Probing self-interaction error of density functionals in one-electron systems with non-integer nuclear charges

Self-interaction error (SIE) is a long-standing limitation of density functional theory calculations where the exchange-correlation (XC) energy has to be approximated. The SIE originates due to the incapability of the XC functionals to exactly cancel out the Coulomb energy best illustrated in the one-electron systems. SIE of density functionals is well demonstrated in the prototypical H₂⁺, for which popular density functionals provide accurate binding energy around equilibrium but develop substantial errors due to SIE. Here, we will extend the H₂⁺system to allow non-integer nuclear charges [1] with multiple nuclear centers while keeping only one electron and conserving the total charge to be +1. We will study SIE of several popular density functionals in these one-electron systems against the exact solution from the Hartree-Fock method and analyze it at the level of XC holes, with a focus on how SIE evolves with the number of nuclear centers.