Basic concepts of potential energy surfaces

The thermal state-to-state rate constant kif T) is specified at a fixed temperature, and it is obtained by averaging over the Maxwell-Boltzmann distribution of relative velocities/(v) given by 

The next step is to obtain the state-averaged thermal rate constant ki T) by summing over all possible final states, one can then write 

The final thermal rate constant (T) c an be obtained by summing over the thermal population of the reagents state. In other words, k T) is given by 

From a theoretical point of view, in the study of atom-atom or atom-molecule collisions one needs to solve the Schrodinger equation, both for nuclear and electronic motions. When the nuclei move at much lower velocities than those of the electrons inside the atoms or molecules, both motions (nuclear and electronic) can be separated via the Born-Oppenheimer approximation. This approach leads to a wave function for each electronic state, which describes the nuclear motion and enables us to calculate the electronic energy as a function of the intemuclear distance, i.e. the potential energy V r). Therefore, V r) can be obtained by solving the electronic Schrodinger equation for each inter-nuclear distance. As a result, the nuclear motion, which we shall see is the way chemical reactions take place, is a dynamical problem that can be solved by using either quantum or classical mechanics. 

Z mt is the reactant molecular partition function for the Diatomic potentials 

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