The one-electron density of a many-electron system is the ground-state density of a one-electron Schrodinger equation. The potential $v$ appearing in this Schrodinger equation can be constructed in two ways: In density functional theory (DFT), $v$ is the sum of the Kohn-Sham (KS) potential and the Pauli potential, where the latter can be explicitly expressed in terms of the KS system of non-interacting electrons. As the KS system is fictitious, this construction is only indirectly related to the interacting many-electron system. In contrast, in the exact electron factorization (EEF), $v$ is a functional of the conditional wavefunction $phi$ that describes the spatial entanglement of the electrons in the interacting system. We compare the two constructions of the potential, provide a physical interpretation of the contributions to $v$ in the EEF, and relate it to DFT. With numerical studies of one-dimensional two- and three-electron systems, we illustrate how features of $phi$ translate to the one-electron potential $v$. We show that a change in $phi$ corresponds to a repulsive contribution to $v$, and we explain step structures of $v$ with a charge transfer encoded in $phi$. Furthermore, we provide analytic formulas for the components of $v$ by using a two-state model. Our work thus presents the mapping of a many-electron system to a one-electron system from another angle and provides insights into what determines the shape of the exact one-electron potential. We expect our findings to be helpful for the search of suitable approximations in DFT and in related theories.
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