Density embedding with constrained chemical potential

We formulate a chemical-potential constrained density embedding method for systems where different fragments can have either continuum or discrete electronic states. We illustrate the method with the simplest model system designed to mimic an atom near a metal surface. It is trivial to separate the full system into two fragments (metal and atom) only when the distance between them is infinite. In this case, a range of metallic chemical potentials, μ, will lead to an identical number of electrons, N, on the atom. Our density embedding method can be used to define fragments even at finite separations. We show that using these definitions for fragments, the typical N(μ) staircase function is partially smoothed out due to the finite-distance interactions, resembling finite-temperature effects. Fractional occupations on the atom occur only for sharply-defined μ's. Because calculating fractional charges is important in various fields, from electrolysis to catalysis, solar cells and organic electronics, we antic...