${\mathrm{Si}}_6$ and ${\mathrm{Si}}_7$ are the two largest clusters with confirmed geometries experimentally. The apex atoms in the pentagonal bipyramidal structure for ${\mathrm{Si}}_7$ have a coordination number of 6, completely different from the tetrahedral structure of the diamond lattice, and never given a definite explanation so far. In this paper, we suggest that atomic bondings in small silicon clusters be the unsaturated covalent, rather than covalent-bond. We introduce the number of bond order and defined it as the ratio of the number of valence electron of an atom to the effective coordination number of the $i\mathrm{th}$ atom, which can be smaller than 1 or greater. An ad hoc tight-binding model (fractional bond model) for silicon clusters is constructed by explicitly incorporating the dependence of hopping integrals between two atoms on the number of bond order, and their ground state structures of silicon clusters are performed. The results indicate that this model is successful in determining the structures of free small silicon clusters.

• Received 20 July 1998

DOI:https://doi.org/10.1103/PhysRevB.60.10703