Quantum mechanical close coupling approach to molecular collisions. <i>jz</i> -conserving coupled states approximation

New coupled equations describing collisions of an atom and a diatomic molecule are derived in this paper. By utilizing a description of the collision in terms of rotating coordinates, all coupling in the z component of angular momentum is isolated into purely kinematic effects. By neglecting these couplings, one is led to approximate equations for which the jz component of angular momentum for the molecule is conserved. In addition, the scattering cross sections are formulated by neglecting the effect on the wavefunction of the rotation of the coordinate axes so that in place of Wigner rotation matrices dmmJ (Θ) appearing, one deals with simple Legendre polynomials and the orbital angular momentum l2 is approximated by l(l + 1) ℏ2. It is noted that the procedure involves no approximations so far as the potential matrix elements are concerned. Furthermore, the number of equations remaining coupled is drastically reduced and a completely quantum mechanical description of the dynamics of both internal states and relative motion is retained. The physical implications of the approximations are examined, and it is seen that the neglect of intermultiplet coupling gives rise to consideration of only transitions where both the orientation and magnitude of the rotor angular momentum change. Further, the neglect of transformation effects on the wavefunction is expected to be least accurate for the inelastic forward scattering and best for backward scattering and the j =0→0 elastic scattering. Finally, the present simplest version of the approximation obviously is not intended for treating processes dependent on mj transitions, e.g., NMR relaxation in He–H2. Next the formalism is applied in test calculations to He–H2 collisions using the Krauss-Mies potential energy surface. Numerical results for elastic and inelastic integral and differential cross sections are compared with exact quantum mechanical close coupling solutions of the standard coupled channel equations. Over the energy range studied (from 0.1 eV up to 0.9 eV), agreement to within a few percent is obtained. Additional coupled states calculations are reported at 1.2 eV and computation times are compared against those required for a full close coupling solution. Calculations for the Roberts He–H2 surface are also reported to illustrate the independence of the approximations on the strength of the coupling (so long as the inelastic scattering is predominantly in the backward direction). The dramatic savings afforded by the present approach are such as to make possible fully converged calculations at collision energies typically studied in molecular beam experiments. Thus, for elastic and inelastic nonreactive collisions, involving a repulsive-type interaction, the approach makes the a priori quantum mechanical description of the scattering of a diatom by an atom practical.