(169u) Perdew-Zunger Self-Interaction Correction for Ionization Energies of Transition Metal Atoms

Density functional theory, commonly used for predicting chemical properties, suffers from self-interaction errors. These errors may result in errors of ionization energies and inaccurate descriptions of various other properties such as reaction barriers, molecular dissociation energies, charge transfer energies, and others. In this work, we employ the Fermi-Löwdin Self-Interaction Correction (FLOSIC) method to implement the Perdew-Zunger self-interaction correction (PZSIC), which is implemented on an orbital-by-orbital basis. Our goal is to assess the performance of PZSIC and locally scaled SIC (LSIC) in predicting the ionization energies of the transition metal atoms from changes of total energy. We compared the performance of uncorrected density functional approximations (DFAs) such as LDA, PBE and r²SCAN against the reference experimental values for ionization energies. We also compared the results of PZSIC-LDA, PZSIC-PBE, PZSIC-r²SCAN and LSIC-LDA with reference experimental values. We found that LDA and PBE overestimated the first three ionization energies of the Cu atom (3d¹⁰4s¹), whereas r²SCAN accurately predicted the ionization energies of Cu. PZSIC-LDA accurately predicted the first and third ionization energies but underpredicted the second ionization energy by approximately 2 eV. PZSIC-PBE and PZSIC-r²SCAN further underestimated the second ionization energy by about 3 eV. To understand these errors in SIC methods, we hypothesized that there is a SIC energy penalty for a full 3d subshell in the PZSIC total energy for transition-metal atoms and ions. This penalty is not observed in the first ionization energy of Cu, because it cancels out when taking the difference between Cu and Cu⁺. We observed this penalty when comparing Cu⁺ and Cu⁺² PZSIC total energies. We found similar energy penalties for atoms with 3d⁵ configurations such as Cr and Mn. Thus, there is an energy penalty when an electron is removed from a 3d⁵ or 3d¹⁰ shell of Cr, Mn, Cu, Zn, Ga, and Ge. On average, LDA, PBE and r²SCAN overestimate the removal of the first 3d electron from a 3d⁵ or a 3d¹⁰ configuration by 1.32, 0.76, and 0.32 eV, respectively. PZSIC-LDA, LSIC-LDA, PZSIC-PBE and PZSIC-r²SCAN underestimate the removal by 1.16, 0.11, 2.96, and 2.43 eV, respectively. LSIC-LDA gave the best performance among all uncorrected DFAs as well as PZSIC-DFA approaches. LSIC works by scaling down the size of the SIC correction, which reduces the energy penalty seen in 3d⁵ or 3d¹⁰ configurations. We suspect that the penalty in the PZSIC total energy is related to the nodal structure of local orbitals. For a filled 3d shell, the hybridization in FLOSIC leaves one or more of the orbitals less hybridized and more 3d-like, thus more noded. Our future work will focus on investigating the source of the SIC energy penalty in more detail and how to correct it within a PZSIC calculation.