  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.
