“…For instance, the HOMO-LUMO gap of supported clusters of metal and semiconductor atoms has been reviewed by Sattler [4]. Recently, we have clearly shown by STM studies the correlation between the HOMO-LUMO gap and the Pt-Pt interatomic distance in unisized platinum clusters, Pt n (n = 5-40), fixed on a silicon(1 1 1) surface; the HOMO-LUMO gap decreases critically at n $ 20 [5] when the platinum atoms are closely packed in a monatomic-layered two-dimensional geometry [6]; a cluster having this geometry is hereafter called a cluster disk. Therefore, the smaller HOMO-LUMO gap in the range of n > 20 plausibly originates from two-dimensional Pt-Pt metallic interaction in the cluster disk [5].…”