Ab initio quantum Monte Carlo simulations of the uniform electron gas without fixed nodes: The unpolarized case

In a recent publication [S. Groth et al., Phys. Rev. B 93, 085102 (2016)], we have shown that the combination of two complementary quantum Monte Carlo approaches, namely configuration path integral Monte Carlo [T. Schoof et al., Phys. Rev. Lett. 115, 130402 (2015)] and permutation blocking path integral Monte Carlo [T. Dornheim et al., New J. Phys. 17, 073017 (2015)], allows for the accurate computation of thermodynamic properties of the spin-polarized uniform electron gas over a wide range of temperatures and densities without the fixed-node approximation. In the present work, we extend this concept to the unpolarized case, which requires nontrivial enhancements that we describe in detail. We compare our simulation results with recent restricted path integral Monte Carlo data [E. W. Brown et al., Phys. Rev. Lett. 110, 146405 (2013)] for different energy contributions and pair distribution functions and find, for the exchange correlation energy, overall better agreement than for the spin-polarized case, while the separate kinetic and potential contributions substantially deviate.

Figure 1

Influence of the relative interslice spacing $t_0$ on the convergence—the potential energy from PB-PIMC simulations of $N=4$ unpolarized electrons at $\theta =0.5$ and $r_s=1$ is plotted versus $t_0$ for the fixed choice $a_1=0.33$.

Figure 2

Density dependence of the relative time step error from PB-PIMC with $a_1=0.33$ and $t_0=0.14$—the relative differences between PB-PIMC results with $P=2,3,4,5$ and reference data from CPIMC are plotted versus $r_s$ for the potential energy (a) and the kinetic energy (b), with $\theta =1$.

Figure 3

Convergence of the pair distribution function for $N=4$ unpolarized electrons at $\theta =1$ and $r_s=4$—shown are PB-PIMC results for the inter- $[g_{\uparrow \downarrow }$, panel (a)] and intraspecies $[g_{\uparrow \uparrow }$, panel (b)] distribution function for different numbers of propagators $P$ and the fixed free parameters $a_1=0.33$ and $t_0=0.14$.

Figure 4

Convergence of the pair distribution function for $N=4$ unpolarized electrons at $\theta =1$ and $r_s=4$—shown is the same information as in Fig. 3, but for a different combination of free parameters, i.e., $a_1=0$ and $t_0=0.04$.

Figure 5

Average sign for PB-PIMC simulations of $N=66$ unpolarized electrons at different temperatures—all PB-PIMC data have been obtained for $P=2$ with $a_1=0.33$ and $t_0=0.14$ and the standard PIMC data (red curve) have been taken from Ref. [21].

Figure 6

Typical closed path of $N=4$ unpolarized particles in Slater determinant (Fock) space. The state with four occupied orbitals $|{\mathbf{k}}_0\uparrow \rangle ,|{\mathbf{k}}_1\downarrow \rangle ,|{\mathbf{k}}_3\downarrow \rangle ,|{\mathbf{k}}_6\uparrow \rangle$ undergoes a two-particle excitation $s_1$ at time ${\tau }_1$ replacing the occupied orbitals $|{\mathbf{k}}_0\uparrow \rangle ,|{\mathbf{k}}_3\downarrow \rangle$ by $|{\mathbf{k}}_2\uparrow \rangle ,|{\mathbf{k}}_5\downarrow \rangle$. Two further excitations occur at ${\tau }_2$ and ${\tau }_3$. The states at the “imaginary times” $\tau =0$ and $\tau =\beta$ coincide. In addition, the total spin projection is conserved at any time. All possible paths contribute to the partition function $Z$, Eq. (18).

Figure 7

Average sign (a) and average number of kinks (b) of direct CPIMC, plotted versus the density parameter for three different particle numbers $N=4,14,66$ in $N_B=2109,4169,5575$ plane wave basis functions, respectively, at $\theta =1$. Shown are the results from the simulation of the polarized (circles) and unpolarized (dots) UEG, where for the unpolarized case $2N_B$ spin orbitals have been used.

Figure 8

Convergence of (a) the internal energy, (b) the average sign, and (c) the average number of kinks with respect to the kink potential parameter $\kappa$ of $N=66$ unpolarized electrons at $r_s=2$ and $\theta =4$ in $N_B=88946$ spin orbitals. The potential parameter $\delta$ has been fixed to one. The blue (green) line show a horizontal (linear) fit to the last converged points. The asymptotic value (black point) in the limit $1/\kappa \to 0$ is enclosed between the blue and green lines and, within error bars, coincides with the PB-PIMC result (orange points).

Figure 9

Convergence of (a) the internal energy, (b) the average sign, and (c) the average number of kinks with respect to the kink potential parameter $\kappa$ of $N=66$ unpolarized electrons at $r_s=0.8$ and $\theta =1$ in $N_B=11150$ spin orbitals. The potential parameter $\delta$ has been fixed to one. The three curves correspond to CPIMC calculations where the kink potential has been cut off at different values $V_c$, i.e., $V_{1,\kappa }\left(K\right)$ [cf. Eq. (23)] is set to zero if it takes values smaller than $V_c$. The blue (green) line shows a horizontal (linear) fit to the last converged red points. The asymptotic value (black point) in the limit $1/\kappa \to 0$ is enclosed between the blue and green lines and, within error bars, coincides with the PB-PIMC result (orange points).

Figure 10

Exchange-correlation energy $E_{\mathrm{xc}}$ times $r_s$ of the unpolarized $N=66$ particle UEG over the density parameter $r_s$ for different temperatures. In graphic (a), only the best results from CPIMC (dots) or PB-PIMC (crosses) calculations are shown; cf. Table 1 in the Appendix. In addition, RPIMC results by Brown et al. [21, 51] are plotted for comparison (lines with light colors and open circles). Graphic (b) also shows PB-PIMC data for $r_s<1$ at $\theta =1$.

Figure 11

Kinetic (a) and potential (b) energy of the unpolarized $N=66$ particle UEG over the density parameter $r_s$ for different temperatures. Panel (c) shows the relative difference between our results and RPIMC data by Brown et al. [21, 51].

Figure 12

Kinetic (a) and potential (b) energy of the unpolarized $N=66$ particle UEG over the density parameter $r_s$ for two different temperatures. As a supplement to Fig. 11, we show all available data points from CPIMC and PB-PIMC to illustrate their agreement where both approaches are available. Panel (c) shows the relative difference between the potential energy from PB-PIMC and CPIMC (filled dots) as well as between PB-PIMC and RPIMC (empty dots).

Figure 13

Pair distribution function of $N=66$ unpolarized electrons at $r_s=4$ and $\theta =1$—the PB-PIMC results have been obtained for $t_0=0.04$ and $a_1=0$, and the RPIMC data are taken from Ref. [21].