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Magic Numbers in Clusters


If we assume that each coinage metal atom contributes one s electron (as a pseudoalkali metal) to, and that each halide ligand withdraws one electron from, the cluster, then Bp can be calculated by Bp 



where M, X, and Q refer to the numbers of metal atoms, halide ligands, and the overall charge, respectively. The above equations are useful in the systematization and rationalization of a series of Au-Ag clusters synthesized and structurally characterized by us. The metal frameworks of these clusters are based on vertex-sharing polyicosahedra, as portrayed in Fig. 2. For example, the 25-metal-atom cluster [( p  Tol3P)10Au13 Ag12Br8] (83) has Bp  (25  8  1)/2  8 electron pairs as predicted for a biicosahedral supracluster (4n  4  2  8 electron pairs). Similarly, and the 25metal-atom dicationic cluster [( p  Tol3P)10Au13Ag12Cl7]2 (84), Bp  (25  7  2)/2  8 electron pairs, also as expected. For the 38-metal-atom cluster [(p  Tol3P)12Au18Ag20Cl14] (85,86) with three icosahedra sharing three vertices, Bp  (38  14)/2  12 electron pairs, which agrees with the triicosahedral model (4n  4  3  12 electron pairs). Similarly, the 37-metal-atom cluster [(p  Tol3P)12Au18Ag19Br11]2 (87) has Bp  (37  11  2)/2  12 pairs of electrons, once again, as expected. For the 25-atom cluster formed by two icosahedra sharing one vertex, the C2 model predicts B  2  13 (icosahedron)  1  3 (sharing one vertex)  23 skeletal electron pairs, T  6Vm  B  6  (25  2)  23  161 total electron pairs or a total valence electron count of N  2  161  322. One example is the [( p  Tol3P)10Au13Ag12Br2(  Br)2(3  Br)4] cluster (83) for which the Nobs of (10  2  25  11  2  1  2  3  4  5  1)  322 valence electrons is in accordance with the calculated value. More examples can be found in the literature (32–34).

D. Jellium Model: Jelliumic Clusters Generating function 1s, 1p, 1d, 2s, 1f, 2p, 1g, 2d, 3s, 1h, . . .

Magic numbers 2, 8, 18, 20, 34, 40, 58, 68, 70, 92

The jellium model (35,88) treats the cluster as a smooth jelly of positively charged ions to which electrons are attracted. In terms of a free-electron picture, the valence electrons interact with a smooth one-particle effective potential and an electron-electron interaction potential. A spherically symmetric potential well is employed, along with parameters derived from the bulk. Solving the Schroedinger equation yielded discrete energy levels characterized by the angular momentum quantum number L in the order 1s, 1p, 1d, 2s, 1f, 2p, 1g, 2d, 3s, 1h, . . ..


Chemistry of metal nanoparticles


Chemistry of metal nanoparticles