  The Perdew-Zunger self-interaction correction(PZ-SIC) improves the
performance of density functional approximations(DFAs) for the properties that
involve significant self-interaction error(SIE), as in stretched bond
situations, but overcorrects for equilibrium properties where SIE is
insignificant. This overcorrection is often reduced by LSIC, local scaling of
the PZ-SIC to the local spin density approximation(LSDA). Here we propose a new
scaling factor to use in an LSIC-like approach that satisfies an additional
important constraint: the correct coefficient of atomic number Z in the
asymptotic expansion of the exchange-correlation(xc) energy for atoms. LSIC and
LSIC+ are scaled by functions of the iso-orbital indicator z{\sigma}, which
distinguishes one-electron regions from many-electron regions. LSIC+ applied to
LSDA works better for many equilibrium properties than LSDA-LSIC and the
Perdew, Burke, and Ernzerhof(PBE) generalized gradient approximation(GGA), and
almost as well as the strongly constrained and appropriately normed(SCAN)
meta-GGA. LSDA-LSIC and LSDA-LSIC+, however, both fail to predict interaction
energies involving weaker bonds, in sharp contrast to their earlier successes.
It is found that more than one set of localized SIC orbitals can yield a nearly
degenerate energetic description of the same multiple covalent bond, suggesting
that a consistent chemical interpretation of the localized orbitals requires a
new way to choose their Fermi orbital descriptors. To make a locally
scaled-down SIC to functionals beyond LSDA requires a gauge transformation of
the functional's energy density. The resulting SCAN-sdSIC, evaluated on
SCAN-SIC total and localized orbital densities, leads to an acceptable
description of many equilibrium properties including the dissociation energies
of weak bonds.
