Accurate Kohn-Sham auxiliary system from the ground-state density of solids

The Kohn-Sham (KS) system is an auxiliary system whose effective potential is unknown in most cases. It is in principle determined by the ground-state density and it has been found numerically for some low-dimensional systems by inverting the KS equations starting from a given accurate density. For solids, only approximate results are available. In this work, we determine accurate exchange-correlation (xc) potentials for Si and NaCl using the ground-state densities obtained from auxiliary field quantum Monte Carlo calculations. We show that these xc potentials can be rationalized as an ensemble of a few local functions of the density, whose form depends on the specific environment and can be well characterized by the gradient of the density and the local kinetic energy density. The KS band structure can be obtained with high accuracy. The true KS band gap turns out to be larger than the prediction of the local density approximation, but significantly smaller than the measurable photoemission gap, which confirms previous estimates. Finally, our findings show that the conjecture that very different xc potentials can lead to very similar densities and other KS observables is true also in solids, which questions the meaning of details of the potentials and, at the same time, confirms the stability of the KS system.

Figure 1

Density and xc potential of bulk silicon along the same path across the unit cell as in Ref. [70]. The positions of atoms are indicated by dotted vertical lines. The iterative inversion follows Equation (1) with the QMC density $n_{\mathrm{QMC}}$ of silicon as reference density. The potential $v_{\mathrm{xc}}^{\mathrm{QMC},20}$ is obtained after $i=20$ iterations. The density $n^{\mathrm{QMC},20}$ is calculated using $v_{\mathrm{xc}}^{\mathrm{QMC},20}$ in the KS equation. The MaAPE at $i=20$ compared to $n_{\mathrm{QMC}}$ is 0.38% and the MeAPE is 0.04%. Top panel: LPD of $n^{\mathrm{QMC},20}$ (blue), self-consistent LDA $n_{\mathrm{LDA}}$ (orange), and PBE $n_{\mathrm{PBE}}$ (green) densities with respect to $n_{\mathrm{QMC}}$. The gray area is the stochastic error bar of the QMC density. Second panel: the QMC density $n_{\mathrm{QMC}}$ (magenta line). The inset shows the chosen path across the crystal from Ref. [70]. Third panel: $v_{\mathrm{xc}}^{\mathrm{QMC},20}$ (blue), $v_{\mathrm{xc}}^{\mathrm{LDA}}$ (orange), $v_{\mathrm{xc}}^{\mathrm{PBE}}$ (green), and $v_{\mathrm{xc}}^{\mathrm{QMC},10}$ (red). Note that the two QMC potentials (blue and red lines) are almost indistinguishable. The average potentials are aligned. Bottom panel: LPD of xc potentials with respect to $v_{\mathrm{xc}}^{\mathrm{QMC},20}$ for LDA (orange), PBE (green), and QMC at $i=10$ (red).

Figure 2

Density and xc potential of NaCl along the same path across the unit cell as in Ref. [70]. The positions of atoms are indicated by dotted vertical lines. The iterative inversion follows Equation (1) with the QMC density $n_{\mathrm{QMC}}$ of NaCl as reference density. The result of the QMC inversion is shown at $i=200$. The MaAPE on the density at $i=200$ compared to $n_{\mathrm{QMC}}$ is 0.29% and the MeAPE is 0.03%. Top panel: LPD of $n^{\mathrm{QMC},200}$ (blue), self-consistent LDA $n_{\mathrm{LDA}}$ (orange), and PBE $n_{\mathrm{PBE}}$ (green) densities with respect to $n_{\mathrm{QMC}}$. The gray area in the top panel indicates the stochastic error bar of the QMC density. Second panel: the QMC density $n_{\mathrm{QMC}}$ (thin magenta line). The inset shows the chosen path across the crystal from Ref. [70]. Third panel: $v_{\mathrm{xc}}^{\mathrm{QMC},200}$ (blue), $v_{\mathrm{xc}}^{\mathrm{LDA}}$ (orange), and $v_{\mathrm{xc}}^{\mathrm{PBE}}$ (green). The averages of the potentials are aligned. Bottom panel: LPD of xc potentials with respect to $v_{\mathrm{xc}}^{\mathrm{QMC},200}$ for LDA (orange) and PBE (green).

