# How is the derivative discontinuity related to steps in the exact Kohn-Sham potential?

## TL;DR

This paper analytically links the derivative discontinuity and steps in the exact Kohn-Sham potential, demonstrating their common origin and proposing a method to extract the fundamental gap from these features.

## Contribution

It establishes the fundamental relationship between derivative discontinuity and steps in the exact xc potential through analytical derivation and numerical validation.

## Key findings

- Derived the common origin of potential steps and derivative discontinuity.
- Validated the relationship with exact numerical solutions for simple systems.
- Proposed a method to extract the fundamental gap from step structures.

## Abstract

The reliability of density-functional calculations hinges on accurately approximating the unknown exchange-correlation (xc) potential. Common (semi-)local xc approximations lack the jump experienced by the exact xc potential as the number of electrons infinitesimally surpasses an integer, and the spatial steps that form in the potential as a result of the change in the decay rate of the density. These features are important for an accurate prediction of the fundamental gap and the distribution of charge in complex systems. Although well-known concepts, the exact relationship between them remained unclear. In this Letter, we establish the common fundamental origin of these two features of the exact xc potential via an analytical derivation. We support our result with an exact numerical solution of the many-electron Schroedinger equation for a single atom and a diatomic molecule in one dimension. Furthermore, we propose a way to extract the fundamental gap from the step structures in the potential.