How Semilocal Are Semilocal Density Functional Approximations? -Tackling Self-Interaction Error in One-Electron Systems

Self-interaction error (SIE), arising from the imperfect cancellation of the spurious classical Coulomb interaction between an electron and itself, is a persistent challenge in modern density functional approximations. This issue is illustrated using the prototypical one-electron system $H_2^+$. While significant efforts have been made to eliminate SIE through the development of computationally expensive nonlocal density functionals, it is equally important to explore whether SIE can be mitigated within the framework of more efficient semilocal density functionals. In this study, we present a non-empirical meta-generalized gradient approximation (meta-GGA) that incorporates the Laplacian of the electron density. Our results demonstrate that the meta-GGA significantly reduces SIE, yielding a binding energy curve for $H_2^+$ that matches the exact solution at equilibrium and improves across a broad range of bond lengths over those of the Perdew-Burke-Ernzerhof (PBE) and strongly-constrained and appropriately-normed (SCAN) semilocal density functionals. This advancement paves the way for further development within the realm of semilocal approximations.

💡 Research Summary

The paper addresses the persistent problem of self‑interaction error (SIE) in density‑functional approximations (DFAs), focusing on the prototypical one‑electron system H₂⁺. While exact Hartree–Fock (HF) theory perfectly cancels the spurious Hartree self‑Coulomb term with the exchange term in a one‑electron system, most semilocal DFAs such as PBE and SCAN fail to achieve this cancellation, leading to overly shallow binding‑energy curves for H₂⁺. The authors ask whether SIE can be reduced further without abandoning the computational efficiency of semilocal functionals.

To this end, they develop a non‑empirical meta‑generalized gradient approximation (meta‑GGA) named RS that explicitly incorporates the reduced Laplacian of the electron density, q = ∇²n /