Orbital structure of vibrations of nanoparticles, clusters, and coordination polyhedra

The necessity of the development of the orbital structure of vibrations of nanoparticles, clusters, and coordination polyhedra is dictated by synthesis of clusters, supermolecules, and other structures of nanoscale dispersion for which translational symmetry is absent and the crystal system is inapplicable. The composition of complicated molecules, polynuclear complexes, and clusters is described, in addition to the chemical formula, by the composition equations derived from analysis of the symmetry properties of molecular structures. This analysis enables the derivation of analytical formulas for the types of molecular orbitals of structures with arbitrary groups of symmetry. Here, we use the representation of nanoscale structures described by the orbital system as a set of concentric nested spherical orbits of atoms, orbits of faces of different order, and orbits of edges. The orbits are grouped into shells shaped as polyhedra with vertices, edges, or faces accommodating atoms with different types of packing. In such a way, the sets of molecular orbitals of all high-, intermediate-, and low-symmetry groups have been determined depending on the number of atoms in the axial, planar, and primitive orbits.