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 re-

gions. 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.