Benzene Ring

The benzene molecule is a ring of six carbon atoms, each C atom having one H atom attached. There is a mystery about the energy contained in this molecule. The benzene ring can be broken up into pieces, and chemists have measured the energies associated with the pieces and with the single bonds and the double bonds by studying ethylene and so on. The expected total energy can be calculated from these data, but the actual total energy of the benzene ring is much lower, telling us that the carbon atoms are much more tightly bound. Therefore, the bond picture would make the benzene ring easily susceptible to chemical attack, yet the molecule is quite resilient to breaking up.

Using the Schrodinger equation by considering each carbon atom on this ring as the potential home for a single electron, one can calculate the possible energy levels for the benzene ring. Why does this method of calculation work?

Answer

The benzene ring has six-fold rotational symmetry about an axis perpendicular to the plane of the ring. One simply requires a wave function solution of the Schrodinger wave equation that has this six-fold symmetry, and such a solution is easy to find. One would expect that knowing this solution would allow one to calculate the energy levels.

However, we are not done! There are two possible configuration base states, as shown in the diagram.

Both states should have the same energy, and they do. Therefore we really have a two-state system, analogous to the hydrogen molecular ion or the ammonia molecule, so the analysis should be for a two-state system. There will be the possibility that configuration A changes into configuration B. As a result, quantum mechanics will reveal that two new stationary states will occur, one state (the new ground state) with energy below the ground (lowest) state determined before, and one state with higher energy. The new ground state will be neither of the two configuration states shown but will be a linear combination of these two configuration states. Only this state is involved in the chemistry of benzene at room temperatures.

Understanding benzene was one of the first verifications of the linear superposition of states that is at the heart of quantum mechanics and also indicated that quantum mechanics will be successful at larger scales than atomic.