On the use of action-angle variables for direct solution of classical nonreactive 3D (Di) atom–diatom scattering problems

Abstract

Application of action-angle variables for the physical description and direct numerical integration of the exact Hamilton’s classical equations of motion for 3D nonreactive atom–diatom collision systems is discussed in detail. The Hamiltonian, action-angle equations of motion, and necessary coordinate transformations are presented for both rigid and vibrating rotor models of the diatom. Generalization to the vibrating-rotor model of the diatom is done via the introduction of internuclear position and momentum variables (r,pr), and subsequent canonical transformation (r,pr) →(V,ψV) to vibrational action-angle variables. We present a quantitatively realistic interatomic diatom potential model U(r) which allows exact solution of diatom energy ‘‘eigenvalues’’ E(V,J), and which facilitates analytical inversion of the transformation equations to obtain virtually exact expressions for r(ψV;V,J) without resort to any dynamical approximations. Use of rotational and especially vibrational action-angle variables (J,ψJ) and (V,ψV) to describe internal diatom motion [in place of their traditional Cartesian counterparts (r,pr)] allows large savings in computation time, a minimization in the number of necessary dynamical variables, exact treatment of vibrational–rotational coupling, and a trivial assignment of final ‘‘quantum’’ numbers (vf, jf ) which is useful for classical trajectory calculations of vib-rotationally inelastic cross sections.