Notes on relative equilibria of isosceles molecules in classical approximation

TL;DR

This paper analyzes the existence and stability of relative equilibria in a classical model of isosceles triatomic molecules using geometric mechanics, without requiring explicit potential functions, and verifies results with Lennard-Jones potentials.

Contribution

It provides a qualitative method to determine relative equilibria and their stability in isosceles molecules based on potential shape and bond lengths, employing geometric mechanics techniques.

Findings

Relative equilibria can be qualitatively determined without explicit potentials.

Stability analysis is performed using Reduced Energy-Momentum and Symplectic Slice methods.

Results are verified with Lennard-Jones potential simulations.

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 in [J.E. Marsden, Lectures in Mechanics, Cambridge University Press, 1992], whereas for linear (trivial isosceles) RE we apply the Symplectic Slice method, a technique based on the findings in the paper [R.M. Roberts, T. Schmah and C. Stoica, Relative equilibria for systems with configurations space isotropy, J. Geom. Phys., 56, 762, (2006)].