Quantum mechanical reactive scattering for three-dimensional atom plus diatom systems. I. Theory

A method is presented for accurately solving the Schrödinger equation for the reactive collision of an atom with a diatomic molecule in three dimensions on a single Born-Oppenheimer potential energy surface. The Schrödinger equation is first expressed in body-fixed coordinates. The wavefunction is then expanded in a set of vibration-rotation functions, and the resulting coupled equations are integrated in each of the three arrangement channel regions to generate primitive solutions. Next, these are smoothly matched to each other on three matching surfaces which appropriately separate the arrangement channel regions. The resulting matched solutions are linearly combined to generate wavefunctions which satisfy the reactance and scattering matrix boundary conditions, from which the corresponding R and S matrices are obtained. The scattering amplitudes in the helicity representation are easily calculated from the body fixed S matrices, and from these scattering amplitudes several types of differential and integral cross sections are obtained. Simplifications arising from the use of parity symmetry to decouple the coupled-channel equations, the matching procedures and the asymptotic analysis are discussed in detail. Relations between certain important angular momentum operators in body-fixed coordinate systems are derived and the asymptotic solutions to the body-fixed Schrödinger equation are analyzed extensively. Application of this formalism to the three-dimensional H+H₂ reaction is considered including the use of arrangement channel permutation symmetry, even-odd rotational decoupling and postantisymmetrization. The range of applicability and limitations of the method are discussed.