Spherical jellium model closed-shell clusters

This model tends to predict more magic numbers than are actually seen in the mass spectrum. The reason for this may have geometric origins since only certain nuclearities can adopt highly spherical structures which coincide with a closed shell of electrons. When the cluster cannot adopt a highly spherical structure, a splitting of the jellium shells occurs, leading to some instability. A further and more crucial limitation is that the spherical jellium model provides no direct information on the structure, even for clusters with closed shells. 

The calculated electric dipole polarizabilities of some Na clusters with closed electronic shells are given in Table 3 (see column LDA-SJM) [52], The calculations employed the spherical jellium model. The results are expressed in units of the classical polarizability R. The enhancement of a over its classical value is directly proportional to the fraction of the electronic charge that extends beyond the positive background in the field-free system. The agreement with experiment is reasonable, although the theory systematically underestimates the polarizabilities. 

The DFTB stability function shows an odd-even oscillatory behavior, so that the even-numbered clusters tend to be more stable than the odd-numbered ones. The most pronounced peaks occur for N= 4, 6,8, 14,18,20, 30, 34,40,50, 56,58, where the bold numbers correspond to the structures with closed electronic shell for the spherical jellium model. The same odd-even alternation was found in earlier theoretical studies [17,22,24,28,37,38], that have been carried through for the smaller clusters with TV < 21. Moreover, our DFTB results are similar to those of the parameter-free study of Itoh et al. [48]. 

The differences in the stable geometries of the AM and AE clusters have been investigated from the electronic structure view point. Ekhardt and Penzar, using a self-consistent jellium model, reported a more stable prolate structure than the spherical one for Na4 (25). The model placed four valence electrons of the Na4 cluster into a spherical potential. Two electrons occupy the I5 shell in the spherical potential and the other two electrons are accommodated in the p shell. Prolate distortion splits the I/7 levels and then the lowered Ip level is filled with two electrons. Therefore, the Na4 cluster prefers the prolate deformation. Using a molecular orbital method, Rao and Jena came to a conclusion which is consistent with the jellium results (13). The Li4 cluster adopts a planar structure while the Be4 cluster has a close packed structure since the latter cluster has eight valence electrons and the molecular orbitals corresponding to the p shell for the Jellium model are completely filled with the electrons. 

Experiments on noble metal clusters (Cun, AgN, Aun) indicate the existence of shell-effects, similar to those observed in alkali clusters. These are reflected in the mass spectrum [10] and in the variations of the ionization potential with N. The shell-closing numbers are the same as for alkali metals, that is N = 2,S,20,40, etc. Cu, Ag and Au atoms have an electronic configuration of the type nd °(n + l)s so the DFT jellium model explains the magic numbers if we assume that the s electrons (one per atom) move within the self-consistent, spherically symmetric, effective jellium potential. 

In the jellium model, one has a closed electronic shell for n = 8, 20, 40, 58,. .. valence electrons. This gives a spherical shape for the cluster, and one dominant line in the... 

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