Local Exchange-Correlation Potentials by Density Inversion in Solids

TL;DR

This study addresses the mismatch between Kohn–Sham band gaps and experimental gaps by density-inverting target densities from a range of DFAs to obtain local exchange-correlation potentials $v_\mathrm{LXC}(\mathbf{r})$. It compares the resulting LXC-band structures with the original GKS band structures to quantify the non-locality across HF, hybrid functionals, LDA+$U$, and meta-GGA densities, and examines the role of the XC derivative discontinuity $\Delta_\mathrm{xc}$ in solids. The main finding is that the LFX potential obtained from HF densities yields band gaps in good agreement with experiment for many materials, while LDA/GGA densities require non-locality (hybrids, HF) to improve gaps; LDA+$U$ inversion mitigates over-localisation, and meta-GGA densities show weak non-locality. Overall, the density-inversion LXC framework provides a diagnostic link between local potentials and non-local XC effects, enabling benchmarking of DFAs and offering insight into Mott physics in TMOs.

Abstract

Following Hollins et al. [J. Phys.: Condens. Matter 29, 04LT01 (2017)], we invert the electronic ground state densities for various semiconducting and insulating solids calculated using several density functional approximations within the generalised Kohn-Sham (GKS) scheme, Hartree-Fock (HF) theory and the LDA+$U$ method, and benchmark against standard (semi-)local functionals. The band structures from the resulting local exchange-correlation (LXC) Kohn-Sham potential for these densities are then compared with the band structures of the original GKS method. We find the LXC potential obtained from the HF density systematically predicts band gaps in good agreement with experiment, even in strongly correlated transition metal monoxides (TMOs). Furthermore, we find that the HSE06 and PBE0 hybrid functionals yield similar densities and LXC potentials, and in weakly correlated systems, these potentials are similar to PBE. For LDA+$U$ densities, the LXC potential effectively reverses the flattening of bands caused by over-localisation by a large Hubbard-$U$ value, while for meta-GGAs, we find only small differences between the GKS and LXC results demonstrating that the non-locality of meta-GGAs is weak.

Figures (13)

• Figure 1: Steepest descent algorithm to calculate the local exchange-correlation (LXC) potential $v_\mathrm{LXC}(\mathbf{r})$. The total potential $v(\mathbf{r})$ is the sum of $v_\mathrm{LXC}(\mathbf{r})$, Hartree $v_\mathrm{H}(\mathbf{r})$ and external (electron-nuclear) $v_\mathrm{ext}(\mathbf{r})$ potentials such that in the $n$-th iteration $v_\mathrm{LXC}^n(\mathbf{r})=v^n(\mathbf{r}) - v_\mathrm{H}[\rho_v^n](\mathbf{r}) - v_\mathrm{ext}(\mathbf{r})$.
• Figure 2: Convergence of the Coulomb energy difference $U_{\rho_\mathrm{t}}[v]$, (refer to Eq. \ref{['eq:Coulomb_diff']}), for inversion of the Hartree-Fock (HF) density of silicon using steepest descent (dotted red line) and Fletcher-Reeves-basedFletcher_Reeves_CG conjugate gradient (solid blue line) algorithms.
• Figure 3: Computed band structures of GaAs using PBE (solid blue) and LXC-PBE (dotted red), the latter obtained via inversion of the PBE density. The Fermi energy has been set to 0 eV. Note that the two band structures are indistinguishable.
• Figure 4: Computed band structures of Si for (a) HF and (b) LFX obtained by inversion of the HF density. The Fermi energy has been arbitrarily set to 0 eV in both band structures. Occupied bands are in blue, valence band in orange, conduction band in green and unoccupied bands in red.
• Figure 5: Comparison of calculated band gaps with experimental band gaps. Local Fock exchange (LFX) denotes the band gap calculated using the LXC potential obtained from the inversion of the Hartree-Fock (HF) density. The straight line indicates perfect agreement between theory and experiment; points above the line indicate that the band gap is overestimated while points below the line indicate the band gap is underestimated.