Auxiliary-field quantum Monte Carlo calculations with multiple-projector pseudopotentials

We have implemented recently developed multiple-projector pseudopotentials into the plane-wave-based auxiliary-field quantum Monte Carlo (pw-AFQMC) method. Multiple-projector pseudopotentials can yield smaller plane-wave cutoffs while maintaining or improving transferability. This reduces the computational cost of pw-AFQMC, increasing its reach to larger and more complicated systems. We discuss the use of nonlocal pseudopotentials in the separable Kleinman-Bylander form, and the implementation in pw-AFQMC of the multiple-projector optimized norm-conserving pseudopotential ONCVPSP of Hamann. The accuracy of the method is first demonstrated by equation-of-state calculations of the ionic insulator NaCl and more strongly correlated metal Cu. The method is then applied to calibrate the accuracy of density-functional theory (DFT) predictions of the phase stability of recently discovered high temperature and pressure superconducting sulfur hydride systems. We find that DFT results are in good agreement with pw-AFQMC, due to the near cancellation of electron-electron correlation effects between different structures.

Figure 1

NaCl DFT/LDA calculated EOS curves (fits to Murnaghan's equation [23]), comparing all-electron LAPW (green dashed line), PAW (red dot-dashed line), and ONCVPSP (blue solid line). Curves are shifted to have the same minimum energy. The experimental lattice constant is indicated by the black dotted vertical line.

Figure 2

NaCl EOS calculated from pw-AFQMC (filled red circles with statistical error bars) using the same ONCVPSPs as in Fig. 1. For comparison, the DFT EOS in Fig. 1 is reproduced (blue solid line, energy shifted for convenient display). The vertical blue dashed and black dotted lines indicate the DFT and experimental equilibrium lattice constants $a_0$, respectively. The pw-AFQMC calculated $a_0$ is indicated by the vertical red arrow, with a horizontal error bar indicating the uncertainty from a fit of the statistical data to Murnaghan's equation [23].

Figure 3

Cu DFT/LDA EOS comparison of ONCVPSP and LAPW. The EOSs are shifted to have the same minimum energy. The inset shows the energy difference vs lattice size.

Figure 4

Cu EOS calculated by pw-AFQMC using the same ONCVPSP as in Fig. 3. AFQMC results are shown by filled symbols, with statistical error bars indicated. For comparison, the DFT EOS with ONCVPSP is reproduced from Fig. 3 (blue solid line, energy shifted). The vertical blue dashed and black dotted lines indicate the DFT and experimental equilibrium lattice constants $a_0$, respectively. The pw-AFQMC calculated $a_0$ is indicated by the vertical red arrow, with a horizontal error bar indicating the uncertainty from a fit to Murnaghan's equation [23].

Figure 5

${\mathrm{H}}_3\mathrm{S}$ ($Im\overline{3}m$) EOS calculated with ONCVPSP, single-projector NCPP, and USPP pseudopotentials, plotted as $\mathrm{\Delta }E\left(V\right)=E\left(V\right)-E_{\mathrm{USPP}}$ (minimum energies aligned). OPIUM generated NCPP EOS are shown for a range of $E_{\mathrm{cut}}$. The upper-right inset shows the the actual EOS. Raw AFQMC results with OPIUM 50 and 100 Ry pseudopotentials are plotted in the lower-left inset, using ONCVPSP results as reference.

Figure 6

Electron density distributions in ${\mathrm{H}}_3\mathrm{S}$ (top panel) and ${\mathrm{H}}_2\mathrm{S}$ (bottom panel) as a function of $r_s$, computed from DFT/PBE with ONCVPSP. The main plots show the difference between the two space group structures for each composition, while the insets show the actual distributions of the two structures, with one shown as negative. $\mathrm{\Delta }{r}_s=0.02$ is chosen as the size of histogram bin.

Figure 7

Comparison of pseudopotentials in (a) DFT and (b) AFQMC FC calculations. Two multiple-projectors ONCVPSPs with $E_{\mathrm{cut}}=64$ and $200\mathrm{Ry}$ and one single-projector NCPP with $E_{\mathrm{cut}}=600\mathrm{Ry}$ are adopted. Finite-size correction is not included in the AFQMC results.