The Pauli potential is an essential quantity in orbital-free density functional theory (DFT) and in the exact electron factorization method for many-electron systems. Knowledge of the Pauli potential allows the description of a system relying on the density alone, without the need to calculate the orbitals. In this work, we explore the behavior of the exact Pauli potential in finite systems as a function of the number of electrons, employing the ensemble approach in DFT. Assuming the system is in contact with an electron reservoir, we allow the number of electrons to vary continuously and to obtain fractional as well as integer values. We derive an expression for the Pauli potential for a spin-polarized system with a fractional number of electrons, and we find that when the electron number surpasses an integer, the Pauli potential jumps by a spatially uniform constant, similarly to the Kohn-Sham potential. The magnitude of the jump equals the Kohn-Sham gap. We illustrate our analytical findings by calculating the exact and approximate Pauli potentials for Li and Na atoms with fractional numbers of electrons.
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