closed shell in a regular hexagonal molecule. However for cyclobutadiene or cyclooctatrene with regular geometries, the highest molecular orbital pair is occupied by only 2 π electrons forming a less stable open shell. The molecules therefore stabilize by geometrical distortions which separate the degenerate orbital energies so that the last two electrons occupy the same orbital, but the molecule as a whole is less stable in the presence of such a distortion. Heteroatoms Hückel's rule can also be applied to molecules containing other atoms such as nitrogen or oxygen. For example pyridine (C5H5N) has a ring structure similar to benzene, except that one -CH- group is replaced by a nitrogen atom with no hydrogen. There are still six π electrons and the pyridine molecule is also aromatic and known for its stability. Polycyclic hydrocarbons Hückel's rule is not valid for many compounds containing more than one ring. For example, pyrene and trans-bicalicene contain 16 conjugated electrons (8 bonds), and coronene contains 24 conjugated electrons (12 bonds). Both of these polycyclic molecules are aromatic, even though they fail the 4n + 2 rule. Indeed, Hückel's rule can only be theoretically justified for monocyclic systems. Three-dimensional rule In 2000, Andreas Hirsch and coworkers in Erlangen, Germany, formulated a rule to determine when a fullerene would be aromatic. They found that if there were 2(n + 1)2 π-electrons, then the fullerene would display aromatic properties. This follows from the fact that an aromatic fullerene must have full icosahedral (or other appropriate) symmetry, so the molecular orbitals must be entirely filled. This is possible only if there are exactly 2(n + 1)2 electrons, where n is a nonnegative integer. In particular, for example, buckminsterfullerene, with 60 π-electrons, is non-aromatic, since 60 ÷ 2 = 30, which is not a perfect square. In 2011, Jordi Poater and Miquel Solà expanded the rule to determine when a fullerene species would be aromatic. They found that if there were 2n2 + 2n + 1 π-electrons, then the fullerene would display aromatic properties. This follows from the fact that a spherical species having a same-spin half-filled last energy level with the whole inner levels being fully filled is also aromatic. See also Baird's rule (for triplet states) References Physical organic chemistry Rules of thumb