Some Quantum-Mechanical Considerations in the Theory of Reactions Involving an Activation Energy

Abstract

The activated complex or transition state method for calculating the absolute rate of a chemical reaction with an activation energy would be rigorously valid if classical mechanics applied to all degrees of freedom. In quantum mechanics, two kinds of limitations must be considered. First, because of Heisenberg's uncertainty principle, the transition state itself can be defined only if the potential surface is sufficiently flat around the highest point of the reaction path. Second, even if this condition is fulfilled, the transmission coefficient can differ from the value expected on the basis of classical mechanics, because a wave packet can be reflected both on its way up, and also on its way down the potential barrier separating the initial and final states. In fact, the transmission coefficient is, in many cases, a rapidly fluctuating function of the energy of the system. If the temperature distribution of the energy is sufficiently broad to cover several periods of this fluctuation, an average transmission coefficient can be defined which nearly agrees with the classical value. For the crossing of a one-dimensional potential barrier, the quantum corrections are surprisingly small. In problems with several degrees of freedom, the transmission coefficient is affected by the interchange of translational and vibrational energy. However, if the vibrational motion is fast as compared with the motion along the reaction path, these degrees of freedom can be treated on a par with the electronic coordinates. In this case, the formulas of Eyring, with a mechanically sensible transmission coefficient, are satisfactory. On the whole, we conclude that quantum-mechanical considerations invalidate the transition state method to a much smaller extent than could be presumed and it is only in the consideration of the relative rates of reactions between isotopes and reactions at very low temperatures that these effects may be important.