Quantum Monte Carlo study of the three-dimensional spin-polarized homogeneous electron gas

We have studied the spin-polarized three-dimensional homogeneous electron gas using the diffusion quantum Monte Carlo method, with trial wave functions including backflow and three-body correlations in the Jastrow factor, and we have used twist averaging to reduce finite-size effects. Calculations of the pair-correlation function, including the on-top pair density, as well as the structure factor and the total energy, are reported for systems of 118 electrons in the density range $r_{\mathrm{s}}=0.5$–20 a.u., and for spin polarizations of 0, 0.34, 0.66, and 1. We consider the spin resolution of the pair-correlation function and structure factor, and the energy of spin polarization. We show that a control variate method can reduce the variance when twist averaging, and we have achieved higher accuracy and lower noise than earlier quantum Monte Carlo studies.

Figure 1

Finite-size effects in the spin-resolved PCF. The upper curves are for antiparallel spins and the lower-right curves are for parallel spins. The data are from VMC twist-averaged simulations at $r_{\mathrm{s}}=5$ a.u. and $\zeta =0$, and at the various sizes shown. The region around the first peak in the antiparallel-spin PCF is shown, where the largest finite-size effects occur.

Figure 2

Spin-resolved PCFs for unpolarized HEGs calculated at the densities shown. The antiparallel-spin PCFs are translated upwards by 0.2 units. The data were obtained using twist-averaged DMC simulations.

Figure 3

Variation in the PCF with spin polarization at $r_{\mathrm{s}}=3$ a.u. Antiparallel-spin PCFs are translated upwards by 0.4 units, parallel-majority-spin PCFs are translated upwards by 0.2 units, and the lower curves give the parallel-minority-spin PCFs. Note that the $\zeta =0$ parallel-spin PCF is shown twice (in black) for comparison with both up-up and down-down spin PCFs in those cases where they differ. The DMC data are shown for unpolarized and fully polarized systems, while extrapolated estimates are used for intermediate polarizations. All of the data are twist averaged.

Figure 4

Changes in the PCF due to twist averaging. We plot parallel-spin PCFs with twist averaging minus the same quantity evaluated without twist averaging, for the densities shown and $\zeta =0$. Only DMC results are shown, as the VMC results are very similar.

Figure 5

Spin-resolved PCFs for unpolarized HEGs at $r_{\mathrm{s}}=5$ a.u. We show our raw twist-averaged DMC data, together with the fit to DMC results from Ortiz and Ballone (Ref. 20) (labeled OB) given in their paper, and a fit due to Gori-Giorgi, Sacchetti, and Bachelet (Ref. 23) using the DMC data of Ortiz, Harris, and Ballone (Ref. 22) (labeled GSB). The inset shows the low-$r$ region, where the differences are largest.

Figure 6

The on-top pair density $g\left(0\right)$ (multiplied by $r_{\mathrm{s}}$) as a function of $r_{\mathrm{s}}$ for unpolarized systems. The data shown are those of our averaged DMC and VMC results, together with the fit of Eq. (2) (solid line); Gori-Giorgi and Perdew (Ref. 13); Holzmann et al. (Ref. 24); Yasuhara (Ref. 17); Qian (Ref. 19); Gori-Giorgi et al. (Ref. 23); Ortiz and Ballone (Ref. 20); and Fantoni (Ref. 12). Error bars are shown for our data, those of Holzmann et al. (Ref. 24), and Fantoni (Ref. 12), but they are mostly smaller than the size of the symbols. Our values are slightly smaller than those of Holzmann et al. (Ref. 24).

Figure 7

The on-top pair density (multiplied by $r_{\mathrm{s}}$) as a function of $r_{\mathrm{s}}$ and $\zeta$. Averaged DMC and VMC data are used for the unpolarized systems, and extrapolated estimates for the polarized systems. Twist averaging was used for all data shown. An error bar is plotted for each data point, although some of them are smaller than the size of the symbols. The partially polarized systems show a larger statistical uncertainty. The lines are fits as discussed in the text.

Figure 8

Structure factors at $\zeta =0$ for three densities. The upper curves show $S_{\uparrow \uparrow }-S_{\uparrow \downarrow }$ and are translated upwards by 0.2 units; the lower curves show $S_{\uparrow \uparrow }+S_{\uparrow \downarrow }$. The lines are visual guides only. The statistical uncertainty is smaller than the size of the symbols. The DMC method with twist averaging was used, as the VMC data are again very similar. Intermediate densities lie between the curves shown.

Figure 9

Effect of twist averaging on the antiparallel-spin SSF. We plot SSFs at $\zeta =0$ for three densities, with twist averaging (TA) and without (PBC). Higher densities are towards the top of the graph; intermediate densities lie between the data shown and similar effects are seen in parallel-spin SSFs. Only DMC data are plotted as the VMC data are very similar.

Figure 10

Correlation energy at $\zeta =0$ and 1 against density parameter $r_{\mathrm{s}}$. The present work is simply labeled “$\zeta =0$” and “$\zeta =1$.” Also shown are the Ceperley-Alder QMC data (Ref. 3); the Perdew-Zunger fit to the latter data (Ref. 45); and the recent high-density result of Loos and Gill (Ref. 46). Our data are corrected for finite-size effects. Time-step extrapolation was not performed. The statistical error bars are shown, but are usually smaller than the symbols.

Figure 11

Correlation energies obtained in DMC calculations together with the fit of Eqs. (3), (4), and (5). Dashed lines show the fit; square symbols the QMC data. Lower densities appear towards the top of the graph; higher densities towards the bottom of the figure. From top-to-bottom, therefore, the densities shown are: $r_{\mathrm{s}}=20$, 10, 5, 3, 2, 1, and 0.5 a.u. Error bars on the QMC data are shown, but are smaller than the size of the symbols. Corrections have been applied for finite-size effects. Time-step extrapolation was not performed.

Figure 12

Cubic-spline fits to our twist-averaged extrapolated pair-correlation function data at $r_{\mathrm{s}}=20$ a.u. and $\zeta =0.66$. Ten knots were used in the spline fits. Square symbols represent the raw QMC data points and lines represent our spline fits to those data. In all other figures, we have shown the raw QMC data rather than the spline fits displayed here.