How accurate are the parametrized correlation energies of the uniform electron gas?

Abstract

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 $\ensuremath{\zeta}$, requiring a parametrization of the correlation energy per electron ${\ensuremath{\epsilon}}_{c}({r}_{s},\ensuremath{\zeta})$. 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-$({r}_{s}\ensuremath{\rightarrow}0)$ and low- $({r}_{s}\ensuremath{\rightarrow}\ensuremath{\infty})$ density limits. For the physically important range $0.5\ensuremath{\le}{r}_{s}\ensuremath{\le}20$, 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 $\ensuremath{\zeta}=0$), is also reasonably close, both in its original version and with corrections for $\ensuremath{\zeta}\ensuremath{\ne}0$. 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.