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Comprehensive study of sodium, copper, and silver clusters over a wide range of sizes 2=<N=<75

The geometric and electronic structures of NaN, CuN, and AgN metal clusters are systematically studied based on the density functional theory over a wide range of cluster sizes 2=<N=<75. A remarkable similarity is observed between the optimized geometric structures of alkali and noble metal clusters over all of the calculated cluster sizes N. The most stable structures are the same for the three different metal clusters for approximately half the cluster sizes N considered in this study. Even if the most stable structures are different, the same types of structures are obtained when the meta-stable structures are also considered. For all of the three different metal clusters, the cluster shapes change in the order of linear, planar, opened, and closed structures with increasing N. This structural type transition leads to a deviation from the monotonic increase in the volume with N. A remarkable similarity is also observed for the N dependence of the cluster energy E(N) for the most stable geometric structures. The amplitude of this energy difference is larger in the two noble metal clusters than in the alkali metal cluster. This is attributed to the contribution of $d$ electrons to the bonds. The magic number is defined in the framework of total energy calculations for the first time. In the case of NaN, a semi-quantitative comparison between the experimental abundance spectra (Knight et al., Phys. Rev. Lett., 52, 2141 (1984)) and the total energy calculations is carried out. The changing aspect of the Kohn-Sham eigenvalues from N=2 to N=75 is presented for the three different metal clusters. The feature of the bulk density of states already appears at N=75 for all of three clusters. With increasing N, the HOMO-LUMO gap clearly exhibits an odd-even alternation and converges to 0.