Potential Energy Surfaces for Triatomic Molecules

For a triatomic molecule such as ozone, there are more nuclear coordinates; two inter-atomic distances, and the O—O—O bond angle. Often the angle is fixed, leaving the two bond lengths as coordinates. Plotting these against the potential energy produces a 3D topological map – a clear visualisation of a potential energy surface.
This is shown in Figure 5.1, where the bond angle of ozone is constrained at 117°, and the two bond lengths form the x- and y-axes of the graph:

Figure 5.1 - Potential energy surface (PES) of the ground electronic state of ozone. Bond angle of 117° [10]

Again, the surface shows a sharp increase (to infinity) as the nuclei come close to each other. The minimum, or equilibrium geometry is where both O—O bond lengths are approximately 2.5au. Stretching the bonds simultaneously increases the potential energy, but stretching only one increases the potential energy only slightly; this is reflected in the channel or valley seen in the PES. A cross section parallel with one of the axes through this PES would be similar to the curve shown for a diatomic in Figure 4.1.

An alternative way of plotting such a system on a 3D graph is, given a symmetric molecule such as ozone, the bond stretches can be calculated symmetrically (that is, simultaneously) and plotted on one axis; the bond angle being plotted on the other. Figure 5.2 shows a small section of the PES for water, with bond length and angle as independent coordinates:

Figure 5.2 - Potential energy surface for water as a function of symmetric bond stretch (length) and angle [11]

The three-dimensional PES is also used for triatomic systems involving the substitution collision reaction of an atom with a diatomic molecule. The most-studied of these is the so-called H₃ system, H + H₂ → H₂ + H.
In this situation, the triatomic molecule is the H—H—H transition state. As this will be higher in energy than the reactants or products, the PES will have a barrier at this point, as opposed to the well seen for the ozone PES in Figure 5.1.

The PES for the H₃ system is shown in Figure 5.3, below:

Figure 5.3 - Potential energy surface for the H3 (H+H2) system (Springer, 1998) [12]

An alternative way of plotting 3D potential energy surfaces such as these is to use a contour plot. This is presented for a H₃-like system in Figure 5.4; the A—B—C transition state barrier is marked by a red cross.

Figure 5.4 - Contour plot PES for a triatomic system. Ligher colours represent greater energies [13]

The minimum energy path is represented by the dashed line in Figure 5.4 – this involves the approach (decreasing inter-atomic distance, d_AB) of the reagents (A+BC), and the subsequent separation of B and C (increasing inter-atomic distance, d_BC) following collision at the transition state point (marked ‘+’). This motion, along one of the axes, corresponds to asymmetric stretching. Symmetric stretching is diagonal to this, and results in complete dissociation, but is energetically prohibited by the steep walls of the minimum energy valley.
The barrier at the transition state is termed a ‘saddle point’ – it is a minimum along the symmetric stretch coordinate (simultaneously increasing d_AB and d_BC), but a maximum along the asymmetric stretching coordinates (the minimum energy path).

3D potential energy plots are also particularly suited to representation using computer technology - below is a VRML object of the solvation potential energy surface of 1-(4-methoxyphenyl)-cyclohexane-3,5-diol.
If you have a suitable VRML client installed, you will be able to rotate a 3D model of the PES. If you're using Internet Explorer, the Cortona VRML client will offer automatic installation if it does not exist.

(Christopher Leach and Henry S. Rzepa, “VRML Models for Analysing Chemical Structure-Activity Relationships”, Imperial College, London)