Efficient method for grand-canonical twist averaging in quantum Monte Carlo calculations

We introduce a simple but efficient method for grand-canonical twist averaging in quantum Monte Carlo calculations. By evaluating the thermodynamic grand potential instead of the ground-state total energy, we greatly reduce the sampling errors caused by twist-dependent fluctuations in the particle number. We apply this method to the electron gas and to metallic lithium, aluminum, and solid atomic hydrogen. We show that, even when using a small number of twists, grand-canonical twist averaging of the grand potential produces better estimates of ground-state energies than the widely used canonical twist-averaging approach.

Figure 1

System-size dependence of the calculated total energy per electron of an $r_s=1$ uniform electron gas in the Hartree-Fock approximation. Results obtained using canonical twist averaging of the total energy, grand-canonical twist averaging of the total energy, and grand-canonical twist averaging of the grand potential (free energy) are shown. In all cases, a $3\times 3\times 3$ grid of twists centered on the $\mathrm{\Gamma }$ point was used.

Figure 2

System-size dependence of the calculated kinetic energy per electron of an $r_s=1$ uniform electron gas in the Hartree-Fock approximation. Results obtained using canonical twist averaging of the kinetic energy, grand-canonical twist averaging of the kinetic energy, and grand-canonical twist averaging of the kinetic component of the grand potential are shown. In all cases, a $3\times 3\times 3$ grid of twists centered on the $\mathrm{\Gamma }$ point was used.

Figure 3

System-size dependence of the calculated exchange energy per electron of an $r_s=1$ uniform electron gas in the Hartree-Fock approximation. Results obtained using canonical twist averaging of the exchange energy, grand-canonical twist averaging of the exchange energy, and grand-canonical twist averaging of the exchange component of the grand potential are shown. In all cases, a $3\times 3\times 3$ grid of twists centered on the $\mathrm{\Gamma }$ point was used.

Figure 4

Relative fluctuations in the number of electrons in the grand-canonical simulation cell: $\delta N_{\text{e}}=N_{\text{e}}^{\text{C}}-N_{\text{e}}^{\text{GC}}$, where $N_{\text{e}}^{\text{C}}$ is the number of electrons occupying the simulation cell in the canonical ensemble and $N_{\text{e}}^{\text{GC}}$ is the number in the grand-canonical ensemble.

Figure 5

Twist dependence of the total DMC energy per atom for metallic H in the $I{4}_1/amd$ structure, fcc Li, and fcc Al. The red triangles are internal energies $E_c$ calculated using canonical simulations in which the number of electrons in the simulation cell is fixed. The black diamonds are internal energies $E_{gc}^{EM}$ calculated using grand-canonical simulations in which the number $N_e$ of electrons in the simulation cell depends on the twist ${\mathbf{k}}_s$. The blue triangles are energies which are calculated by $E_{gc}^{\mathrm{GPM}}=\mathrm{\Omega }({\mathbf{k}}_s,N_s)+\mu \langle N_e\rangle$ where $\mathrm{\Omega }({\mathbf{k}}_s,N_s)$ is the grand-canonical potential defined as $E_{gc}^{EM}-\mu N_e\left({\mathbf{k}}_s\right)$ and $\langle N_e\rangle$ is the average number of electrons. The statistical errors in all data points are smaller than the symbols.