Multipole splats for optimized and inverted effective potentials
Approximate single-particle descriptions of correlated matter remain the principal source of actionable predictions in quantum science. In modern density functional theory (DFT), hybrid effective Hamiltonians are constructed to reproduce equilibrium densities and forces. However, their nonlocal and orbital-dependent potentials leave systematic errors in response properties and real-time dynamics. Local optimized effective potentials (OEP) or inverted Kohn-Sham (IKS) potentials would remove this limitation, but the numerical fragility in finite orbital bases has long prevented their wide adoption. We introduce multipole splats, a class of trial potentials that carry the correct asymptotic decay by construction. By showing that OEP and IKS map to variational and supervised variants of Hamiltonian learning, we recast both problems as stable nonlinear optimization formulated directly in standard orbital basis sets and applicable to any hybrid functional approximation. We show improvements to three canonical DFT failure modes. Firstly, by comparing approximate and near-exact inverted effective correlation potentials, we isolate the spatial signatures of self-interaction, delocalization, and static correlation errors. Secondly, we recover the Rydberg series without empirical asymptotic corrections, demonstrating access to precise spectral properties. Finally, we compute exact-exchange potentials for larger $\pi$-conjugated systems, establishing scaling capacity. Multipole splats extract new first-principles insights from established approximations and provide capacity for robust dataset generation for downstream processing and learning.