Classical S-Matrix Theory of Reactive Tunneling: Linear H+H2 Collisions

Complex‐valued classical trajectories (computed by direct numerical integration of Hamilton's equations) are found for linear reaction collisions of $\mathrm{H}+{\mathrm{H}}_2\to {\mathrm{H}}_2+\mathrm{H}$ (on the Porter‐Karplus potential surface) at collision energies for which all ordinary real trajectories are nonreactive, and from such trajectories classical S‐matrix elements are constructed. This analytically continued classical‐limit theory is seen to be an accurate description of reactive tunneling for the $\mathrm{H}+{\mathrm{H}}_2$ system. At each collision energy there is only one classical trajectory that contributes to the reaction, so that various features of the reaction dynamics are easily elucidated by looking specifically at this one trajectory. It is also shown how a Boltzmann average of the reaction probability can be carried out semiclassically, and this leads to an interesting relation between the imaginary part of the time increment of the complex‐valued trajectory at a given energy and the absolute temperature at which this is the dominant energy in the Boltzmann average: $\mathrm{Im}{\mathrm{(t}}_2{\mathrm{-t}}_1\mathrm{)=-}\frac{1}{2}\mathrm{\hslash \; /(kT)}$ . It is seen, for example, that for $\text{T ≲ 1000°}\mathrm{K}$ the dominant energy region is below the classical threshold, i.e., in the tunneling region.