Correlation energies of the high-density spin-polarized electron gas to meV accuracy

We present a combination of quantum Monte Carlo methods and a finite-size extrapolation framework with which we calculate the thermodynamic limit of the exact correlation energy of the polarized electron gas at high densities to meV accuracy, $-40.44\left(5\right)$ and $-31.70\left(4\right)$ mHa at $r_{\mathrm{s}}=0.5$ and 1, respectively. The fixed-node error is characterized and found to exceed 1 mHa, and we show that the magnitude of the correlation energy of the polarized electron gas is underestimated by up to 6 meV by the Perdew-Wang parametrization, for which we suggest improvements.

Figure 1

Correlation energy of the polarized UEG as a function of $r_{\mathrm{s}}$. Shown are the CA data [6], the PW92 parametrization [12], and the DPI parametrization [13, 14, 15]. The inset magnifies the region around $r_{\mathrm{s}}=0.5$. The width of the PW92 curve represents its statistical uncertainty.

Figure 2

VMC and DMC energies of the polarized 19-electron gas at $r_{\mathrm{s}}=1$ (at $\mathrm{\Gamma }$) relative to FCIQMC, as a function of the number of determinants in the wave function, both without and with backflow transformations.

Figure 3

Finite-size corrected fixed-node energies of the polarized UEG at $r_{\mathrm{s}}=0.5$ as a function of $N^{-1}$. Our results correspond to the solid circles and the solid line. Data replicating the finite-size treatment of Ceperley and Alder [6] (open diamonds and dotted line) and Spink et al. [34] (open squares) are also plotted, along with an extrapolation of the latter using $N^{-4/3}$ and $N^{-5/3}$ terms.

Figure 4

Fixed-node correlation energies of the polarized UEG at $r_{\mathrm{s}}=0.5$, 1, and 5 relative to the thermodynamic limit as a function of $\xi$. The line represents a combined fit of the data at $r_{\mathrm{s}}=0.5$ and 1 to Eq. (2), with density-dependent $c_0$ and density-independent ${\tilde{c}}_4$, ${\tilde{c}}_5$, and ${\tilde{c}}_6$ coefficients. The $r_{\mathrm{s}}=5$ data demonstrate the breakdown of the approximate $\xi$ dependence at low densities.

Figure 5

Fixed-node error for the polarized UEG at $r_{\mathrm{s}}=0.5$ and 1 as a function of $N^{-1}$. The curves are obtained by simultaneously fitting the data at both densities to Eq. (3) with density-dependent $f_0$ and density-independent ${\tilde{f}}_3$ and ${\tilde{f}}_4$ coefficients. The linewidth represents the statistical uncertainty in the fit. The inset shows the combined fit against $\xi$.

Figure 6

Correlation energy of the polarized UEG at (a) $r_{\mathrm{s}}=1$ and (b) $r_{\mathrm{s}}=0.5$ from our calculations (dotted lines), and values of the PW92 and DPI parametrizations, our unweighted PW92 fit to the CA data (uPW92), and our revised unweighted PW92 fit to the CA data and our present results (rPW92).