Scattering Matrix and Chemical Reaction Rates

The connection between macroscopic bimolecular chemical reaction rate expressions and microscopic collision properties is usually based formally on collision cross sections. An alternative formulation based on spherical waves, or on classical trajectories classified according to the initial angular momentum L of the collision, leads to a direct connection between reaction rates and the scattering matrix, from which the collision cross sections can always be derived. Because of the ease of expressing the scattering matrix for many‐body collisions, this formulation can be used for three‐body or many‐body reactions, as well as for two‐body ones. For a gas otherwise in thermodynamic equilibrium, the rate coefficient for a reaction taking n+1 reactant molecules in internal states collectively labeled by i to n′+1 products in internal states j is $$k_{\mathrm{ij}}^{\text{(n+1)}}={\left(\frac{h^2}{\text{2πμkT}}\right)}^{\text{3n/2}}{h}^{\text{−1}}{\displaystyle {\int }_0^{\infty }}{\displaystyle \sum _{\gamma }}{\displaystyle \sum _{{\gamma }^{\prime }}}{\mathrm{|S}}_{{\mathrm{i\gamma ,j\gamma }}^{\prime }}{\mathrm{(E)|}}^2\exp \mathrm{(-E/kT)dE.}$$ Here μ is the common reduced mass, $${\mu }^n{\displaystyle \sum _{\text{k=1}}^{\text{n+1}}}{m}_k={\displaystyle \prod _{\text{k=1}}^{\text{n+1}}}{m}_k,$$ and γ and γ′ are the 3n—1 or 3n′—1 orbital quantum numbers of ordinary and generalized angular momentum which, together with E, specify the initial and final relative motion of the particles. The close connection between the scattering matrix and the lifetime matrix makes it possible to bring out effects, such as those characterizing unimolecular reactions, depending on the lifetime of a metastable intermediate.