Figure 3

LPD of self-consistent LDA $n_{\mathrm{LDA}}$ (orange), PBE $n_{\mathrm{PBE}}$ (green), and TBmBJ (light blue) densities with respect to $n_{\mathrm{QMC}}$. The gray area is the stochastic error bar of the QMC density. (Top panel) Silicon. (Bottom panel) NaCl.

Figure 4

Map of the xc potential of silicon with respect to the local density at all points in the unit cell. Color codes reflect the modulus of the local gradient of the density (left column) or the local kinetic energy density (right column). Upper figures are for PBE, middle figures for QMC, and bottom figures for TBmBJ. The LDA is shown in green.

Figure 5

Same as Fig. 4, for NaCl. Note the much larger range for $n\left(\mathbf{r}\right)$ here. Upper panels are for PBE, middle panels for QMC, and bottom panels for TBmBJ. The LDA is shown in green. The inset in the middle right panel shows the QMC density along a part of the path across the unit cell. Note that a data point in the main graph (marked by 1) falls into the inset area.

Figure 6

Convergence with the number of iterations of the direct band gap at $\mathrm{\Gamma }$ obtained from the inversion, for silicon (upper panel) and NaCl (lower panel). Shown are the results of LDA (orange), noisy LDA (purple), PBE (green), TBmBJ (light blue), and QMC (blue). The horizontal continuous lines are the reference results, which are obtained directly from the KS calculation except for the noisy LDA where the reference is the result at $i_0=24$ for Si and at $i_0=200$ for NaCl and, for QMC, where the reference is the result at $i_0=20$ for Si and at $i_0=200$ for NaCl. The clean and noisy LDA results are almost entirely overlapping and converge to the reference result of clean LDA, 2.55 eV for Si and 4.59 eV for NaCl. The QMC result converges to 2.72 eV for Si and 5.25 eV for NaCl.

Figure 7

Errors of the iteration procedure as a function of the number of iterations $i$ in silicon. Top panel: MaAPE of the density for the inverted LDA (orange), inverted noisy LDA (purple), and QMC (blue) xc potentials. Each inverted density is compared to its corresponding reference result. In the inset: MeAPE of the density for the inverted LDA, noisy LDA, and QMC xc potentials. Bottom panel: MaAPE of the xc potential for the inverted LDA (orange) and inverted noisy LDA (purple). In these cases the error is defined with respect to $v_{\mathrm{xc}}$ of KS LDA, since the xc potential that yields the noisy density is unknown. In the inset: MeAPE of the xc potential for the inverted LDA and inverted noisy LDA potentials.

Figure 8

Upper panel: the LPD of the density: in orange, LPD of $n^{\mathrm{LDA},500}$ with respect to $n_{\mathrm{LDA}}$ (where the MaAPE is $6.55\times {10}^{-4}\%$); in green LPD of the self-consistent PBE and in blue QMC densities with respect to $n_{\mathrm{LDA}}$. Middle panel: self-consistent LDA density $n_{\mathrm{LDA}}$ (thin magenta line). Lower panel: LPD of $v_{\mathrm{xc}}^{\mathrm{LDA},500}$ (orange) and self-consistent PBE potential $v_{\mathrm{xc}}^{\mathrm{PBE}}$ (green) with respect to $v_{\mathrm{xc}}^{\mathrm{LDA}}$.

Figure 9

Map of the xc potential of silicon with respect to the local density at all points in the unit cell. The results of the KS cycle (circles) are compared to the result of inversion (triangles) for PBE (green) and TBmBJ (light blue).

Figure 10

Map of the xc potential of NaCl with respect to the local density at all points in the unit cell. The results of the KS cycle (circles) are compared to the result of inversion (triangles) for PBE (green) and TBmBJ (light blue).

