The optimized effective potential of meta-generalized gradient approximations in solids

Unlike the local density approximation(LDA) and the generalized gradient approximation(GGA), calculations with meta-GGAs are usually done according to the generalized Kohn-Sham(gKS) formalism. The exchange-correlation potential of the gKS equation is non-multiplicative, which prevents systematic comparison of meta-GGA bandstructures 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 popular meta-GGAs, including the new SCAN functional[Phys. Rev. Lett. 115, 036402(2015)]. We find that the KS gaps of meta-GGAs are close to those of GGAs, and they are smaller than the gKS gaps of meta-GGAs. The well-known grid sensitivity of meta-GGAs is much more severe in OEP calculations.