Frost Circles

by James

in Aromaticity

We’ve spent a lot of time in previous posts “building up” and drawing out the molecular orbitals for various species. In this post we’ll learn an extremely useful shortcut that will help us draw the energy levels of cyclic pi-systems extremely quickly.

The trick is called “Frost Circles”, or, sometimes, the “Polygon method”.

First:  Recall that we saw the energy levels of the molecular orbitals of benzene look like this:

Useful observation: these energy levels can be superimposed on a hexagon with the vertex pointing down. Like this!

Let’s do another one.

We’ve seen what the molecular orbitals for cyclobutadiene look like; the two highest energy levels are each singly occupied (and non-bonding, to boot) which helps to explain why cyclobutadiene is so spectacularly unstable.

The energy levels for cyclobutadiene can be superimposed on a diamond, which is just a fancy name for a square with its vertex pointing down.

That’s pretty cool. You might wonder…. does it work for other polygons too?

Indeed it does. Back in 1953, Frost published an article describing this method for drawing out the energy levels in cyclic systems, with a simplified version as follows [note]

“A circle… is inscribed with a polygon with one vertex pointing down; the vertices represent energy levels with the appropriate energies”. 

Vertices below the halfway mark of the circle are considered bonding orbitals, and vertices above the halfway mark are considered antibonding orbitals. If vertices are exactly in the middle (as they are for 4- and 8- membered rings) they represent non-bonding orbitals.

This idea is presented in the diagram below for 3, 4, 5, 6, 7, and 8-membered rings:

This is an extremely useful mnemonic! This saves us a lot of work in building energy levels from the ground up.

To draw the molecular orbitals of a cyclic pi system, all we have to do is draw the appropriate polygon, vertex-down, and then fill it up with electrons.

Let’s see how we can apply these “Frost Circles”  for 3, 4, 5, 6, 7, and 8 membered rings. (Although we’re going to skip drawing the actual circles, and just focus on the positions of the orbitals).

3-Membered Rings

There are two important configurations of energy levels for 3-membered cyclic pi systems, depending on the number of pi electrons.

With two pi electrons, we’d expect an aromatic molecule.  One example is the cyclopropenium cation (below left), which is indeed aromatic.

With 4 pi electrons, an antiaromatic molecule is expected.  Oxirene (below right) which has never been isolated,  is in this category.

4-Membered Rings

Cyclobutadiene was covered above, and in the last post. Note that the molecular orbital diagram predicts that if you rip off two of the pi-electrons, the resulting cyclobutene di-cation should be aromatic,  (Substituted cyclobutene dications have indeed been synthesized and found to be aromatic, by the group of the estimable, late, George Olah[note]

5-Membered Rings

Cyclic 5-membered pi systems with 6 pi electrons are predicted to be aromatic. Examples are abundant. such as the cyclopentadienyl anion (below left), furan (below right), pyrrole, thiophene, imidazole, and many others. And yes, arsoles are aromatic too, but you probably didn’t need me to tell you that.

Although not drawn here, removing 2 pi electrons would result in an antiaromatic system. The cyclopentadienyl cation is a classic example.

6-Membered Rings

Already covered above – but note that the rules can be applied not only for benzene, but also “heterocycles” (i.e. aromatic rings with at least one non-carbon ring atom) such as pyridine, pyrimidine, and even the “pyrylium cation”.

7-Membered Rings

Cyclic 7-membered pi systems with 6 pi electrons are predicted to be aromatic.

For a ring entirely comprised of carbon atoms, this corresponds to the cycloheptatrienyl cation (sometimes known as the “tropylium ion”).

We’re used to thinking of carbocations as being unstable intermediates with a short lifetime. But the aromatic tropylium ion is such a stable salt you can actually put it in a bottle. Aldrich sells it.

The Frost circle method gives us the energy levels for the tropylium ion:

8-Membered Rings

With 8 pi electrons, cyclooctatetraene is predicted to be antiaromatic, and its molecular orbitals are predicted to look like this:

If you read the earlier post on antiaromaticity, however, you’ll recall that cyclooctatetraene can “escape” from antiaromaticity by adopting a tub-like shape. The molecular orbital levels we’d predict from the polygon method thus do not exactly correspond to the actual energy levels of cyclooctatetraene itself – it’s not a diradical, for instance.

That’s not quite the end of the story, however. Just like anti aromatic cyclobutadiene can be made aromatic through the removal of two electrons, the stability of (theoretically antiaromatic) cyclooctatetraene can likewise be adjusted by the removal or addition of two pi-electrons to give 6 or 10 pi-electrons, respectively.

Chemical relatives (“derivatives”) of the cyclooctatetraene dication (6 pi electrons) and the cyclooctatetraene dianion (10 pi electrons) have been synthesized and found to have aromatic character.

Therefore we would predict that these molecules are planar, unlike cyclooctatetraene itself. [In fact, this has been confirmed for the cyclooctatetraene dianion by X-ray crystallography – reference]


While there’s no end of interesting things to learn about aromaticity (homoaromaticity, anyone?) this post concludes our treatment of it for the time being.

In the next series of articles we’ll finally get in to discussing the reactions of aromatic compounds, beginning with the most important reaction type of all: electrophilic aromatic substitution.

Many thanks to Matt Knowe for assistance with this article. 


[Note 1]  The more complex version is:

 A circle centered at α with radius 2β is inscribed with a polygon with one vertex pointing down; the vertices represent energy levels with the appropriate energies

What are these terms “α” and “2β”, you may ask? The answer is beyond the scope of what we’ll discuss here, but the essence is that they are important terms in Hückel Molecular Orbital Method, a way of simplifying the Schrödinger Wave Equation for pi systems by treating the pi electrons and sigma electrons separately.  It’s a powerful and useful method for calculating the energies of pi systems and understanding their reactions.

[Note 2] The stability arising from antiaromaticity in the cyclobutadiene dication should be reduced considerably, however, by the Coulombic repulsion arising from the di-cationic nature of the molecule (like charges repel, after all). Similarly, the stability arising from aromaticity of the cyclooctatetraene dianion is lessened somewhat by the repulsion of its two negative charges.

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