Valence Bond Theory and the Chemical Bond - Methods of Molecular Quantum Mechanics - page 169

9.6.1 Ethylene (One p -Bond, n ¼ 1)

We have n ¼ i ¼ 1, so that the (1 1) secular equation is

H cc ES cc ¼ Q E þ K ¼ 0
j x þ 1 j¼ 0 ) x ¼ 1 ) E ¼ Q þ K

ð 9 : 68 Þ

which is Pauling's energy of the double bond.

9.6.2 Cyclobutadiene (n ¼ 2)

With reference to the second row of Figure 9.8, we can write for the two
fully equivalent Kekul

e structures 12

c 1 and c 2

Y ¼ c 1 c 1 þc 2 c 2 ð 9 : 69 Þ

The distinct matrix elements are

<

H 11 ES 11 ¼ H 22 ES 22 ¼ Q E þ 2K K ¼ Q E þ K

1
2 ð Q E þ 4K Þ¼

1
2 ð Q E Þþ 2K

ð 9 : 70 Þ

H 12 ES 12 ¼ H 21 ES 21 ¼

:

giving the (2 2) secular equation

x
2 þ 2

x þ 1

¼ 0 ð 9 : 71 Þ

x
2 þ 2

x þ 1

2

x
2 þ 2

2

) x 2 ð x þ 1 Þ ¼ 4 ¼ 0 ) x ¼ 2 ð 9 : 72 Þ

12 Pauling VB theory distinguishes between Kekule structures (all p bonds equal) and Dewar
structures (containing one or more long, i.e. weaker, p bonds).