Abstract: The Hückel eigenvalues of three-dimensional rectangular clusters of arbitary size are explored on concerning not only the nearest but also those second and third neighboring interactions. For better specifying these interactions, three sets of parameters namely (β_1,β_2,β_3), (η_1,η_2,η_3) and ζ are introduced which can be illustrated schematically by 4_~.
Based on the well-known solution of the linear Hückel chain governed by a tri-diagonal matrix A~((1)) and the commutability between matrices A~((1))'s of equal dimension, we efficently derive the genera solutions for the planar and rectangular clusters respectively which are expressed by Eqs.(11) and (16).
The three-dimensional solution can be reduced to various special cases previously obtained by Messmer~[2] and Bilek & Skala~[3] but that for the face center rectangular lattice is new due to the introduction of parameters (η_1, η_2, η_3). The energy levels spread symmetricaly around α in terms of parameters (β_1, β_2, β_3) and ζ but the antibonding partners would be increased depending on the magnitude of η′s as indicated by 7_~. This means that the second neighbors play a role of destabilization in more or less extent much like the Madelung contribution in ionic crystals. As a consequence, the face-center (fc) lattice would be less favorable than the body-center (bc) one for alkali crystals, which seems really the case that most alkali metals are of be lattice except for lithium which is of either lattice form.
What is the minimum size of a cluster behaving like a crystal? In other words, when does the discrete energy levels change into continuous bands? By dint of the present results, we try to propose a criterion of maximum energy level separation, 0.02 eV that a finite cluster might behave approximately as a crystal. In order to illustrate the estimation, numerical calculation are given in Table 2.