The Chemical Bond - Models for Bonding in Chemistry - page 84

with:

<

32 b s h cos 2 v ð

A ¼ cos 2 vþ 1 Þ

32 b sh b 3 cos 2 v þ

B ¼

s h sin2 v ð

1 Þ

2 2

4sin 2 v

4 b s h fða b a h Þ 2 b sh ½ð

C ¼ þ sin2 vÞ þ

ð 2 : 209 Þ

:

2

cos 4 v ½ða p 8 b s h g a h Þ þ

2

4 b sh b s h sin2 v cos 2 v ½ða p a h Þ 8 b s h D ¼ þ

2 2

¼b sh ð 8 b s h E sin2 vÞ ½ða p a h Þ þ

Putting b sh ¼ b s h ¼ b in the spirit of H € uckel theory, the coefficients

become, after dividing by b 2 :

<

32 b 2 cos 2 v ð

A ¼ cos 2 vþ 1 Þ

32 b 2 sin2 v ð cos 2 vþ

B ¼ 1 Þ

2 2 2

2 b 2 4sin 2 v cos 4 v½ða p 8 b 2 C ¼ 4 fða b a h Þ þ ½ð sin2 vÞ þ a h Þ þ g

:

2

4 sin2 v cos 2 v½ða p a h Þ 8 b 2 D ¼ þ

2 2

8 b 2 E ¼ð sin2 vÞ ½ða p a h Þ þ

ð 2 : 210 Þ

Two special cases can be immediately analysed.

(i) For v ¼ 0 (no hybridization), a b a h ¼ a p a h , coefficients B, D, E

64 b 2 32 b 2 , and Equation (2.208) gives:

¼ ; ¼

vanish, A C

1

45

2x 2 ¼ 1 ; x ¼ cos u ¼

p Y u ¼

ð 2 : 211 Þ

as it must be.
(ii) If we put:

a b a h ¼ a p a h ¼ 0 ; ð 2 : 212 Þ

(which is tantamount to assuming nonpolar bonds) dividing through-
out by 8 b 2 , the coefficients will depend only on v :
A

<
:

4 cos 2 vð cos 2 vþ ¼ 1 Þ

cos 2 vþ B ¼ 4 sin2 vð 1 Þ

2

4 sin 2 v

4 cos 4 v

ð 2 : 213 Þ

C ¼ð sin2 vÞ þ

4 sin 2 v cos 2 v D ¼

2

E ¼ð sin2 vÞ