Abstract

The ability of density-functional calculations to accurately describe materials critically depends on the approximation to the generally unknown exchange-correlation (xc) potential. Common xc approximations, such as the local-density approximation (LDA) or the generalized-gradient approximations (GGAs), although being successful in many contexts in chemistry, materials science and solid-state physics, lack some of the properties possessed by the exact xc potential. Among them are the uniform "jump" experienced by the xc potential as the number of electrons infinitesimally surpasses an integer and the abrupt spatial steps that form in the xc potential, e.g., in stretched diatomics, far from both nuclei. These features are crucial for an accurate prediction of the fundamental gap and for the correct distribution of electrical charge in complex systems. 

Although both the uniform "jump" and the spatial steps are well-known concepts, the exact relationship between them remained elusive. In our work, we establish a clear connection between these two features of the exact xc potential via an analytical derivation. We then support our result with exact numerical solutions of the many-electron Schroedinger equation for simple systems, such as a single atom and diatomic molecules. For the first time, the three-dimensional step structure of the exact potential of a molecule is visualized. We discuss the ways of incorporating the achieved results into advanced approximations for the exchange and correlation energies and the improvement it brings to the description of materials.