Background

Sometimes there can be a strong coupling between molecular vibrations and the motion of the electrons. The electrons in the C₆₀ molecule are known to be sensitive to the molecular vibrations.  It is believed that this vibronic coupling (known as the Jahn-Teller [JT] effect) plays an important role in determining various aspects of the behaviour of C₆₀ ions and related fullerene compounds. Modes of vibration between fullerene molecules (intramolecular modes) probably play an important role in the relatively high-temperature superconductivity of certain fulleride compounds.

As icosahedral symmetry is extremely rare in nature, vibronic coupling effects were not been investigated in this symmetry until relatively recently. Many interesting new effects due to the vibronic coupling are possible from a theoretical point of view due to the existence of quantum-mechanical states that are four and five fold degenerate.  The ground state of a neutral C₆₀ molecule contains a completely filled orbital, so is not subject to a Jahn-Teller effecy.  However, the ground states of its cationic and anionic states do potentially exhibit strong Jahn-Teller effects.  Vibronic coupling is also important from an experimental point of view. However, attempts to explain all the observed data using physically acceptable vibronic coupling models remain in a very underdeveloped state.

Static and dynamic Jahn-Teller effects

The Jahn-Teller effect tells us that a molecule and ions with degenerate states, such as C₆₀ anions, can spontaneously distort, such that the distorted configuration has lower energy than the undistorted one. The distortions that are allowed are certain combinations of the normal modes of vibration, and group theory can be used to determine the symmetry of the distortions. For C₆₀^- anions, theory tells us that coupling is possible to the eight h_g modes of vibration, and that the distortions can be of D_5d or D_3d symmetry.

Coupling to one of the modes of vibration results in D_5d distortions that are an elongation/compression along a 5-fold symmetry axis (in other words, an axis from the centre of the molecule to the centre of a pentagonal face), as shown below:

The image in the centre shows the conversion from the undistorted icosahedrom to the two extremes on the left and right, and is essentially just one particular combination of normal modes. If the Jahn-Teller coupling is very strong, the molecule could be permanently in a distorted state, such as in the left an right-hand images above. However, the displacements of the atoms from their undistorted positions is greatly exaggerated in these images. It is highly unlikely that the distortions could ever be seen in experiments that show submolecular resolution, such as atomic force microscopy (AFM) or scanning tunnelling imaging (STM).

A further complication is that there will be a number of possible distortions with the same energy. For example, for the distortion shown above, the distortion could be along any of the six equivalent 5-fold axes.This is shown below.

The central image shows the 6 equivalent 5-fold axes of a (truncated) icosahedron, and the outer 6 images show equivalent distortions along the 6 axes. Although the six outer images look different to each other when drawn from a common viewpoint (as in the figure), they can easily be shown to be identical if they are rotated into the same orientation.

There are 10 equivalent three-fold axes (through the centres of hexagons), which mean that there are 10 equivalent distortions of D_3d symmetry for each mode of vibration.

In order to get from one (equivalent) distortion to another, the C₆₀ ion will have to pas through distortions that are higher in energy. In other words, there are energy barriers separating the equivalent distortions. However, quantum mechanics tells us that systems can tunnel through energy barriers. In this case, this means that the C₆₀ ion can, and will, move from one distortion to another. On average, the original icosahedral symmetry is restored, although at any give instant the ion will not have icosahedral symmetry. This is known as the dynamic Jahn-Teller effect.

Because the Jahn-Teller effect will probably be dynamic, and the displacements of the atoms are small anyway, the presence of the Jahn-Teller effect can only be detected indirectly. For C₆₀, this could be through the analysis of spectroscopic data. We are also currently working on the likely signatures of the Jahn-Teller effect in STM images of fullerene ions. In other Jahn-Teller systems, such as magnetic ion impurities in semiconductors, the Jahn-Teller effect can be implied through parameters in effective or spin Hamiltonians used to model a system.