The proper description of step structures in the exchange correlation potential, of charge localization, and a reasonable prediction of band gaps have been long-standing, serious challenges for semilocal density functionals. In practice, obtaining all of these properties from the functional derivative of an energy functional was possible only at the price of incorporating exact exchange. We here show that they can be achieved at significantly lower, semilocal computational expense by using kinetic-energy density-dependent functionals. The key to obtaining these features is a functional construction strategy that focuses on the derivative discontinuity and the density response.

Nodal surfaces of orbitals, in particular of the highest occupied one, play a special role in Kohn-Sham density-functional theory. The exact Kohn-Sham exchange potential, for example, shows a protruding ridge along such nodal surfaces, leading to the counterintuitive feature of a potential that goes to different asymptotic limits in different directions. We show here that nodal surfaces can heavily affect the potential of semilocal densityfunctional approximations. For the functional derivatives of the Armiento-Kümmel (AK13) 2421 (1994)] model potentials, we explicitly demonstrate exponential divergences in the vicinity of nodal surfaces. We further point out that many other semilocal potentials have similar features. Such divergences pose a challenge for the convergence of numerical solutions of the Kohn-Sham equations. We prove that for exchange functionals of the generalized gradient approximation (GGA) form, enforcing correct asymptotic behavior of the potential or energy density necessarily leads to irregular behavior on or near orbital nodal surfaces. We formulate constraints on the GGA exchange enhancement factor for avoiding such divergences.

have been shown to mimic features of the exact Kohn-Sham exchange potential, such as step structures that are associated with shell closings and particle-number changes. A key element in the construction of these functionals is that the potential has a limiting value far outside a finite system that is a system-dependent constant rather than zero. We discuss a set of anomalous features in these functionals that are closely connected to the nonvanishing asymptotic potential. The findings constitute a formidable challenge for the future development of semilocal functionals based on the concept of a nonvanishing asymptotic constant.

Range-separated hybrid functionals with a fitted or tuned global range-separation parameter are frequently used in density functional theory. We here explore the concept of local range separation, i.e., of turning the range-separation parameter into an explicit semilocal density functional. We impose three simple constraints on the local range-separation parameter that are frequently used in density functional construction: uniform density scaling, the homogeneous electron gas limit, and freedom from one-electron self-interaction. We further discuss different ways of how to model the spin dependence in combination with local range separation. We evaluate our local range-separation energy functionals exactly for closed-shell atoms using the previously suggested hypergeneralized gradient approximation for molecules and assess the quality of this approximation. We find a local range-separated hybrid functional that yields accurate binding energies for a set of small molecules.

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