More realistic band gaps from meta-generalized gradient approximations: Only in a generalized Kohn-Sham scheme

Unlike the local density approximation (LDA) and the generalized gradient approximation (GGA), calculations with meta-generalized gradient approximations (meta-GGA) are usually done according to the generalized Kohn-Sham (gKS) formalism. The exchange-correlation potential of the gKS equation is nonmultiplicative, which prevents systematic comparison of meta-GGA band structures to those of the LDA and the GGA. We implement the optimized effective potential (OEP) of the meta-GGA for periodic systems, which allows us to carry out meta-GGA calculations in the same KS manner as for the LDA and the GGA. We apply the OEP to several meta-GGAs, including the new SCAN functional [Phys. Rev. Lett. 115, 036402 (2015)]. We find that the KS gaps and KS band structures of meta-GGAs are close to those of GGAs. They are smaller than the more realistic gKS gaps of meta-GGAs, but probably close to the less-realistic gaps in the band structure of the exact KS potential, as can be seen by comparing with the gaps of the EXX+RPA OEP potential. The well-known grid sensitivity of meta-GGAs is much more severe in OEP calculations.

Figure 1

Comparison of $v_{\mathrm{XC}}$'s of the Ne atom for SCAN, LDA, and GGA. The exact $v_{\mathrm{XC}}$ is from Ref. [46].

Figure 2

Comparison of $v_{\mathrm{XC}}$'s of bulk Si along the Si-Si bond for SCAN, LDA, and GGA. The Si atoms are located at $r=0$ and $r=2.35\text{Å}$. The vertical dashed line is a numerical artifact and does not affect the band structure.

Figure 3

The band structure of Si calculated with PBE, gKS SCAN, and KS(KLI) SCAN.

Figure 4

The band structure of GaAs calculated with PBE, gKS SCAN, and KS(KLI) SCAN.

Figure 5

The absolute errors of PBE, KS(KLI) SCAN, and gKS SCAN (comparing with the experimental gap) in the band gap. The EXX+RPA OEP gaps for Si, LiF, and Ar, from Ref. [56], can be considered very close to the gaps for the corresponding exact KS potential.

Figure 6

The KS(ELP) SCAN $v_{\mathrm{XC}}$ (a.u.), the radial component of $\mathbf{\nabla }\partial e_{\mathrm{XC}}/\partial \tau$ (a.u.), and $\alpha$ of the Ne atom evaluated with two different grids. The solid lines are results obtained by using 504 radial grid points, and the dots and dashed lines are results obtained by using 65 radial grid points.

Figure 7

The KS(KLI) SCAN $v_{\mathrm{XC}}$ (a.u.), the $x$ component of $\mathbf{\nabla }\partial e_{\mathrm{XC}}/\partial \tau$ (a.u.), and $\alpha$ of the Ar dimer evaluated with two different grids, plotted along the Ar-Ar axis. The Ar atoms are located at $x=\pm 1.88\text{Å}$, so that comparison with Ref. [63] is possible. Since the system is symmetric, only the $x\in [0,4]$ part is plotted. The solid lines are results obtained by using 806 radial grid points, and the dots and dashed lines are results obtained by using 134 radial grid points.

Figure 8

The KS(KLI) SCAN $v_{\mathrm{XC}}$ (a.u.), the $x$ component of $\mathbf{\nabla }\partial e_{\mathrm{XC}}/\partial \tau$ (a.u.), and $\alpha$ of bulk Si evaluated with two different grids, plotted along the Si-Si bond. The Si atoms are located at $r=0$ and $r=2.35\text{Å}$. The solid lines are results obtained by using 571 radial grid points, and the dots and dashed lines are results obtained by using 134 radial grid points.

Figure 9

The radial component of $\mathbf{\nabla }\partial e_{\mathrm{XC}}/\partial \tau$ of the Ne atom of different functionals. $\alpha$'s of different functionals are similar, so only that of SCAN is plotted.

Figure 10

KS(KLI) SCAN $v_{\mathrm{XC}}$ of bulk Si along the Si-Si bond. The Si atoms are located at $r=0$ and $r=2.35\text{Å}$.