Potential Energy Surfaces for Structure and Dynamics

(There are three problems)

Fully specifying the location of a diatomic molecule, AB, would require specifying six numbers, the x, y, and z coordinates of each of its atoms.

This would be hard to graph on two dimensional paper.

But if one is not interested in motion of the molecule as a whole, nor in its rotation, it is easy to draw on paper an explicit graph to show the potential energy (PE) of the molecule as a function of the distance between the two atoms dAB. The red line on this graph is a crude version of such a potential energy curve (the Morse potential) for a bond that can break.

The violet circle could be thought of as a ball poised to roll back and forth between the two stationary positions in which all of its energy is potential energy. When the ball reaches the bottom of the violet curve, the potential energy has been converted into kinetic energy. Motion of the ball can be used to visualize the vibration of the diatomic molecule as the bond stretches and shrinks interconverting potential and kinetic energy.

One can include a third dimension and go from a Potential Energy line to a Potential Energy Surface (PES), by using contours and/or colors to code the Potential Energy as a function of two geometric variables.

For example, the plot to the right shows the Potential Energy for a LINEAR triatomic molecule A-B-C, with darker colors corresponding to lower energies. (Allowing the molecule to bend at B would require a third geometric dimension and make plotting difficult.)

One could imagine this plot as a topographic map with altitude corresponding to potential energy. There are two valleys meeting at a pass (denoted by the +) with steep cliffs along the bottom and left edges and a plateau at the top right.

When both dAB and dBCare long (top right) one has three separate atoms at an energy that ceases to depend on further increase in these distances.

At the lower right is a dark valley corresponding to a BC molecule with the A atom far away (BC is short and AB is long).

At the upper left is the corresponding valley for an AB molecule with the C atom far away.

Molecular Vibration

Slicing this three dimensional PES along the horizontal red line and folding the top section up and back gives a profile (at the top) that is exactly the two-dimensional plot of potential energy for stretching of the AB molecule that was shown at the top of this page.

The double-headed violet arrow shows the path of the ball rolling back and forth to display vibration of AB.

The only difference between this motion and the one in the first two-dimensional plot is that now it is understood that C is at a given substantial distance from B.

In fact one can use motion of a single ball on the potential energy surface to show motion of two atoms (changes of both dAB and dBC), as shown in the next figure

A Reactive Collision
Substitution at B

A + BC -> AB + C

At first motion of a ball along the red line in this figure corresponds to A approaching BC (i.e. dAB shrinks from a great distance while dBC stays at the bonding distance for the BC molecule). As A gets close to BC, the BC bond begins to stretch until at the pass (+), dAB = dBC. From this point on the C atom flies away as A finishes bonding to B.

The red path is thus a trajectory for the reaction A + BC -> AB + C.

Note how different this trajectory plot is from the trajectory plot of Heller. Heller's plot used a separate point for each atom. In this plot a single point denotes the position of all atoms.

Slicing out a thin section of the PES along the curved trajectory of the previous figure and flattening it to look at the potential energy profile would yield the "Reaction Coordinate Profile" below. This plot shows the minimum amount of energy that would be required to get from one valley to the other across the pass marked by +.

Obviously there would be many other paths that would pass through the same + point, but this one seems as good as any for a start. One way to define such a path unambiguously is to descend into both valleys from the + pass making a 90° angle with every contour as it is crossed. This "Steepest Descent" path is well defined, but it may not be dynamically realistic (see below).

A Nonreactive Collision

In this trajectory the ball has more than enough energy to gain the altitude of the pass and react, but reaction fails because the ball caroms off the walls at the head of the valley and heads back out to the right. In this case too much of the energy was in vibration of the BC molecule (initial vertical motion) and too little in translation of the A atom toward the BC molecule (initial horizontal motion).

Problems:
1) Think about how the velocity of the ball changes as it follows this trajectory, remember that the sum of potential and kinetic energy should be constant. (Note - the curve was sketched by hand and may not be correct in detail).

2) Consider how a change in the phase of the vibration of BC (whether it is stretching or shrinking at the time that A collides with it) could influence the probability of reaction.

3) Sketch two cross section of the PES passing through the red +. One section should run diagonally from lower left to upper right; the other from upper left to lower right.

Vibrationally Excited Product

It would be very difficult, maybe impossible, to roll a ball along the Reaction Coordinate trajectory on the PES. Here is a more likely trajectory where the ball ends up snaking back and forth as it rolls down the AB + C valley. Some of the energy that was originally in the motion of A toward BC ends up in vibration of the AB product molecule. Even a marble that rolls directly across the + point is unlikely to follow the Reaction Coordinate.

Realism

Just how fanciful is this way of looking at reaction trajectories as the rolling of a ball over a PES?

In 1935 Henry Eyring at Princeton showed that if one "distorted" the surface by making the dAB and dBC axes form an angle of 60° instead of the customary 90°, the motion of a rolling ball would precisely track the behavior of the two-dimensional linear triatomic system.

Here is the surface he drew for H3 to illustrate this approach.

Of course there were still important weaknesses:

A profound weakness was assuming that a trajectory could be strictly deterministic rather than probabilistic and quantized as quantum mechanics demands. For example the "ball" could never roll straight along a valley floor, because a true, quantum mechanical molecule has a finite minimum amount of vibration - it would always have a snake-like path.

A less profound weakness was that Eyring used a relatively simple function to reckon the altitude of the PES rather than using accurate quantum-mechanical values of the energy for the various nuclear positions. This function resulted in a shallow well at the pass between the two valleys, which came to be known derisively as Lake Eyring. The lake disappears to become a saddle or "potato chip" point (like the + above) on a more realistic PES.

(Click for more detail)

People now calculate more sophisticated Potential Energy Surfaces (though still not good enough to treat vibrations really accurately) and take quantum mechanics into better account, but this is not our business. Our purpose was to see how knowing the potential energy as a function of structure would allow us to get a foothold on understanding molecular dynamics and reactivity.

For a real triatomic system there is an additional geometric variable (bending) that makes drawing an easily visualized PES impossible. But in our mind's eye (or the computer's) we can dream about the motion of a single marble over a 39-dimensional surface showing the coordinated motion of 13 atoms as they vibrate and exchange bonding partners.

We can think about valleys corresponding to stable molecules, and cliffs where atoms run into one another, and mountain passes through which the "ball" must squeeze to get from one valley to another.

We can see one reason why a system with more energy reacts more easily - because it can get to higher altitudes, where the pass is wider and the chance of hitting it during a trajectory is greater (the pass typically widens faster than the valleys with increasing energy). A more significant reason for a reaction to accelerate with increasing temperature is that the Boltzmann distribution gives a much higher probability of the system's having an energy in excess of the minimum necessary to reach the + point.

We need a more convenient name for the + point. Eyring referredto it as the "transition state", though others have pointed out that it is not really a "state" with a finite lifetime, since there are two directions of motion along which it is unstable. They call it a "transition structure".

As in the Heller case, individual trajectories provide too much detail, and one would need to consider an "Avogadrian" number of trajectories with various initial conditions to model a bulk reaction. With high-speed computers people do this kind of thing with a significant measure of success (for example one experimental challenge nowadays it to predict how proteins fold into their active conformations).

But we want a simple scheme for predicting reactivity, so we will lump things together statistically using the ideas of enthalpy and entropy. We will use the PES to gain insight into these quantities.


Return to Chem 125 Homepage
Go to Heller's 13-Atom Trajectory

copyright 2001 J.M.McBride