From Densities to Potentials: Benchmarking Local Exchange-Correlation Approximations

TL;DR

The authors benchmark local exchange-correlation approximations by inverting high-quality QMC densities to obtain exact-like KS potentials $v_s(\mathbf{r})$ for insulators and semiconductors, then compare these to DFAs across several metrics. They show that while electron densities from DFAs are similar, the corresponding $v_{xc}(\mathbf{r})$ differ substantially, and the KS gaps from QMC densities generally exceed those of most DFAs except HF, with gaps highly sensitive to semicore-state treatment in pseudopotentials. They introduce an integrated density-potential metric $\mathcal{E}$ and an IAE to quantify errors, and apply Burke-style DE/FE analysis to assess XC energy functionals. The work highlights the critical role of $v_{xc}$ and derivative discontinuities in predicting band gaps and underscores pseudopotential effects as a practical limitation when benchmarking against experiment. Overall, QMC-density inversions provide a valuable standard for evaluating and guiding development of DFAs, especially for gap predictions and core-valence treatment in pseudopotentials.

Abstract

Using the Kohn-Sham (KS) inversion method of Hollins et al. [J. Phys.: Condens. Matter 29, 04LT01 (2017)], we invert densities from variational and diffusion quantum Monte Carlo (QMC) calculations to obtain benchmark QMC-KS potentials for a range of insulators and semiconductors, which we then compare to the KS potentials of popular density functional approximations (DFAs). Our results show that different DFAs yield similar electron densities, despite differences in their KS potentials, which originate primarily from the exchange and correlation contribution. We also find that the KS gap from the QMC density is typically larger than the KS gaps of most DFAs, with the exception of Hartree-Fock. Finally, the KS gap is sensitive to the inclusion of semicore states in the pseudopotentials, such that comparison with experiment should be done with caution.

Figures (13)

• Figure 1: Fractional difference between extrapolated estimates of the DMC charge density in a 111b cell with SJB and SJ trial wave functions ($\rho_\text{SJB}$ and $\rho_\text{SJ}$), and between extrapolated estimates of the DMC charge density in 333b and 111b cells with an SJ trial wave function ($\rho_\text{SJ,333b}$ and $\rho_\text{SJ}$). Results are shown for (a) Si and (b) Ge. $r$ is the distance along a straight line from the origin through the corner of the conventional unit cell, passing through its center. $a_\mathrm{cell}$ is the lattice parameter.
• Figure 2: Difference between the $n$th Fourier component of the density of Si as evaluated by QMC in a supercell containing $N_\text{p}$ primitive cells with the supercell Bloch vector at the Baldereschi MVP ($\rho_{\mathbf{G},n}^\mathrm{QMC}$) and the $n$th Fourier component of the density as evaluated by PBE using a $9\times 9\times 9$ k-point grid centered on the Baldereschi MVP ($\rho_{\mathbf{G},999\text{b}}^\text{PBE}$), plotted against the reciprocal of $N_\text{p}$. Note that $\rho_{\mathbf{G},2}^\mathrm{QMC}$ and $\rho_{\mathbf{G},4}^\mathrm{QMC}$ are symmetry equivalent, as are $\rho_{\mathbf{G},6}^\mathrm{QMC}$ and $\rho_{\mathbf{G},8}^\mathrm{QMC}$. The differences between the symmetry equivalent Fourier coefficients are indicative of the random errors in the QMC results.
• Figure 3: KS band gaps $E_{\mathrm{g},N}$ calculated via inversion of the Si density obtained from various QMC simulation supercells consisting of $N_\text{p}$ primitive cells (equivalently, the number of k-points within the inversion/DFT calculation). The dashed red line shows a linear fit of the gap $E_{\mathrm{g},N}$ as a function of $N_\text{p}^{-1}$, yielding an extrapolated band gap of 0.81 eV at infinite system size.
• Figure 4: Calculated KS band structure for Si using the inverted density from various supercells. The color scheme is as follows: blue 444b supercell, red 333b supercell, green 222b supercell.
• Figure 5: (a) Isosurfaces of the (unsymmetrized) QMC density $\rho_\mathrm{QMC}(\mathbf{r})$ and $v_{xc}^\mathrm{QMC}(\mathbf{r})$ in bulk Si. (b) In the first and third panels, the density of each method $\rho(\mathbf{r})$ and the corresponding $v_{xc}(\mathbf{r})$, respectively, are plotted along the path through the unit cell shown by the coloured arrows in (a) (the colours of the arrows are help distinguish portions of the path). This respective portion of the path is indicated by the coloured circles in (a) and in the x-axis of (b). In the second and fourth panels, we plot the density difference $\Delta\rho_\mathrm{DFA}(\mathbf{r})$ and the XC potential difference $\Delta v_{xc}^\mathrm{DFA}(\mathbf{r})$ from the respective QMC results for various DFAs. The isosurface for the density has been chosen to coincide with regions of high electron density associated with bonding.