Analyses of mechanical stability for systems of ions and atoms associated with rings, cages and crypts

Abstract

There exist examples of systems in which small, simple metallic cations are associated with ring-like molecules (e.g. aromatic radical anions), with aggregates of molecules which encage the ion (e.g. solvation and coordination), or with special molecules which trap ions within the molecular structure (crypts). For each of these examples, the harmonic oscillations of the ion in the presence of the molecular structure can be observed. Moreover, in many instances, the stability of the ionic motions in the presence of the molecular structure is important in considering some from of transport process which is associated with the ion. In this paper we present an analysis of the mechanics of the motion of an ion under the influence of various molecular structures. For our analyses, we use a form of symmetry-adapted Taylor series which we have recently developed. We determine the positions of equilibrium for the ion in the presence of a continuous ring, a polygon and a polyhedron of sources for the pair-wise forces. We also determine the states of stability for the positions of equilibrium through an examination of the second-order terms in the Taylor series. We show that the general form of the Taylor series for a continuous ring is of the same form as that for a discrete polygon. We also show that small numbers of terms in the Taylor series can duplicate the exact potential for an atom within a tetrahedron or octahedron of sources to a high degree of accuracy.