Electron localization in noncompact covalent bonds captured by the r²SCAN+V approach

In density functional theory, the SCAN (Strongly Constrained and Appropriately Normed) and r²SCAN (regularized–restored SCAN) functionals significantly improve over GGA (Generalized Gradient Approximation) functionals such as PBE (Perdew–Burke–Ernzerhof) in predicting electronic, magnetic, and structural properties across various materials, including transition-metal compounds. However, there remain puzzling cases where SCAN/r²SCAN underperform, such as in calculating the band structure of graphene, the magnetic moment of Fe, the potential energy curve of the Cr₂ molecule, and the bond length of VO₂. This research identifies a common characteristic among these challenging materials: noncompact covalent bonding through s-s, p-p, or d-d electron hybridization. While SCAN/r²SCAN excel at capturing electron localization at local atomic sites, they struggle to accurately describe electron localization in noncompact covalent bonds, resulting in a biased improvement. To address this issue, we propose the r²SCAN+V approach as a practical modification that improves accuracy across all the tested materials. The parameter V is 4 eV for metallic Fe, but substantially lower for the other cases. Our findings provide valuable insights for the future development of advanced functionals.