The Chemical Bond - Models for Bonding in Chemistry - page 94

A root of the quartic equation (2.241) is given in this case by the simple
trigonometric relation:

p cot

g ¼ cos v ð 2 : 246 Þ

74 g 90 ð 45 u 60 Þ provided 54 0

the interbond angle resulting fromenergy optimization opens beyond 90 .
The four p AOs directed according to C 3v symmetry:
p 1

: . This shows that for v =

<
:

¼ z cos g þ x sin g

p
2

1
2 x sin

p 2 ¼ z cos g g þ y sin g

ð 2 : 247 Þ

p

3

1
2 x sin

p 3 ¼ z cos g g y sin g

2

p 4 ¼ z

are still normalized, but no longer orthogonal to each other:

<
:

1

2 sin 2

cos 2 h p 1 j p 2 i¼h p 1 j p 3 i¼h p 2 j p 3 i¼ g g

ð 2 : 248 Þ

h p 1 j p 4 i¼h p 2 j p 4 i¼h p 3 j p 4 i¼ cos g

Orthogonality can be restored by mixing in a certain amount of s
(Magnasco, 2007; Torkington, 1951):

8
<
:

p p

1 1

3

b 1 ¼ as þ bp 1 ¼

p ð

s sin v z cos vÞþ

p

x ¼ p b þ

p

x

1 1 1 1

3

1
6

1

¼ þ ¼

p ð

s sin v z cos vÞ

p

þ

p

¼ p p þ

p

b 2 as bp 2 x y b x y

1 1 1

2

1
3

1
6

1

p ð p

p p p

p

b 3 ¼ as þ bp 3 ¼ s sin v z cos vÞ x y ¼ b x y

ð : Þ

2 249

l ¼ cs þ dp 4 ¼ scos vþ z sin v ð 2 : 250 Þ

since:

p

2

1 = 2

p cot

1 = 2

3cos 2 3cos 2 1

3

1 g 1

sin

1 g

a ¼ p ; b ¼ p

3

g ;

c ¼ g; d ¼

sin 2 sin 2

g g

ð 2 : 251 Þ