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

Semi-local 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 semi-local functionals, it is often worse for equilibrium properties of sp-bonded molecules and solids. Non-empirical semi-local 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 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 local spin density approximation (LSDA-SIC), and about -3.5% for nonempirical generalized gradient (PBE-SIC) and meta-generalized gradient (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.