Two new theoretical developments are presented in this article. First an energy corrected sudden (ECS) approximation is derived by explicitly incorporating both the internal energy level spacing and the finite collision duration into the sudden S-matrix. An application of this ECS approximation to the calculation of rotationally inelastic cross sections is shown to yield accurate results for the H+–CN system. Second, a quantum number and energy scaling relationship for nonreactive S-matrix elements is derived based on the ECS method. A few detailed illustrations are presented and scaling predictions are compared to exact results for R–T, V–T, and V–R, T processes in various atom–molecule systems. The agreement is uniformly very good — even when the sudden approximation is inaccurate. An important result occurs in the analysis of V–T processes: the effects of anharmonic wave functions (coupling) and decreasing vibrational energy gaps (energetics) are separated. Each factor makes significant contributions to the deviation of the anharmonic from the harmonic scaling relationship.

Time-dependent treatments of molecular collisions have frequently employed the combination of quantum equations for the internal degrees of freedom (i.e., vibration, rotation) with a classical description of the translational motion. In this paper, it is shown how energy-conserving classical path equations may be derived from first principles, thereby obtaining quantum correction terms of arbitrary order. These expressions have interesting implications for the question of how the classical limit is approached. If the translational degree of freedom is assumed to be described by a very narrow wave packet, the correction term to first order in fz is purely imaginary and therefore non-Hermitian. These techniques can be generalized to a classical treatment of other degrees of freedom, and the case of rotation in an atom-vibrotor collision is explicitly considered. For both the translational and rotational examples, the full series of corrections takes on an interesting and suggestive exponential form. The failure of classical path methods to satisfy microscopic reversibility is ascribed to a difficulty with the boundary conditions for the translational motion.

The formalism necessary for the application of ``classical S-matrix'' theory to collisions of an atom with a rigid asymmetric rotor is derived. This is applied to rotational excitation of formaldehyde by H2 (taken to be spherically symmetric) at energies from 10 to 15°K. Classical Monte Carlo trajectory calculations were also carried out for the same system in the energy range 10–40°K. The results support the proposal of Townes and Cheung that a collisional mechanism is responsible for the 111 →110 anomalous absorption of formaldehyde in cool interstellar dust clouds.

Conventional quantum mechanical treatments of many systems have worked with coordinates and momenta that are not canonically conjugate. In this work it is shown how the quantum expressions may be reformulated in terms of the canonical set of action-angle variables, and specific examples of the harmonic oscillator, linear rotor, and triaxial rotor are presented. When expressed in these terms, the quantum mechanics take on a form which can be directly related to analogous results from classical mechanics. In addition, it becomes possible to express the Hamiltonian in the minimum number of coordinates. It is also shown that the common assumption of an exponential form for the overlap of canonical coordinate and momentum eigenstates is false for an asymmetric rotor. This has important implications for the quantization rules applicable to nonseparable systems.