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 factorization (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 are proposed. 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.
There are different ways to obtain an exact one-electron theory for a many-electron system, and the exact electron factorization (EEF) is one of them. In the EEF, the Schrödinger equation for one electron in the environment of other electrons is constructed. The environment provides the potentials that appear in this equation: A scalar potential v H representing the energy of the environment and another scalar potential v G as well as a vector potential that have geometric meaning. By replacing the interacting many-electron system with the non-interacting Kohn-Sham (KS) system, we show how the EEF is related to density functional theory (DFT) and we interpret the Hartree-exchangecorrelation potential as well as the Pauli potential in terms of the EEF. In particular, we show that from the EEF viewpoint, the Pauli potential does not represent the difference between a fermionic and a bosonic non-interacting system, but that it corresponds to v G and partly to v H for the (fermionic) KS system. We then study the meaning of v G in detail: Its geometric origin as a metric measuring the change of the environment is presented. Additionally, its behavior for a simple model of a homoand heteronucler diatomic is investigated and interpreted with the help of a two-state model. In this way, we provide a physical interpretation for the one-electron potentials that appear in the EEF and in DFT.
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