Ensemble Generalization\nof the Perdew–Zunger\nSelf-Interaction Correction: A Way Out of Multiple Minima and Symmetry\nBreaking

The Perdew–Zunger (PZ) self-interaction correction\n(SIC)\nis an established tool to correct unphysical behavior in density functional\napproximations. Yet, the PZ-SIC is well-known to sometimes break molecular\nsymmetries. An example of this is the benzene molecule, for which\nthe PZ-SIC predicts a symmetry-broken electron density and molecular\ngeometry, since the method does not describe the two possible Kekulé\nstructures on an even footing, leading to local minima [Lehtola et al. J. Chem. Theory Comput. 2016, 12, 3195]. The PZ-SIC is often implemented with Fermi–Löwdin\norbitals (FLOs), yielding the FLO-SIC method, which likewise has issues\nwith symmetry breaking and local minima [Trepte et al. J. Chem. Phys. 2021, 155, 224109]. In this work, we propose a generalization of the PZ-SICthe\nensemble PZ-SIC (E-PZ-SIC) methodwhich shares the asymptotic\ncomputational scaling of the PZ-SIC (albeit with an additional prefactor).\nThe E-PZ-SIC is straightforwardly applicable to various molecules,\nmerely requiring one to average the self-interaction correction over\nall possible Kekulé structures, in line with chemical intuition.\nWe showcase the implementation of the E-PZ-SIC with FLOs, as the resulting\nE-FLO-SIC method is easy to realize on top of an existing implementation\nof the FLO-SIC. We show that the E-FLO-SIC indeed eliminates symmetry\nbreaking, reproducing a symmetric electron density and molecular geometry\nfor benzene. The ensemble approach suggested herein could also be\nemployed within approximate or locally scaled variants of the PZ-SIC\nand its FLO-SIC versions.