[25]. The DMC energy depends much less sensitively than the VMC energy on the value of C [24]. However, the energy difference between DMC results obtained with C = C F DMC and C = C F VMC is significant, while the fixed-node error that results from using C = C F H is very substantial. We find that the DMC-optimized exponents of 64-electron crystals at different densities are given by C F DMC = 0 . 071 r $_{−}$3 / 2 s and that the DMC-optimized exponents for 16-, 64-, and 196-electron crystals at r$_{s}$ = 40 a.u. are very similar. We have therefore used C = C F DMC at all densities and system sizes. For antiferromagnetic crystals we have used C = C AF DMC = 0 . 082 r $_{−}$3 / 2 s . 

Simulations were performed with up to 162, 109, 100, and 121 electrons for the paramagnetic fluid, fully spin- polarized (ferromagnetic) fluid, antiferromagnetic crystal, and ferromagnetic crystal, respectively [24]. We eliminated single-particle finite-size effects from the fluid energies by twist averaging within the canonical ensemble [26]. Every so often during VMC or DMC simulations, an offset to the grid of G vectors was chosen at random in the first Brillouin zone of the simulation cell, the lowest-energy plane-wave orbitals were selected, and a short period of re-equilibration was carried out. The finite-size errors in the energy per electron resulting from the compression of the exchange- correlation hole into the simulation cell and the neglect of long-ranged correlations in the kinetic energy fall off as N $_{−}$5 / 4 [19]. We therefore extrapolated our fluid energies to infinite system size by fitting our data at each density to E$_{N}$ = E ∞ − c/N 5 / $^{4}$, where E$_{N}$ is the energy per electron of the N -electron system and c and E ∞ are fitting parameters [24]. This differs from the form of bias that has been incorrectly assumed in previous studies of the 2D HEG [6, 7, 14]. 

Our DMC results for the different phases are given in Table I. Our Wigner-crystal energy data were fitted to the first five terms in the low-density expansion of the crystal energy ( E = c 1 r $_{−}$1 s + c 3 / 2 r $_{−}$3 / 2 s + c 2 r $_{−}$2 s + c 5 / 2 r $_{−}$5 / 2 s + c 3 r $_{−}$3 s ) [27]. The first term is the Madelung energy of the static lattice, while the second is the quasiharmonic zero-point phonon energy. The corresponding coefficients can be determined analytically: c 1 = − 1 . 106103 and c 3 / 2 = 0 . 814 [28]. The remaining three coefficients were determined by fitting to our QMC data, giving c 2 = 0 . 113743, c 5 / 2 = − 1 . 184994, and c 3 = 3 . 097610 for the ferromagnetic crystal and c 2 = 0 . 2662977, c 5 / 2 = − 2 . 63286, and c 3 = 6 . 246358 for the antiferromagnetic crystal. We fitted our fluid energy data to the parameterization of the correlation energy suggested by Rapisarda and Senatore [Eq. (28) of Ref. 7]. For the paramagnetic fluid we find a 0 = − 0 . 1863052 a.u., a 1 = 6 . 821839, a 2 = 0 . 155226, and a 3 = 3 . 423013, where a 0 , a 1 , a 2 , and a 3 are the free parameters in Rapisarda and Senatore’s fitting form. For our ferromagnetic data we obtain a 0 = − 0 . 2909102 a.u., a 1 = − 0 . 6243836, a 2 = 1 . 656628, and a 3 = 3 . 791685. We find a greater difference between the energies of the paramagnetic and ferromagnetic Fermi 

