The Approximate Electronic Solutions in a Closed Form, for f.c.c., b.c.c. and h.c.p. Clusters

A cluster made of $N_A,\;N_B\;and\;N_C$ atoms in the x, y and z directions respectively, is treated with Huckel method. We obtain the approximate expressions for the eigenvalues and eigenvectors of f.c.c., b.c.c. and h.c.p. clusters in closed forms. The maximum and minimum values of the band so obtained converge to those derived from the Bloch sum in the limit of infinite extension. For a small cluster (of $9{\times}9{\times}5$ atoms, for instance), LDOS from the analytical (approximate) solution manifests better agreement at the surface, than inside the bulk.