Balancing the Contributions to the Gradient Expansion: Accurate Binding and Band Gaps with a Nonempirical Meta-GGA

The gradient expansion has been a long-standing guide rail in density-functional theory. We here demonstrate that for exchange-correlation approximations that depend on the gradient of the density and the kinetic energy density, i.e., for meta-generalized gradient approximations (meta-GGAs), there is a so far unexploited degree of freedom in the gradient expansion that allows to shift the relative weight of gradient and kinetic energy contributions. As the dependence on the kinetic energy density determines the derivative discontinuity, this allows to construct meta-GGAs that adhere to the known exact constraints, yet have new properties. We demonstrate this with the construction of a meta-GGA that describes both electronic bonds and band gaps with remarkable accuracy.