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

## TL;DR

This paper analyzes the limitations of the Perdew-Zunger self-interaction correction, revealing significant errors for uniform densities and large-Z atoms, and suggests the need for a more accurate generalized SIC.

## Contribution

The study provides analytical and extrapolated estimates of PZ SIC errors for large atoms, highlighting its inaccuracies for uniform densities and proposing directions for improved SIC methods.

## Key findings

- PZ SIC introduces large errors in uniform gas correlation energies.
- Estimated errors for large atoms are +5.5% (LSDA-SIC) and -3.5% (PBE/SCAN-SIC).
- Errors may explain PZ SIC's poor performance in equilibrium properties.

## Abstract

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.