Gradient-expansion of the inhomogeneous electron-gas revisited

TL;DR

This paper reevaluates the gradient expansion for the inhomogeneous electron gas under a weak external potential by employing a regulated Coulomb interaction and a diagrammatic analysis of the static polarization $\Pi^{xc}(q,0)$. It shows that while the individual exchange ($b_x$) and correlation ($b_c$) contributions to the $q^2$ coefficient $b_{xc}$ depend on the regulator, their sum $b_{xc}$ is regulator-independent, implying that separating $b_x$ and $b_c$ is conceptually flawed for constructing a GGA. The authors provide explicit regulator-dependent expressions for $b_x$, $b_c$, and their primed components, and report a robust, regulator-independent value for $b_{xc}^{\lambda\to 0}$ and the corresponding $B_{xc} = 0.029116/r_s^4$, which should govern gradient corrections in GGAs. Consequently, the work challenges common GGA design principles that constrain exchange and correlation pieces separately and advocates using the combined $B_{xc}$ constraint for accurate density-functional approximations. These results have significant implications for the development of future GGAs and the interpretation of existing functionals like PBE/PBEsol.

Abstract

In the present work, we revisit the problem of the inhomogeneous electron gas under the influence of a weak external potential, which allows us to calculate the gradient corrections to the density functional within linear response, an approach known as the gradient expansion approximation. To obtain the exchange ($b_x$) and correlation ($b_{c}$) contributions to the coefficient $b_{xc}$, i.e., to the prefactor of the $q^2$ term of the proper-polarization function, we revisited all the previous calculations and expose misconceptions which led to incorrect conclusions. We used various ways to apply a necessary regularization to the singular Coulomb interaction potential. We found that the separate exchange ($b_x$) and correlation ($b_c$) contributions to the coefficient $b_{xc}$ have regularization-scheme dependent values even though the regulator is set to zero at the end of the calculation. This implies that it is impossible to define such a separation meaningfully. On the contrary, we found that when the regulator is set to zero at the end of the calculation, the combination $b_{xc}$ is regularization-scheme independent and, thus, has a unique value. We conclude that it is incorrect to separate those two terms when constructing a generalized-gradient-approximation (GGA) contribution to the density functional. This appears to be a common approach in most popular GGA functionals, where various constraints are applied to each contribution separately.

Figures (2)

• Figure 1: Diagrammatic contributions to the irreducible polarization function up to first order of an expansion in $\tilde{V}_{\lambda}(p)$ given by Eq. \ref{['yukawa2']}.
• Figure 2: The rest of the diagrams contributing to $\Pi^{xc}(q,0)$ that need to be combined with the diagrams from Fig. \ref{['GA_Fock']}. These diagrams contribute to the same order in $r_s$ in the long-wavelength limit. The fuchsia color represents the Regularized-Coulomb interaction line, while the green color represents the RPA renormalized interaction line. The solid blue (red) lines represent the fermionic non-interacting electron (hole) propagator, while the dashed line represents the insertion due to the weak external potential.