Large-Z Atoms in the Strong-Interaction Limit of DFT: Implications for Gradient Expansions and for the Lieb-Oxford Bound

We numerically study the strong-interaction limit of the exchange-correlation functional for neutral atoms and Bohr atoms as the number of electrons increases. Using a compact representation, we analyze the second-order gradient expansion, comparing it with the one for exchange (weak interaction limit). The two gradient expansions, at strong and weak interaction, turn out to be very similar in magnitude but with opposite signs. We find that the point-charge plus continuum model is surprisingly accurate for the gradient expansion coefficient at strong coupling, while generalized gradient approximations, such as Perdew-Burke-Ernzerhof (PBE) and PBEsol, severely underestimate it. We then use our results to analyze the Lieb-Oxford bound from the point of view of slowly varying densities, clarifying some aspects on the bound at a fixed number of electrons.