# Chemical Potential of Integer Electron Systems

## TL;DR

This paper introduces a density embedding method to analyze how finite-distance interactions between an atom and a metal surface affect the atom's electron count and chemical potential, revealing smoothed electron number steps similar to temperature effects.

## Contribution

The paper presents a novel density embedding approach that accounts for finite-distance interactions, reducing the range of chemical potentials leading to integer electron counts.

## Key findings

- Finite-distance interactions smooth out the N(μ) staircase.
- Fractional occupations occur only at sharply-defined μ.
- Method demonstrated on a model atom-metal surface system.

## Abstract

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.