Efficient method for the quantum Monte Carlo evaluation of the static density response function of a many-electron system

In a recent letter we introduced Hellmann-Feynman operator sampling in diffusion Monte Carlo calculations. Here we derive, by evaluating the second derivative of the total energy, an efficient method for the calculation of the static density-response function of a many-electron system. Our analysis of the effect of the nodes suggests that correlation is described correctly and we find that nodal issues can be dealt with.

Figure 1

The dashed line shows the density response function ${\chi }_0$ of an infinitely large unpolarized noninteracting homogeneous electron gas (Lindhard function) at $r_s=2$. The black lines close to the Lindhard function show the exact finite-size Lindhard function ${\chi }_0^{\text{fs}}$ with 114 and 4218 electrons, the latter following the Lindhard function closer. The dotted line shows the RPA response function ${\chi }_{\text{RPA}}$, and the three remaining dot-dashed lines show (from top to bottom) the frozen-node Lindhard function ${\chi }_0^{\text{fn}}$ at 114 electrons, the corresponding frozen-node density response of an interacting system ${\chi }^{\text{fn}}$, and the frozen-node RPA ${\chi }_{\text{RPA}}^{\text{fn}}$. The “wiggles” are not noise [see Fig. 2 and the text for details] as they correspond to the shell structure seen in the exact noninteracting finite size ${\chi }_0^{\text{fs}}$.

Figure 2

${\chi }^{\text{fn}}$ (solid line) and ${\chi }_0^{\text{fn}}$ (dashed line) at $r_s=5$ including error bars calculated as described in the text. The inset shows relative errors. Note that these seem to be independent of $k$. The results for $r_s=2$ and 10 are similar.

Figure 3

The density response at $r_s=5$ for an unpolarized 114 particle homogeneous electron gas. The dotted lines show the Lindhard function ${\chi }_0$ (top) and ${\chi }_{\text{RPA}}$ (bottom). The dashed lines are ${\chi }_0^{\text{fn}}$ (top) and ${\chi }_{\text{RPA}}^{\text{fn}}$ (bottom). Finally, the solid lines show the fixed node ${\chi }^{\text{fn}}$ (bottom) and extrapolated interacting $\chi$. The dots correspond to the results in Ref. 2. Our data reaches to smaller $k$ values as our system is larger (114 vs up to 66 electrons).

Figure 4

The ratio $\frac{{\chi }_0}{{\chi }_0^{\text{fn}}}$ at $r_s=5$ for $N_{el}=66$ and $N_{el}=114$ electrons. Except for the omnipresent DMC noise [see Fig. 2] this graph is independent of the value of $r_s$ due to scaling. Note the pronounced shell structure for $k<2k_f$.

Figure 5

$f_{\text{xc}}^{\text{fn}}$ at $r_s=2,5,10$. The value at $k=0.4189k_F$ clearly is an outlier. Interestingly, there is a slight dip in $f_{\text{xc}}^{\text{fn}}$ for $k/k_F<2$ as demonstrated in Ref. 12. The dots correspond to the data in Ref. 2.

Figure 6

As Fig. 5 but showing the local field factor.

Figure 7

RPA (dotted), fn RPA (dashed), fn DMC (solid), and extrapolated (dot-dashed) density response function calculated using a 114 particle homogeneous electron gas at $r_s=2$, 5, and 10. The dots correspond to the results in Ref. 2. Note that for $r_s=2$ the uncorrected DMC data is lower still than standard RPA.

Figure 8

A comparison of our results for $f_{\text{xc}}$ at $r_s=5$ and the finite size corrected $\chi$ with the data in Ref. 2 where comparison is possible, i.e., at $N_{el}=66$ electrons. All our other data uses a larger simulation cell with 114 electrons and more configurations in the DMC run. The solid dots are the four data points for $N_{el}=66$ electrons of Ref. 2 and the solid line corresponds to our data (just under 100 data points in the shown region). The dashed line shows our data for the $N_{el}=114$ electron system. As suggested in Ref. 2 there is no significant difference between $N_{el}=66$ and $N_{el}=114$. Note that for $N=66$ we use fewer walkers than for our main results at $N=114$.