Magic numbers for metallic clusters and the principle of maximum hardness.

Abstract

It is shown that for relatively more stable metallic clusters (those with magic number of atoms) the chemical hardness (I-A) too is relatively larger. Thus the occurrence of magic numbers for metal clusters whose stability is determined by their electronic shell structure can be understood as a manifestation of the principle of maximum hardness. This may also represent a possible way of delineating clusters with stability dominated by their electronic shell structure from those for which the magic numbers occur as a result of their geometric structure.Mass spectra of clusters, both nonmetallic (1-3) and metallic (1, 2, 4, 5), show pronounced intensity for certain numbers of atoms. Points at which these intensity anomalies occur have been termed magic numbers. For nonmetallic clusters, in particular for the rare gas ones, the occurrence of magic numbers is a consequence (3, 6) of their geometric structure. For example, in the case of rare gas atoms it is argued (3, 6, 7) that clusters are formed by packing of atoms around a central atom and the most stable ones are those for which the icosahedral shells are completely filled. For metal clusters, on the other hand, magic numbers have been shown (5,8) to be dominated by their electronic structure for up to about 1500-atom (8) clusters. Here, I show that the relative stability of these clusters is related to their chemical hardness (9). Magic numbers appear at those points where the cluster hardness has a local maximum. This is thus a manifestation of the principle of maximum hardness (10). In the following, I first discuss the energetics of metal clusters and in particular the relationship between their energies and the magic numbers. I then review the definition of hardness and the principle of maximum hardness. Finally, I calculate hardnesses for lithium clusters within the spherical jellium background model (5, 11, 12) (SJBM) with numbers of atoms in the range 2-67. 1 show that hardness is maximum at the magic numbers and conclude with a discussion of the results. Energetics of Metal ClustersMetal clusters have been studied (1, 2, 4) extensively using the spherical jellium background model. In this model the ionic charge is spread uniformly over a sphere ofradius R = r. (NZ) 13 where rs is the Wigner-Seitz radius ofthe metal and N and Z are the number and valence, respectively, ofthe constituent atoms. The advantage of using this model is that the potential it gives rise to is a simple central potential so that the solution of the Schrodinger equation is greatly facilitated. The justification (4) for using this model comes from the weak pseudopotential for the alkali metals for which this model is most frequently used. Electron-electron interactions are usually accounted for by using the density-functional (13-15) approach of Kohn and coworkers (16,17) within the local density approximation (13)(14)(15) for the exchange and correlation. For the purposes discussed here, however, the local density approximation is not applicable-neither for atomic (18)(19...

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