Static and Restricted Rigid Rotor Configurations of Three Classical 12-6-Lennard-Jones Particles - Rupp, Florian (Few-body systems, 2015)
Motivated by the continuous search for stable geometric configurations of atom and molecule clusters, we analyse the planar evolution of two freely movable point particles around a third immovable one subject to the 12-6-Lennard-Jones potential. This tailors our discussion to systems with one very heavy particle that can be assumed to be permanently at rest in the moving reference frame for the whole ensemble. Relating to Lennard-Jones interactions, we allow all three point particles to take different parameters. This breaks the symmetry conditions that are usually imposed on such systems. Through a classical non-regularized Hamiltonian description of our restricted three particle system, we study the existence of genuine equilibria and rigid rotor solutions around a single axis of rotation. We prove, depending on the choice of the Lennard-Jones parameters, that for these genuine equilibria, collinear alignments and triangular configurations of any shape can occur. Moreover, for the discussed type of relative equilibria a complete classification is provided by proving the existence of rigid rotor configurations in the plane of rotation (collinear cis and trans as well as triangle shaped configurations) and out of the plane of rotation (triangle shaped and flag-like configurations). Furthermore, we show that there are no further rigid rotor solutions of the underlying equations of motion.