Theory of finite size effects for electronic quantum Monte Carlo calculations of liquids and solids

Concentrating on zero temperature quantum Monte Carlo calculations of electronic systems, we give a general description of the theory of finite size extrapolations of energies to the thermodynamic limit based on one- and two-body correlation functions. We introduce effective procedures, such as using the potential and wave function split up into long and short range functions to simplify the method, and we discuss how to treat backflow wave functions. Then we explicitly test the accuracy of our method to correct finite size errors on example hydrogen and helium many-body systems and show that the finite size bias can be drastically reduced for even small systems.

Figure 1

Energy per atom for b.c.c. hydrogen at $r_s=1.31$ using a Slater-Jastrow wave function with plane wave orbitals occupied up to the Fermi surface for various system sizes $N$. We show the uncorrected twist averaged results, $E_{\mathrm{TABC}}$ and $E_{\mathrm{GC}\text{−}\mathrm{TABC}}$, together with the size corrected ones, $E_{\infty }\left(\mathrm{TABC}\right)=E_{\mathrm{TABC}}+\mathrm{\Delta }{T}_{\mathrm{TABC}}+\mathrm{\Delta }{T}_U^{lr}+\mathrm{\Delta }{V}_{lr}$ and $E_{\infty }\left(\mathrm{GC}\text{−}\mathrm{TABC}\right)=E_{\mathrm{GC}\text{−}\mathrm{TABC}}+\mathrm{\Delta }{T}_{\mathrm{GC}\text{−}\mathrm{TABC}}+\mathrm{\Delta }{T}_U^{lr}+\mathrm{\Delta }{V}_{lr}$. In the inset we show the corresponding values of kinetic energy.

Figure 2

Energy per atom for b.c.c. hydrogen at $r_s=1.31$ using a backflow wave function with plane wave orbitals occupied up to the Fermi surface for various system sizes $N$. We show the uncorrected twist averaged results, $E_{\mathrm{TABC}}$ and $E_{\mathrm{GC}\text{−}\mathrm{TABC}}$, together with the size corrected ones, $E_{\infty }\left(\mathrm{TABC}\right)=E_{\mathrm{TABC}}+\mathrm{\Delta }{T}_{\mathrm{TABC}}+\mathrm{\Delta }{T}_U^{lr}+\mathrm{\Delta }{T}_{bf}+\mathrm{\Delta }{V}_{lr}$ and $E_{\infty }\left(\mathrm{GC}\text{−}\mathrm{TABC}\right)=E_{\mathrm{GC}\text{−}\mathrm{TABC}}+\mathrm{\Delta }{T}_{\mathrm{GC}\text{−}\mathrm{TABC}}+\mathrm{\Delta }{T}_{bf}+\mathrm{\Delta }{T}_U^{lr}+\mathrm{\Delta }{V}_{lr}$. In the inset we show the corresponding values of potential energy.

Figure 3

Kinetic energy per atom for b.c.c. hydrogen at $r_s=1.31$ using a backflow wave function with plane wave orbitals occupied up to the Fermi surface for various system sizes $N$. We show the uncorrected twist averaged results, $T_{\mathrm{TABC}}$ and $T_{\mathrm{GC}\text{−}\mathrm{TABC}}$ in the main figure, the size corrected ones are presented in the inset, $T_{\infty }\left(\mathrm{TABC}\right)=T_{\mathrm{TABC}}+\mathrm{\Delta }{T}_{\mathrm{TABC}}+\mathrm{\Delta }{T}_U^{lr}+\mathrm{\Delta }{T}_{bf}$ and $T_{\infty }\left(\mathrm{GC}\text{−}\mathrm{TABC}\right)=T_{\mathrm{GC}\text{−}\mathrm{TABC}}+\mathrm{\Delta }{T}_{\mathrm{GC}\text{−}\mathrm{TABC}}+\mathrm{\Delta }{T}_{bf}+\mathrm{\Delta }{T}_U^{lr}$ with and without backflow corrections.

Figure 4

Energy per atom for b.c.c. hydrogen at $r_s=1.31$ using a Slater-Jastrow wave function with DFT orbitals for various system sizes $N$. We show the uncorrected twist averaged results, $E_N$, together with the size corrected ones, $E_{\infty }=E_N+\mathrm{\Delta }{T}_U^{lr}+\mathrm{\Delta }{V}_{lr}$. In the inset we show the corresponding values of kinetic energy. We also show the linear fits to the uncorrected data and the corresponding extrapolated values in the thermodynamic limit.

Figure 5

Differences of the potential energy per electron $V_N$ in units of $Ha$, relative to the thermodynamic limit, $V_{\infty }$ for the homogeneous electron gas at $r_s=1$ within the Hartree-Fock approximation. The black symbols are GCTABC results of the finite system with $N$ electrons, red symbols correspond to the finite size corrected ones with long and short range corrections, $\mathrm{\Delta }{V}_{lr}$ and $\mathrm{\Delta }{V}_{sr}$, using the Ewald method described in appendix pp4. In the inset we compare the long and short range potential corrections, $\mathrm{\Delta }{V}_{lr}$ (pink symbols), and $\mathrm{\Delta }{V}_{sr}+\mathrm{\Delta }{V}_{lr}$ (red symbols) with the leading order corrections (blue symbols) of Ref. [13]. Lines are guides to the eyes. For all size corrections, the value of the slope of the structure factor at the origin is estimated from its values at the discrete $k$ mesh; imposing the exact value, the energy difference can be further improved and $V_N+\mathrm{\Delta }{V}_{lr}+{\mathrm{\Delta }}_{sr}$ becomes equal to the thermodynamic limit on the scale of the figures.