1A: The reaction coordinate and the potential energy surface

The simplest way to describe a reaction coordinate is the one-dimensional plot,
which shows the energy level of reactants and products, and a maximum in between, the transition state.
It is the energy difference between the TS and the reactants, the activation energy E(A), which determines the rate of the reaction. If, in a kinetically controlled reaction, more than one product can be formed, then the product ratio is determined by the energy difference(s) between the respective TS's.

Electronic and/or steric contributions in these structures can be responsible for the difference in energy. (energy vs enthalpy, ignoring entropy differences)

(If an equilibrium exists between reactants and products, then in the long run the product ratio will be determined by the relative energies of the products only, which is called thermodynamic control)
Such a one-dimensional plot is not only used for reactions; for conformational changes the same picture and nomenclature are valid.

The horizontal axis of the plot depicts what we call the reaction coordinate. Sometimes this one dimension suffices to describe a reaction or other change. The rotation about a bond, or the bond length in an SN1 reaction for instance. But this does not imply that only one degree of freedom is changing during the proces. Other bond lengths, angles or torsions will adapt themselves to the changing situation. So, instead of a uni-dimensional curve, we should think of the energy function as a multi-dimensional surface.
The curve is a cross-section along one coordinate.

The two-dimensional case is used very often (two coordinates vs. energy), as it bears a nice analogy to a geographical landscape with valleys (energy minima), hilltops (maxima) and passes (saddle points, the transition states) to get from one valley to another.
There are many examples where two coordinates describe a reaction: the SN2 reaction, the Diels-Alder and other cyclo-addition reactions.

The higher dimensions are impossible to imagine, but mathematically equivalent. We still speak of minima, maxima and saddle points as stationary points on the PES, the potential energy surface.

As stated above, almost all reactions can be described by one or two variables, the other coordinates adapting themselves to the changing geometry. The selection of the proper variables, in order to get a continuous variation of the energy, is very important.

So far we have used 'variables' and 'coordinates' as more or less equivalent ways to describe the molecular geometry. Although we may store structures as atoms in cartesian space (x,y,z coordinates), organic chemists use (intuitively) bond lengths or torsion angles as coordinates (variables, degrees of freedom) as well. From vibration (infrared) spectroscopy we also know the normal coordinates for a particular structure.
For a structure with N (>2) atoms there are 3N-6 normal coordinates (three translations and three rotations of the whole molecule do not change the energy of the system), so the PES has 3N-6 dimensions.
For linear molecules there are two rotations, and 3N-5 normal coordinates.

This chapter is continued in:
1B: Normal coordinates

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