A modified version of the exchange potential proposed by Becke and Johnson [ J. Chem. Phys. 124, 221101 (2006)] is tested on solids for the calculation of band gaps. The agreement with experiment is very good for all types of solids we considered (e.g., wide band gap insulators, $sp$ semiconductors, and strongly correlated $3d$ transition-metal oxides) and is of the same order as the agreement obtained with the hybrid functionals or the $GW$ methods. This semilocal exchange potential, which recovers the local-density approximation (LDA) for a constant electron density, mimics very well the behavior of orbital-dependent potentials and leads to calculations which are barely more expensive than LDA calculations. Therefore, it can be applied to very large systems in an efficient way.

The WIEN2k program is based on the augmented plane wave plus local orbitals (APW+lo) method to solve the Kohn–Sham equations of density functional theory. The APW+lo method, which considers all electrons (core and valence) self-consistently in a full-potential treatment, is implemented very efficiently in WIEN2k, since various types of parallelization are available and many optimized numerical libraries can be used. Many properties can be calculated, ranging from the basic ones, such as the electronic band structure or the optimized atomic structure, to more specialized ones such as the nuclear magnetic resonance shielding tensor or the electric polarization. After a brief presentation of the APW+lo method, we review the usage, capabilities, and features of WIEN2k (version 19) in detail. The various options, properties, and available approximations for the exchange-correlation functional, as well as the external libraries or programs that can be used with WIEN2k, are mentioned. References to relevant applications and some examples are also given.

A comparison of DFT methods: Density functional theory (DFT) is now routinely used for simulating material properties. Many software packages are available, which makes it challenging to know which are the best to use for a specific calculation. Lejaeghere et al. compared the calculated values for the equation of states for 71 elemental crystals from 15 different widely used DFT codes employing 40 different potentials (see the Perspective by Skylaris). Although there were variations in the calculated values, most recent codes and methods converged toward a single value, with errors comparable to those of experiment. Science , this issue p. 10.1126/science.aad3000 ; see also p. 1394

The modified Becke-Johnson exchange potential [F. Tran and P. Blaha, Phys. Rev. Lett. 102, 226401 (2009)] (TB-mBJ) is tested on various types of solids which are difficult to describe theoretically: nonmagnetic semiconducting transition-metal oxides and sulfides, metals (Fe, Co, Ni, and Cu), and (anti)ferromagnetic insulators (e.g., YBa2Cu3O6). The results for the band gap and other quantities such as the magnetic moment or electric field gradient are analyzed in detail, in particular to have a better understanding of the mechanism which leads to improved (or sometimes worse) results with the TB-mBJ potential compared to the standard local density and generalized gradient approximations.

The modified Becke-Johnson exchange potential [F. Tran and P. Blaha, Phys. Rev. Lett. 102, 226401 (2009)] (TB-mBJ) yields very accurate electronic band structures and gaps for various types of semiconductors and insulators (e.g., sp semiconductors, noble-gas solids, and transition-metal oxides). However, the TB-mBJ potential has, for a few groups of solids, the tendency to underestimate the band gap. This has led us to examine the possibility to further improve over the original TB-mBJ potential by either reparametrizing its coefficients using a larger test set of solids or defining a parametrization for small-/medium-size band-gap semiconductors only. We also checked alternatives to the average of |$\nabla${}$\rho${}|/$\rho${} in the unit cell for the determination of parameter c, which determines the amount of the screening contribution. Among these different possibilities, the best one seems to be a reparametrization of the coefficients, which leads to a much more balanced description of the band gaps.

The exchange-correlation functionals of the generalized gradient approximation (GGA) are still the most used for the calculations of the geometry and electronic structure of solids. The PBE functional [J. P. Perdew et al., Phys. Rev. Lett. 77, 3865 (1996)], the most common of them, provides excellent results in many cases. However, very recently other GGA functionals have been proposed and compete in accuracy with the PBE functional, in particular for the structure of solids. We have tested these GGA functionals, as well as the local-density approximation (LDA) and TPSS (meta-GGA approximation) functionals, on a large set of solids using an accurate implementation of the Kohn-Sham equations, namely, the full-potential linearized augmented plane-wave and local orbitals method. Often these recently proposed GGA functionals lead to improvement over LDA and PBE, but unfortunately none of them can be considered as good for all investigated solids.