The electronic structures of small copper, nickel, palladium, and platinum clusters having simple-cubic and cubo-octahedral geometries have been calculated, using the self-consistent-field-$X\alpha$ scattered-wave (SCF-$X\alpha$-SW) approach to molecular-orbital (MO) theory. As the cluster size and coordination number are increased, the MO results show increasing similarity to the electronic structures of the corresponding crystalline metals, with the results for 13-atom cubo-octahedral clusters exhibiting all the main features of the bulk band structures, e.g., overlap of the "$d$ band" by the "$s$, $p$ band," a sharp peak in the density of states around the Fermi level in the cases of Ni, Pd, and Pt, increasing $d$ band width through the series Cu, Ni, Pd, and Pt, and spin polarization of the levels in the case of Ni. The calculated ionization potentials decrease gradually with increasing cluster size and for the 13-atom cubo-octahedral clusters are approximately 2 eV greater than the corresponding average bulk work functions. Localized $d$ electron states, split off from the top and bottom of the manifold of closely spaced $d$ levels, are also observed for the cubo-octahedral transition-metal clusters. These states have no counterparts in the bulk band structures but arise because of the finite cluster size and presence of the cluster "surface". They appear to be cluster analogs of the "surface states" postulated for crystalline transition metals and probably play an important role in the chemisorption and catalytic activities of small metal aggregates of the type utilized as the active centers of heterogeneous catalysts. The SCF-$X\alpha$-SW MO results for transition-metal clusters are critically compared with those obtained for similar clusters by the extended-Hückel (EH) and complete-neglect-of-differential-overlap (CNDO) methods. The implications of these results in surface-cluster studies of chemisorption on transition metals are discussed.

• Received 13 October 1975

DOI:https://doi.org/10.1103/PhysRevB.13.1396