Perdew-Zunger self-interaction correction: How wrong for uniform densities and large-<i>Z</i> atoms?

Semilocal density functionals for the exchange-correlation energy of a many-electron system cannot be exact for all one-electron densities. In 1981, Perdew and Zunger (PZ) subtracted the fully nonlocal self-interaction error orbital-by-orbital, making the corrected functional exact for all collections of separated one-electron densities and making no correction to the exact functional. Although the PZ self-interaction correction (SIC) eliminates many errors of semilocal functionals, it is often worse for equilibrium properties of sp-bonded molecules and solids. Nonempirical semilocal functionals are usually designed to be exact for electron gases of uniform density and, thus, also make 0% error for neutral atoms in the limit of large atomic number Z, but PZ SIC is not so designed. For localized SIC orbitals, we show analytically that the local spin density approximation (LSDA)-SIC correlation energy per electron of the uniform gas in the high-density limit makes an error of -50% in the spin-unpolarized case and -100% in the fully spin-polarized case. Then we extrapolate from the Ne, Ar, Kr, and Xe atoms to estimate the relative errors of the PZ SIC exchange-correlation energies (with localized SIC orbitals) in the limit of large atomic number: about +5.5% for the LSDA-SIC and about -3.5% for nonempirical generalized gradient [Perdew-Burke-Ernzerhof (PBE)-SIC] and meta-generalized gradient strongly constrained and appropriately normed (SCAN)-SIC approximations. The SIC errors are considerably larger than those that have been estimated for LSDA-SIC by approximating the localized SIC orbitals for the uniform gas and may explain the errors of PZ SIC for equilibrium properties, opening the door to a generalized SIC that is more widely accurate.