  When a molecule dissociates, the exact Kohn-Sham (KS) and Pauli potentials
may form step structures. Reproducing these steps correctly is central for the
description of dissociation and charge-transfer processes in density functional
theory (DFT): The steps align the KS eigenvalues of the dissociating subsystems
relative to each other and determine where electrons localize. While the step
height can be calculated from the asymptotic behavior of the KS orbitals, this
provides limited insight into what causes the steps. We give an explanation of
the steps with an exact mapping of the many-electron problem to a one-electron
problem, the exact electron factorizaton (EEF). The potentials appearing in the
EEF have a clear physical meaning that translates to the DFT potentials by
replacing the interacting many-electron system with the KS system. With a
simple model of a diatomic, we illustrate that the steps are a consequence of
spatial electron entanglement and are the result of a charge transfer. From
this mechanism, the step height can immediately be deduced. Moreover, two
methods to approximately reproduce the potentials during dissociation suggest
themselves. One is based on the states of the dissociated system, while the
other one is based on an analogy to the Born-Oppenheimer treatment of a
molecule. The latter method also shows that the steps connect adiabatic
potential energy surfaces. The view of DFT from the EEF thus provides a better
understanding of how many-electron effects are encoded in a one-electron theory
and how they can be modeled.
