Chemical Potential of Integer Electron Systems

A truly isolated atom always has an integer number of electrons. If placed in contact with a far-away metallic reservoir, a {\em range} of metallic chemical potentials $\mu$ will lead to an identical number of electrons, $N$, on the atom. We formulate a density embedding method in which the range of $\mu$ leading to integer $N$ decreases due to finite-distance interactions between the metal and the atom. The typical $N(\mu)$ staircase function is smoothed out due to these finite-distance interactions, resembling finite-temperature effects. Fractional occupations on the atom occur only for sharply-defined $\mu$'s. We illustrate the new method with the simplest model system designed to mimic an atom near a metal surface. Because calculating fractional charges is important in various fields, from electrolysis to catalysis, solar cells and organic electronics, we anticipate several potential uses of the proposed approach.