# Video: Hückel Theory: Energy of the Highest Occupied Molecular Orbital (HOMO) for the Benzene Molecule

Fill in the blank: Based on Hückel molecular orbital theory, the highest occupied molecular orbital (HOMO) for the benzene molecule has an energy of ＿.

05:15

### Video Transcript

Fill in the blank: Based on Hückel molecular orbital theory, the highest occupied molecular orbital, HOMO, for the benzene molecule has an energy of blank.

Let’s start by reminding ourselves of the structure of benzene. We know that benzene has six carbons in its ring. It also has six 𝜋 electrons, which are all conjugated around the ring. We also know that the molecule is planar. In order to use Hückel molecular orbital theory, the system that we’re working on has to be conjugated and planar. Luckily, these are both true for benzene. Hückel molecular orbital theory, sometimes shortened to HMO, is used to determine the energies and shapes of 𝜋 molecular orbitals. When using this theory, we can completely ignore the sigma bonding framework. This is because they are orthogonal to the 𝜋 framework that we’re interested in. This is why Hückel theory only works on planar molecules.

So how can we use this theory to answer the question? We could calculate the eigenvalues from the Hamiltonian matrix associated with benzene. However, this is really much easier to do if you had a computer. Luckily for us, there’s a shortcut. We can use frost circles, sometimes called frost cycles, as a shortcut. The first step is to draw our circle. We’re going to place the center of our circle at a point which we will denote as 𝛼. We will also draw our circle so that the radius can be denoted as negative two 𝛽. This is negative because 𝛽 is less than zero. We can then label this so that the bottom of the circle is at the point 𝛼 plus two 𝛽 with the top of the circle at 𝛼 minus two 𝛽. This will come in handy later on.

Next, we need to draw the shape of our cyclic molecule inside the circle with one of the corners at the bottom. Luckily for us, the way we’ve already drawn benzene is actually the correct orientation. Now that we’ve drawn benzene inside our frost circle, we can add the energy levels for the molecular orbitals. We need to add one orbital at each point that a vertex of benzene touches the frost circle. This gives us six molecular orbitals, which is what we expect.

Next, we can label the energies of each of these molecular orbitals. The top and bottom of the circle are straightforward to label. We can then use a little logic and maths to work out the energies of the remaining orbitals. These orbitals lie halfway between the center, which is at 𝛼, and the top and bottom. This makes them correspond to energies of 𝛼 minus one 𝛽 and 𝛼 plus one 𝛽. So now we can label these. Notice that the two orbitals with energy 𝛼 plus 𝛽 are degenerate. Likewise, the two orbitals at 𝛼 minus 𝛽 are also degenerate. Orbitals with an energy less than 𝛼 are bonding orbitals and those with an energy greater than 𝛼 are antibonding.

To complete our frost circle, we should add in the six 𝜋 electrons. We can add the first two to the lowest energy orbital. And the last four can occupy the 𝛼 plus 𝛽 degenerate orbitals This question is asking us for the energy of the highest occupied molecular orbital. For this molecule, it is 𝛼 plus 𝛽. So the answer to the question is simply 𝛼 plus 𝛽.

While the Hückel molecular orbital theory is quite approximate and involves several assumptions, it still produces an awful lot of information. From this simple frost circle, we can start to understand why benzene is quite so stable. Notice that all of its electrons are in bonding orbitals and there are no nonbonding electrons. You can also use this theory to work out the shape of these orbitals. So it’s really a very valuable tool.