Good-$\mathrm{\Gamma }$ Numbers: A Shell-Optimized Route to the Thermodynamic Limit in Quantum Monte Carlo Calculations of the High-Density Two-Dimensional Electron Gas

Single-particle finite-size errors remain a central obstacle in quantum Monte Carlo (QMC) simulations of periodic systems, conventionally reduced via twist averaging (TA). Here we introduce Good-$\mathrm{\Gamma }$ (GG) particle numbers as system sizes $N$ at which the $\mathrm{\Gamma }$-point Hartree–Fock (HF) energy of the high-density two-dimensional uniform electron gas (2D UEG) lies anomalously close to its thermodynamic-limit (TL) value. All results in this work concern the 2D UEG in its high-density Fermi-liquid regime, $r_s=1$–$4$. We systematically investigate the kinetic- and exchange-energy errors, $\mathrm{\Delta }KE(N)$ and $\mathrm{\Delta }EX(N)$, that underlie this one-body finite-size-error cancellation across four densities and both spin polarizations. We then compare diffusion Monte Carlo (DMC) energies extrapolated to the TL from GG-$N$ sequences against twist-averaged extrapolations at the same densities and polarizations, finding agreement to within a fraction of a millihartree in every case, independent of whether a $1\mathrm{/}N$ or an $N^{-5\mathrm{/}4}$ extrapolation form is used.