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 S* _{N}*1 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 S* _{N}*2 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

This is the first chapter: Types of coordinates, z-matrices, input files

Next chapter How to locate a Transition State.

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