Comment on "Is there a density-gradient expansion for the electronic exchange energy?"

Abstract

Perdew et al. [arXiv:2610.07201 (2026)] dispute our finding [Phys. Rev. B 113, 115136 (2026)] that the separate exchange ($b_x$) and correlation ($b_c$) contributions to the $q^2$ coefficient of the long-wavelength proper polarization function depend on the infrared regularization of the Coulomb interaction, although their sum $b_{xc}=b_x+b_c$ does not. We question their claim that the conventional exchange coefficient is uniquely determined. In Qian's [J. Math. Phys. 56, 111901 (2015)] derivation, which Perdew et al. cite in support of their claim of a unique exchange-gradient coefficient, an integral of the form $\int_{-a}^{b}dx/x$ crosses a singularity and does not exist as an ordinary improper integral. Its finite Cauchy principal value (which is a particular way of regularizing the integral) is obtained only by excluding equal intervals on the two sides of the singularity; unequal exclusions give different answers. Independently, Sham's calculation with the screened interaction $e^{-λ_0 r}/r$ gives an exchange-gradient coefficient different from the bare-Coulomb calculations of Svendsen–von Barth [Int. J. Quantum Chem. 56, 351 (1995)] and Engel–Vosko [Phys. Rev. B 42, 4940 (1990)]. The discrepancy remains when $λ_0>0$ is so small that $1/λ_0$ is comparable to the size of the observable universe, making the two interactions experimentally indistinguishable on laboratory scales. Why should the exactly unscreened result be preferred, even if its calculation is accepted as valid? Our calculation shows that the ambiguity of the separate contributions cancels in the combined, regulator-independent coefficient $b_{xc}$.