Distance-dependent Hückel-type model for the study of sodium clusters

Abstract

A distance-dependent extension of the Hückel model is proposed and applied to sodium clusters. It consists primarily of a two-band monoelectronic formulation expressed in an s+p basis set. The s+p Hamiltonian is reduced into an s-only Hamiltonian by means of quasidegenerate perturbation theory, with the p band treated perturbatively. The parametrization is taken from accurate calculations of ${\mathrm{Na}}_2$ and ${\mathrm{Na}}_4$. This formulation allows a very quick determination of the potential-energy surfaces, and the use of the Monte Carlo simulated-annealing technique for determining the stable isomers of clusters. For the smallest clusters (${\mathrm{Na}}_3$–${\mathrm{Na}}_8$), the model provides stabilities and geometries in very good agreement with previous studies involving more sophisticated calculations (ab initio configuration-interaction or density-functional theory). Optimization results without constraint are also presented for clusters in the range ${\mathrm{Na}}_9$–${\mathrm{Na}}_{19}$. Larger clusters in the range ${\mathrm{Na}}_{55}$–${\mathrm{Na}}_{561}$ are examined with restricted symmetry constraints (icosahedra, cuboctahedra, and cubic clusters). Beyond n=147, the cuboctahedral structure is preferred.