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##### Metal-cluster ionization energy: a profile insensitive exact expression for the size effect

**Authors**: M. Seidl, J. P. Perdew, M. Brajczewska, and C. Fiolhais

**Ref.**: Physical Review B **55**, 13288-13292 (1997)

**Abstract**: The ionization energy of a large spherical metal cluster of radius R is I(R) = W + (1/2 + c)/R, where W is the bulk work function and c approx 0.1 is a material-dependent quantum correction to the electrostatic size effect. We present "Koopmans" and "displaced-profile change-in-self-consistent-field" expressions for W and c within the ordinary and stabilized-jellium models. These expressions are shown to be exact and equivalent when the exact density profile of a large neutral cluster is employed; these equivalences generalize the Budd-Vannimenus theorem. With an approximate profile obtained from a restricted variational calculation, the "displaced-profile" expressions are the more accurate ones. This profile insensitivity is important, because it is not pratical to extract c from solutions of the Kohn-Sham equations for small metal clusters.