  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.
