Improving the modified Becke-Johnson exchange potential

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 $|\ensuremath{\nabla}\ensuremath{\rho}|/\ensuremath{\rho}$ 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.