Enlarged symmetry groups of finite-size clusters with periodic boundary conditions (Q1181258)

The paper deals with the consequences of approximating the space group symmetry of an ideal crystal by a finite space group \(G_ 0\) by means of Periodic Boundary Conditions (PBC). Quantum mechanically the PBC assumption leads in reciprocal space to consider a finite number of wavevectors only and in direct space to a finite-cluster description involving \(M\) different cluster sites, each site representing an equivalence class with respect to a PBC-unit-cell. In analogy with the symmetry characterization of non-rigid molecules, the full symmetry group \(G_ f\) of such a cluster is the subgroup of the permutation group \(S_ M\) of the cluster sites leaving the Hamiltonian invariant. One can then distinguish among three cases for the group \(G_ f\) which may: A) be a proper subgroup of \(G_ 0\), B) contain elements not belonging to \(G_ 0\), C) be identical with \(G_ 0\). The relations between these three cases are discussed. In case B) the elements of \(G_ f\) not belonging to \(G_ 0\) represent hidden symmetries giving possibly rise to sticking together of irreducible representations of \(G_ 0\). These accidental degeneracies have been observed experimentally as well as in computer simulation. The ideas are analyzed further on a group theoretical base for the case of simple, body-centered, and face-centered cubic lattices and for the 2-dimensional square lattice.