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

Akilan Ramasamy, Lin Hou, Jorge Vega Bazantes, Tom J. P. Irons, Andrew M. Wibowo-Teale, Jianwei Sun

Abstract

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.

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

Abstract

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 . 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 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.

Paper Structure

This paper contains 8 equations, 4 figures, 1 table.

Figures (4)

  • Figure 1: $\mathrm{H}_2^+$ binding energy curves of HF and exchange-only density functionals evaluated on HF orbitals.
  • Figure 2: System- and spherically-averaged exact exchange holes versus the inter-electron distance $u$ for H, the Gaussian electron density, and $\mathrm{H}_2^+$ at various bond lengths $R$.
  • Figure 3: (a) Contour Plot of the RS enhancement factor $F_\mathrm{x}^\mathrm{RS}$ versus s and q. (b) $F_\mathrm{x}^\mathrm{RS}$ as a function of ($q-q_0(s)$) for various values of s.
  • Figure 4: (a) Contour plot of the difference in the RS and SCAN-i enhancement factors against $s$ and $q$, overlaid by $(s, q)$ points existing in $\mathrm{H}_2^+$ at the equilibrium bond length ($R=1.058$Å). The inset zooms in the range of $1.4 < s < 2.2$. (b) Same as (a) but for $\mathrm{H}_2^+$ with $R=4.76$Å.