Gradient-expansion of the inhomogeneous electron-gas revisited

TL;DR

This paper revisits the gradient expansion approximation for the inhomogeneous electron gas, revealing that separating exchange and correlation contributions is scheme-dependent and proposing that only their combined effect is well-defined.

Contribution

The study demonstrates that individual exchange and correlation gradient corrections are scheme-dependent, challenging common practices in constructing GGA functionals.

Findings

Separate exchange and correlation contributions depend on regularization scheme.

The combined $b_{xc}$ coefficient is scheme-independent and unique.

Separating exchange and correlation in GGA functionals is theoretically unjustified.

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.