An Introduction to Bonding in Solids - Models for Bonding in Chemistry - page 122

3.1.1 Butadiene N ¼

4

k 5 ; 2 cos k 5 ;

u k ¼

x k ¼

¼ ; ; ; ð : Þ

k 1 2 3 4 3 15

The roots in ascending order are:

(i) Bonding levels

8
<

2 cos 5 ¼ 2 cos36 ¼ x 1 ¼ 1 : 618

ð 3 : 16 Þ

:

2 cos 2

5 ¼

2 cos72 ¼ x 2 ¼ 0 : 618

(ii) Antibonding levels

<

2 cos 3

5 ¼

2 cos108 ¼ x 3 ¼ 0 : 618

ð : Þ

3 17

:

2 cos 4

5 ¼

2 cos144 ¼

x 4 ¼ 1 : 618

which coincide with those of Equations (2.272) of Chapter 2.

For the MOs, we have:

X

C X
m

x m sinm k

5

f k ¼ x m c mk ¼ ð : Þ

3 18

m

where C is a normalization factor.
Then:

<
:

!

C X

4

x m sinm 5 ¼ x 1 sin 5 þx 2 sin 2

5 þx 3 sin 3 5 þx 4 sin 4 5

f 1 ¼ C

m ¼ 1

¼ C 0 ð : 5878 x 1 þ 0 : 9510 x 2 þ 0 : 9510 x 3 þ 0 : 5878 x 4 Þ ð 3 : 19 Þ

¼ 0 : 3718 x 1 þ 0 : 6015 x 2 þ 0 : 6015 x 3 þ 0 : 3718 x 4