Notes on relative equilibria of isosceles molecules in classical approximation

Abstract

We study a classical model of isosceles triatomic “A-B-A” molecules. The atoms, considered mass points, interact mutually via a generic repulsive-attractive binary potential. First we show that the steady states, or relative equilibria (RE), corresponding to rotations about the molecule symmetry axis may be determined qualitatively assuming the knowledge of (1) the shape of the binary interaction potential, (2) the equilibrium diatomic distances (i.e., the equilibrium bond length) of the A-A and A-B molecules, and (3) the distance at which the RE of the diatomic A-A molecule ceases to exist. No analytic expression for the interaction potentials is needed. Second we determine the stability of the isosceles RE modulo rotations using geometric mechanics methods and using Lennard-Jones diatomic potentials. As a by-product, we verify the qualitative results on RE existence and bifurcation. For isosceles RE, we employ the Reduced Energy-Momentum method presented by Marsden [Lectures in Mechanics (Cambridge University Press, 1992)], whereas for linear (trivial isosceles) RE, we introduce the Symplectic Slice method, a technique based on the findings in the paper of Roberts et al. [J. Geom. Phys. 56, 762 (2006)].