On the nonclassical asymptotic behavior of electronic properties in metal clusters

The ionization potential I(R) of small metal spheres (of radius R) as well as the electronic chemical potential μ(R) in such particles are considered within a three-parameter variational local-density-functional calculation. The asymptotic (R→∞) deviations of I(R) and μ(R) from their bulk values behave as C/R and C_μ/R, respectively, where within the computational accuracy C + C_μ = 0.5. These results are quantitatively similar to those obtained from a recent variational calculation by Engel and Perdew (EP), and identify the origin of the deviation of C from its classical value of 0.5 in the size dependence of μ(R). While EP show that this size dependence originates from the gradient terms in the energy functional, we find that its magnitude results from a delicate balance between different contributions. The classical limit C = 0.5 is approached when both Z and R are large, where Z is the number of electrons involved hi the transition. These results also lead to the resolution of an apparent paradox recently described by van Staveren et al.