hohenberg-Kohn (HK) theorem establishes a rigorous one-to-one mapping between the ground-state electron density and the external potential. While Quantum Deep Field (QDF) architectures successfully parameterize latent molecular fields using a Linear Combination of Atomic Orbitals (LCAO) framework, they typically approximate the density-to-potential mapping through a pointwise evaluation functional. We propose LAMP-QDF that replaces the pointwise map with a distance-aware graph attention operator. This local representation provides a more mathematically consistent approximation of the HK inverse while preserving the unsupervised density-generation capabilities of the underlying QDF framework. On the QM9 dataset of small organic molecules, LAMP-QDF reduces the atomization energy prediction error by 26.1\% in the interpolation regime ($N_{\text{Atoms}} \le 14$, test MAE of 1.06 kcal/mol) and by 28.5\% in the size-extrapolation regime ($N_{\text{Atoms}} \ge 15$, MAE of 2.14 kcal/mol) when trained on $N_{\text{Atoms}} \le 14$.