From finite clusters of atoms to the infinite solid. I. Solution of the eigenvalue problem of a simple tight binding model for clusters of arbitrary size

Abstract

The following model is considered. A cluster of atoms having a rectangular parallelepiped geometry with ${N}_{A}$, ${N}_{B}$, and ${N}_{C}$ atoms in the $x$, $y$, and $z$ directions, respectively, with one atomic orbital per atom, is treated within a linear combination of atomic orbitals framework in which only nearest-neighbor interactions are included. General expressions for the eigenvalues and eigenvectors of this model are obtained for arbitrary values of ${N}_{A}$, ${N}_{B}$, and ${N}_{C}$ and of the transfer intergrals in the three directions, ${t}_{A}$, ${t}_{B}$, and ${t}_{C}$. Furthermore, general expressions are also obtained for the case of including nearest-neighbor overlap integrals ${S}_{A}$, ${S}_{B}$, and ${S}_{C}$. Under the following three conditions: (i) the number of atoms in all directions $\ensuremath{\rightarrow}\ensuremath{\infty}$, (ii) ${t}_{A}={t}_{B}={t}_{C}=t$, and (iii) ${S}_{A}={S}_{B}={S}_{C}=0$, the solutions reduce to those of a simple-cubic $s$-band solid.