Ensemble Generalization of the Perdew–Zunger Self-Interaction Correction: A Way Out of Multiple Minima and Symmetry Breaking

Abstract

The Perdew–Zunger (PZ) self-interaction correction (SIC) is an established tool to correct unphysical behavior in density functional approximations. Yet, the PZ-SIC is well-known to sometimes break molecular symmetries. An example of this is the benzene molecule, for which the PZ-SIC predicts a symmetry-broken electron density and molecular geometry, since the method does not describe the two possible Kekulé structures on an even footing, leading to local minima [J. Chem. Theory Comput.2016123195]. The PZ-SIC is often implemented with Fermi–Löwdin orbitals (FLOs), yielding the FLO-SIC method, which likewise has issues with symmetry breaking and local minima [J. Chem. Phys.2021155224109]. In this work, we propose a generalization of the PZ-SICthe ensemble PZ-SIC (E-PZ-SIC) methodwhich shares the asymptotic computational scaling of the PZ-SIC (albeit with an additional prefactor). The E-PZ-SIC is straightforwardly applicable to various molecules, merely requiring one to average the self-interaction correction over all possible Kekulé structures, in line with chemical intuition. We showcase the implementation of the E-PZ-SIC with FLOs, as the resulting E-FLO-SIC method is easy to realize on top of an existing implementation of the FLO-SIC. We show that the E-FLO-SIC indeed eliminates symmetry breaking, reproducing a symmetric electron density and molecular geometry for benzene. The ensemble approach suggested herein could also be employed within approximate or locally scaled variants of the PZ-SIC and its FLO-SIC versions.