Quantum mechanical scattering theory
Classical trajectory calculations do not recognize the fact that the motion of atoms, electrons, and nuclei is governed by quantum mechanics. The concept of trajectory then fades and is replaced by the unfolding of a wavefunction that represents initially the reactants and finally products. Complete quantum mechanical calculations of trajectories and rate constants are very onerous because it is necessary to take into account all the allowed electronic, vibrational, and rotational states populated by each atom and molecule in the system at a given temperature. It is common to define a ‘channel’ as a group of molecules in well-defined quantum mechanically allowed states. Then, at a given temperature, there are many channels that represent the reactants and many channels that represent possible products, with some transitions between channels being allowed but others not allowed. Furthermore, not every transition leads to a chemical reaction. For example, the process H2*+OH→H2+(OH) *, where the asterisk denotes an excited state, amounts to energy transfer between H2 and OH, whereas the process H2 * + OH →H2O+H represents a chemical reaction. What complicates a quantum mechanical calculation of trajectories and rate constants even in this simple four-atom system is that many reacting channels present at a given temperature can lead to the desired products H2O + H, which themselves may be formed as many distinct channels. The cumulative reaction probability, N(E), at a fixed total energy E is then written as
N(E)= ∑ i, j Pij (E)
Fig. 24.24 An example of the trajectories calculated for a complex-mode reaction, KCl+NaBr→KBr+NaCl, in which the collision cluster has a long lifetime. (P. Brumer and M. Karplus, Faraday Disc. Chem. Soc., 55, 80 (1973).)
where Pi,j(E) is the probability for a transition between a reacting channel i and a product channel j and the summation is over all possible transitions that lead to product. It is then possible to show that the rate constant is given by
where Qr (T) is the partition function density (the partition function divided by the volume) of the reactants at the temperature T. The significance of eqn 24.74 is that it provides a direct connection between an experimental quantity, the rate constant, and a theoretical quantity, N(E).
