Density functional approximations to the exchange-correlation energy are designed to be exact for an electron gas of uniform density parameter $r_s$ and relative spin polarization $\zeta$, requiring a parametrization of the correlation energy per electron ${\epsilon }_c(r_s,\zeta )$. We consider three widely used parametrizations [J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981) or PZ81, S. H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58, 1200 (1980) or VWN80, and J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 (1992) or PW92] that interpolate the quantum Monte Carlo (QMC) correlation energies of Ceperley-Alder [Phys. Rev. Lett. 45, 566 (1980)], while extrapolating them to known high-$(r_s\to 0)$ and low- $(r_s\to \infty )$ density limits. For the physically important range $0.5\le r_s\le 20$, they agree closely with one another, with differences of 0.01 eV (0.5%) or less between the latter two. The density parameter interpolation (DPI), designed to predict these energies by interpolation between the known high- and low-density limits, with almost no other input (and none for $\zeta =0$), is also reasonably close, both in its original version and with corrections for $\zeta \ne 0$. Moreover, the DPI and PW92 at $r_s=0.5$ are very close to the high-density expansion. The larger discrepancies with the QMC of Spink et al. [Phys. Rev. B 88, 085121 (2013)], of order 0.1 eV (5%) at $r_s=0.5$, are thus surprising, suggesting that the constraint-based PW92 and VWN80 parametrizations are more accurate than the QMC for $r_s<2$. For $r_s>2$, however, the QMC of Spink et al. confirms the dependence upon relative spin polarization predicted by the parametrizations.

• Received 3 January 2018
• Revised 2 May 2018

DOI:https://doi.org/10.1103/PhysRevB.97.195128