In organic chemistry, Hückel's rule estimates whether a planar ring molecule will have aromatic properties. The quantum mechanical basis for its formulation was first worked out by physical chemist Erich Hückel in 1931. The succinct expression as the 4n + 2 rule has been attributed to von Doering (1951), although several authors were using this form at around the same time.
Contents
A cyclic ring molecule follows Hückel's rule when the number of its π-electrons equals 4n + 2 where "n" is zero or any positive integer, although clearcut examples are really only established for values of n = 0 up to about n = 6. Hückel's rule was originally based on calculations using the Hückel method, although it can also be justified by considering a particle in a ring system, by the LCAO method and by the Pariser–Parr–Pople method.
Aromatic compounds are more stable than theoretically predicted by alkene hydrogenation data; the "extra" stability is due to the delocalized cloud of electrons, called resonance energy. Criteria for simple aromatics are:
- the molecule must follow Hückel's rule, having 4n + 2 electrons in the delocalized, conjugated p-orbital cloud;
- the molecule must be able to be planar;
- the molecule must be cyclic; and,
- every atom in the ring must be able to participate in delocalizing the electrons by having a p-orbital or an unshared pair of electrons.
Monocyclic hydrocarbons
The rule can be used to understand the stability of completely conjugated monocyclic hydrocarbons (known as annulenes) as well as their cations and anions. The best-known example is benzene (C6H6) with a conjugated system of six pi-electrons, which equals 4n + 2 for n = 1. The molecule undergoes substitution reactions which preserve the six pi-electron system rather than addition reactions which would destroy it. The stability of this pi-electron system is referred to as aromaticity. Still, in most cases, catalysts are necessary for substitution reactions to occur.
The cyclopentadienyl anion (C
5H–
5) with six pi electrons is considerably more stable than either the neutral cyclopentadienyl radical with five pi electrons or the corresponding cation with four. Similarly the tropylium cation (C
7H+
7) has six pi electrons and is more stable than either the cycloheptatrienyl radical (C
7H•
7) or its anion. The cyclopropenyl cation (C
3H+
3) and the triboracyclopropenyl dianion (B
3H2–
3) are considered examples of a 2π-electron system
Planar ring molecules with 4n pi electrons do not obey Hückel's rule, and theory predicts that they are less stable and have triplet ground states with two unpaired electrons. In practice such molecules distort from planar regular polygons. Cyclobutadiene (C4H4) with four pi electrons is stable only at temperatures below 35 K and is rectangular rather than square. Cyclooctatetraene (C8H8) with eight pi electrons has a nonplanar "tub" structure. However the dianion C
8H2–
8 with ten pi electrons obeys the 4n + 2 rule for n = 2 and is planar.
Refinement
Hückel's rule is not valid for many compounds containing more than three fused aromatic nuclei in a cyclic fashion. For example, pyrene contains 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 an spherical species having a same-spin half-filled last energy level with the whole inner levels being fully filled is also aromatic.