Minimal energy configurations of finite molecular arrays

Abstract

In this paper we consider the problem of characterizing the minimum energy configurations of a finite system of particles interacting between them due to attracting or repulsive forces given by a certain inter molecular potential. We limit ourselves to the cases of three particles arranged in a triangular array and that of four particles in a tetrahedral array. The minimization is constrained to fixed area in the case of the triangular array, and to fixed volume in the tetrahedral case. For a general class of inter molecular potentials we give conditions for the homogeneous configuration (either an equilateral triangle or a regular tetrahedron) of the array to be stable, that is, a minimizer of the potential energy of the system. To determine whether or not there exist other stable states, the system of first order necessary conditions for a minimum is treated as a bifurcation problem with the area or volume variable as the bifurcation parameter. Because of the symmetries present in our problem, we can apply the techniques of equivariant bifurcation theory to show that there exist branches of non--homogeneous solutions bifurcating from the trivial branch of homogeneous solutions at precisely the values of the parameter of area or volume for which the homogeneous configuration changes stability. For the triangular array, we construct numerically the bifurcation diagrams for both a Lennard--Jones and Buckingham potentials. The numerics show that there exist non--homogeneous stable states, multiple stable states for intervals of values of the area parameter, and secondary bifurcations as well.