Analyzing Perdew-Zunger self-interaction correction for reaction barrier heights beyond the LDSA: Unraveling the Evolution of Stretched Bond Errors

Incorporating self-interaction corrections (SIC) leads to significant improvements in the prediction of chemical reaction barrier heights using density functional theory methods. We present a thorough analysis of these corrections on an orbital-by-orbital basis for three semi-local density functional approximations positioned at the lowest rung of Jacob's Ladder of approximations. We conducted detailed Fermi-Löwdin Orbital SIC (FLOSIC) calculations at various steps along the reaction pathway, spanning from the reactants (R) to the transition state (TS) to the products (P), focusing on four representative reactions selected from the BH76 benchmark set. Across all three functionals, we observed that the primary contribution to SIC of the barrier heights originates from stretched bond orbitals that emerge near the TS configuration. We introduce the XC/H ratio, representing the ratio of the self-exchange-correlation energy to the self-Hartree energy, as an indicator of one-electron self-interaction error. A value of XC/H = 1.0 indicates that an orbital's self-exchange-correlation energy exactly cancels its self-Hartree energy, rendering the orbital neutral to the SIC in the FLOSIC scheme. Our analysis reveals that XC/H varies within a range of values for the practical DFAs studied here, with higher values typically associated with stretched or strongly lobed orbitals. We demonstrate that significant disparities in XC/H for corresponding orbitals across the R, TS, and P configurations can be leveraged to identify the principal contributors to the SIC of barrier heights and reaction energies. Based on these comparisons, we suggest that barrier height predictions achieved using the SCAN meta-generalized gradient approximation may have attained the best accuracy achievable for a semi-local functional employing the Perdew-Zunger SIC approach.