Meta-generalized gradient approximation made in the Hartree gauge

In density functional theory (DFT), exact constraints, fundamental mathematical properties of the exchange-correlation (XC) energy and its underlying XC hole, along with paradigm systems such as the uniform electron gas and the hydrogen atom have been instrumental in developing exchange- correlation (XC) density functional approximations (DFAs). However, since the spatial XC energy density is not uniquely defined, its exact constraints can only be formulated within a chosen gauge and are therefore seldom utilized in DFA construction. Here, we propose a meta-generalized gradient approximation for the exchange energy, explicitly constructed within the Hartree gauge, using the hydrogen atom’s exchange energy density for gauge alignment in core and asymptotic regions. By formulating DFAs at the XC energy density level, this approach expands reference datasets for machine learning and establishes a foundation for more accurate nonlocal density functionals requiring gauge alignment.

💡 Research Summary

This paper introduces a new meta‑generalized gradient approximation (meta‑GGA) exchange functional, named SORFKL, which is explicitly constructed in the Hartree gauge (HG). In density‑functional theory (DFT) the exchange‑correlation (XC) energy density is not uniquely defined; therefore, exact constraints on the XC energy can only be formulated within a chosen gauge. The authors argue that most existing functionals are built without regard to a specific gauge, which hampers the integration of non‑local corrections (e.g., hybrids, self‑interaction corrections) that require a consistent energy‑density representation.

The Hartree gauge is defined by the relation eₓ(r)=½ n(r)∫ nₓ(r,r′)/|r−r′| dr′, where the exchange hole nₓ(r,r′) is directly linked to the exact exchange energy density. In this gauge the exchange energy density of the uniform electron gas (UEG) and that of the hydrogen atom are known analytically, making them ideal reference systems. The authors therefore construct SORFKL so that it reproduces the exact HG exchange density for both the UEG and the hydrogen atom, especially in the core (near‑nucleus) and asymptotic (far‑tail) regions where the hydrogen‑atom density dominates molecular and solid‑state behavior.

The functional form starts from the hydrogen‑atom enhancement factor F_Hydₓ(s), a function of the reduced density gradient s=|∇n|/(2(3π²)¹⁄³ n⁴⁄³). F_Hydₓ(s) captures the correct asymptotic behavior (∝ s ln s) but becomes unphysical for s<s₀ (negative and divergent). To regularize it, the authors replace s by a parametrized mapping g(s,β), where β is an iso‑orbital indicator β=(τ−τ_vW)/(τ+τ_uni) that distinguishes single‑orbital, slowly varying, and rapidly varying regions (β∈