Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 3 / Section F / 46 / b / ii / I / Solution

Take $s=x-x^{‡}$ with $0<x^{‡}<a$. A trajectory crossing to the right at the barrier top goes downhill on exit and cannot return in this model. Its initial Boltzmann weight contains $e^{-\beta V_0}$; negative-velocity trajectories are on the reactant side at the commitment time. ThereforeDividing by the same $Q_r$ gives the result in the energy-integral calculation. No width appears in the classical rate because every classically allowed trajectory passes through the flat barrier.