Harmonium as a Benchmark for DFA Energies in the Classical Limit

The harmonium model is characterized by a two-electron system confined by a harmonic potential and offers a valuable opportunity to study electron correlation effects with exact solutions. We investigate the use of the two-electron harmonium atom as an effective benchmark for density functional approximations (DFAs) across the full range of effective Planck's constant. Using the exact one-body density from Li and Li(2025) and the special-hbar scaling relation, we compute exact total energies and decompose them into kinetic, external, Hartree, and exchange–correlation contributions. These exact densities serve as inputs for evaluating LSDA, PBE, and r2SCAN functionals within the per-hbar mapping framework, enabling direct comparison of approximate and exact energies without self-consistency errors. We further assess the performance of Perdew–Zunger self-interaction correction (PZ-SIC) in both unbroken and symmetry-broken regimes, demonstrating how broken-symmetry SIC solutions recover the exact strong-correlation limit. Our results provide a density-driven benchmark for DFA accuracy in strongly correlated two-electron systems and highlight the importance of symmetry considerations in self-interaction corrections.