Computational Chemistry

The propenyl cation, Fig.4.19b; cf. Fig.4.16. If we take the totalpelectronic
energy of a molecule to be simply the number of electrons in apMO times the
energy level of the orbital, summed over occupied orbitals (a gross approximation,
as it ignores interelectronic repulsion), then for the propenyl cation

Epðprop:cationÞ¼ 2 ðaþ 1 : 414 bÞ¼ 2 aþ 2 : 828 b

We want to compare this energy with that of two electrons in a normal molecule
with no special features (the propenyl cation has the special feature of a formally
emptyporbital adjacent to the formal C/C double bond), and we choose neutral
ethene for our reference energy (Fig.4.15)

EpðreferenceÞ¼ 2 ðaþbÞ¼ 2 aþ 2 b

The stabilization energy is then

E(stab, cation)¼Epðprop:cationÞ"EpðreferenceÞ

¼ð 2 aþ 2 : 828 bÞ"ð 2 aþ 2 bÞ¼ 0 : 828 b

.... nonbonding level,

a - b a - b a - b

a - 2 b

a - b

a + 2b a + 2b

a + b a + b

a + 2b

a a a

a + 2b

a – 2 b

a – 1.618b

a + 0.618b a +^ 0.618b

a – 1.618b

Fig. 4.22 A useful mnemonic for getting the simple H€uckel method pattern for cyclicpsystems.
Setting the radius of the circle at 2|b|, the energy separations from the nonbonding level can even
be calculated by trigonometry

a – b a – b

a + 2 b a + 2 b

a + 0.618 b

a – 1.618 b

a – 2 b

a + 2 b

a + 0.618 b

a + 2 b

a + b a + b

a

a – b a – b

a – 2 b

a – 1.618 b

+ +

aa

Fig. 4.23 H€uckel’s rule says that cyclicpsystems with 4nþ 2 pelectrons (n¼0, 1, 2,...;4nþ
2 ¼2, 6, 10,...) should be especially stable, since they have all bonding levels full and all
antibonding levels empty. The special stability is usually equated with aromaticity. Shown here are
the cyclopropenyl cation, the cyclobutadiene dication, the cyclopentadienyl anion, and benzene;
formal structures are given for these species – the actual molecules do not have single and double
bonds, but rather electron delocalization makes all C/C bonds the same

4.3 The Application of the Schr€odinger Equation to Chemistry by H€uckel 139