Figure 11

Relation between noise and errors of the iterative procedures that use $n_{\mathrm{QMC}}$ or $n_{\mathrm{LDA}+\mathrm{noise}}$ as reference densities. Top panel: in yellow, LPD of the LDA density from the KS calculation decorated with a pointwise Gaussian noise $n_{\mathrm{LDA}+\mathrm{noise}}$ with respect to the clean LDA density $n_{\mathrm{LDA}}$. The noise lies within the stochastic error bar of the QMC calculation (gray area). In purple, LPD of $n^{\mathrm{LDA}+\mathrm{noise},39}$ at the iteration $i=39$, where the MaAPE on the density with respect to $n_{\mathrm{LDA}+\mathrm{noise}}$ has its first minimum. Here, the MaAPE is 0.56% and the MeAPE is 0.15%. In red, LPD of $n^{\mathrm{LDA}+\mathrm{noise},150}$ at the iteration $i=150$, where the MaAPE is 0.54%. In blue, LPD of $n^{\mathrm{QMC},500}$ at the iteration $i=500$, where the MaAPE is 0.21% and the MeAPE is 0.02%, with respect to $n_{\mathrm{QMC}}$. Bottom panel: the LDA $v_{\mathrm{xc}}^{\mathrm{LDA}}$ potential (orange) is compared to the xc potentials obtained by inversion of the QMC density $v_{\mathrm{xc}}^{\mathrm{QMC},500}$ (blue), and by inversion of the noisy LDA density $v_{\mathrm{xc}}^{\mathrm{LDA}+\mathrm{noise},24}$ (brown dashed line), $v_{\mathrm{xc}}^{\mathrm{LDA}+\mathrm{noise},39}$ (purple), and $v_{\mathrm{xc}}^{\mathrm{LDA}+\mathrm{noise},150}$ (red) at $i=150$.

Figure 12

Map of the xc potential versus the local density in silicon. Main panel: results of the inversion of QMC for different $i$. Each $v_{\mathrm{xc}}$ is plotted against its own density. QMC inversion results are shown at 10 (smooth potential) and 20 (minimum of MaAPE of the density) iterations. For one density below 0.01, additional results at $i=50$ and $i=100$ are given to illustrate the stability of the extra branch. In the inset, KS LDA is compared to the inversion of noisy LDA at 24 (smooth potential) and 39 (minimum of MaAPE of the density) iterations.

Figure 13

Analog of Fig. 11, but for NaCl. Here spikes in noisy LDA and QMC are less developed than in the case of silicon. Note that the QMC error bar is smaller than in the case of silicon, in particular in regions of significant density.

Figure 14

Map of the xc potential versus the local density in NaCl. The result of the inversion of LDA with Gaussian noise for $i=200$ is compared to the clean LDA xc potential resulting from the KS calculation.

Figure 15

Exchange-correlation potentials obtained by inversion of the QMC density of bulk silicon, at three different iteration steps. All three potentials yield a very similar, accurate density. The MeAPE of the density is 0.04% at $i=20$, 0.02% at $i=500$, and the same value at $i=1000$. For the MaAPE, we get 0.38% at $i=20$, 0.21% at $i=500$, and 0.18% at $i=1000$.

Figure 16

Silicon: error of the inverted QMC density as a function of the iteration number for three different choices of the shift of the starting point. The starting point is $0.1v_{\mathrm{xc}}^{\mathrm{LDA}}+\mathrm{shift}$.

Figure 17

Map of the xc potential of silicon with respect to the local density for three different choices of the starting point. The three xc potentials have the same MeAPE of 0.02%. This precision is reached at $i=11,20$, or 38 when the starting point is a shifted $v_{\mathrm{xc}}^{\mathrm{LDA}}$, $0.3v_{\mathrm{xc}}^{\mathrm{LDA}}$, or $0.1v_{\mathrm{xc}}^{\mathrm{LDA}}$, respectively.