Computational Chemistry

unreactive noble gases (Fig.4.23). The archetype of such molecules is, of course,
benzene, and the stability is associated with the general collection of properties
called aromaticity [ 17 ]. These results, which were first perceived by H€uckel [ 19 ]
(1931–1937), are summarized in a rule called the 4n+ 2 rule or H€uckel’s rule,
although the 4n+ 2 formulation was evidently actually due to Doering and Knox
(1954) [ 42 ]. This says that cyclic arrays ofsp^2 -hybridized atoms with 4n+2p
electrons are characteristic of aromatic molecules; the canonical aromatic molecule
benzene with sixpelectrons corresponds ton¼1. For neutral molecules with
formally fully conjugated perimeters this amounts to saying that those with an odd
number of C/C double bonds are aromatic and those with an even number are
antiaromatic (seeSection 4.3.5.3).
H€uckel’s rule has been abundantly verified [ 17 ] notwithstanding the fact that the
SHM, when applied without regard to considerations like the Jahn–Teller effect
(see above) incorrectly predicts 4nspecies like cyclobutadiene to be triplet diradi-
cals. The H€uckel rule also applies to ions; for example, the cyclopropenyl system
twopelectrons, the cyclopropenyl cation, corresponds ton¼0, and is strongly
aromatic. Other aromatic species are the cyclopentadienyl anion (sixpelectrons,n
¼1; H€uckel predicted the enhanced acidity of cyclopentadiene) and the cyclohep-
tatrienyl cation. Only reasonably planar species can be expected to provide the AO
overlap need for cyclic electron delocalization and aromaticity, and care is needed
in applying the rule. Electron delocalization and aromaticity within the SHM have
recently been revisited [ 43 ].

Jahn-Teller-

type distotion

(pseudo-Jahn-Teller

distortion)

i.e.

bond order 1.5

square

bond order 1

bond order 2

rectangular

energy

nonbonding level

Fig. 4.20Cyclic systems with degenerate energy levels tend to undergo a geometric distortion
to remove the degeneracy, a consequence of the Jahn–Teller theorem

4.3 The Application of the Schr€odinger Equation to Chemistry by H€uckel 137