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5 INSULATORS, CONDUCTORS, SEMICONDUCTORS AND SUPERCONDUCTORS

5 INSULATORS, CONDUCTORS, SEMICONDUCTORS AND SUPERCONDUCTORS

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ẳ 0



1



x



3:1ị



The homogeneous system corresponding to Equation (3.1) is:

8

xc1 ỵ c2 ẳ 0

>

>

>

>

>

>



>

<

cm1 xcm ỵ cm ỵ 1 ẳ 0

>

>

>

>

>

>

>

:

cN1 xcN ẳ 0



3:2ị



The general equation is:

cm1 xcm ỵ cm ỵ 1 ẳ 0



m ẳ 1; 2; ; N



3:3ị



with the boundary conditions

c0 ẳ cN ỵ 1 ẳ 0



3:4ị



The general solution is the standing wave

cm ẳ A expim uị ỵ B expim uị



3:5ị



where i is the imaginary unit i2 ẳ 1ị, provided

x ẳ 2 cos u



ð3:6Þ



(i) From the first boundary condition it is in fact obtained:

c0 ẳ A ỵ B ẳ 0 Y B ẳ A

cm ẳ Aẵexpim uịexpim uị ẳ 2iA sin mu ¼ C sin mu



ð3:7Þ

ð3:8Þ



THE LINEAR POLYENE CHAIN



121



where C ¼ 2iA is a normalization factor.The general equation then

gives:

8

Afexpẵim1ịux expim uị ỵ expẵim ỵ 1ịug

>

>

>

>

>

>

ỵ Bfexpẵim1ịuxexpim uị ỵ expẵim ỵ 1ịug

>

>

>

>

< ẳ Aexpim uịẵexpiuịx ỵ expiuị

3:9ị

> ỵ Bexpim uịẵexpiuịx ỵ expiuị

>

>

>

>

>

>

ẳ ẵAexpim uị ỵ B expim uịẵexpiuịx ỵ expiuị

>

>

>

:

ẳ cm 2cosuxị ẳ 0

so that, for cm = 0:

2 cos uÀx ¼ 0 Y x ẳ 2 cos u



3:10ị



as required.

(ii) From the second boundary condition it is obtained:

cN ỵ 1 ẳ C sinN ỵ 1ịu ẳ 0



3:11ị



therefore it must be:

N ỵ 1ịu ẳ kp



k ẳ 1; 2; 3; ; N



3:12ị



with k a quantum number:

uk ẳ



kp

Nỵ1



3:13ị



so that angle u is quantized.

In conclusion, we see that the general solution for the linear chain will

be:

8

kp

>

>

k ¼ 1; 2; Á Á Á ; N

xk ẳ 2 cos

>

>

Nỵ1

<

3:14ị

>

kp

>

> cmk ẳ ck sin m

>

:

Nỵ1

the first being the p bond energy of the kth level (in units of b), the second

the coefficient of the mth AO in the kth MO. All previous results for

ethylene, allyl radical and butadiene given in Section 2.8 of Chapter 2 are

easily rederived from the general formula (Equation 3.14).

We give below the derivation of the detailed formulae for the open

linear chain with N ¼ 4 (butadiene).



122



AN INTRODUCTION TO BONDING IN SOLIDS



3.1.1



Butadiene N ¼ 4

p

uk ¼ k ;

5



p

xk ¼ 2 cos k ;

5



k ¼ 1; 2; 3; 4



ð3:15Þ



The roots in ascending order are:

(i) Bonding levels

8

p



>

>

> x1 ¼ 2 cos 5 ¼ 2 cos36 ¼ 1:618

>

<

2p

>

>

>

x ẳ 2 cos

ẳ 2 cos72 ẳ 0:618

>

: 2

5



3:16ị



(ii) Antibonding levels

8

3p

>

>

¼ 2 cos108 ¼ À0:618

x3 ¼ 2 cos

>

>

5

<

>

4p

>



>

>

: x4 ¼ 2 cos 5 ẳ 2 cos144 ẳ 1:618



3:17ị



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

For the MOs, we have:

fk ẳ



X

m



xm cmk ẳ C



X

m



x m sin m



kp

5



3:18ị



where C is a normalization factor.

Then:

8

!

4

X

>

p

p

2p

3p

4p

>

>

>

xm sin m ¼ C x 1 sin þ x 2 sin

þ x 3 sin

þ x4 sin

f1 ¼ C

>

>

5

5

5

5

5

>

mẳ1

<

>

>

>

>

>

>

>

:



ẳ C0:5878x 1 ỵ 0:9510x2 ỵ 0:9510x3 ỵ 0:5878x 4 ị

ẳ 0:3718x1 ỵ 0:6015x 2 ỵ 0:6015x 3 ỵ 0:3718x4



3:19ị



THE CLOSED POLYENE CHAIN



123



the deepest bonding MO (no nodal planes);

!

8

4

X

>

2p

2p

4p

6p

8p

>

>

x m sin m ẳ C x1 sin ỵ x 2 sin þ x 3 sin þ x4 sin

> f2 ¼ C

>

5

5

5

5

5

>

>

m¼1

<

>

>

>

>

>

>

>

:



¼ C0:9510x1 ỵ 0:5878x 2 0:5878x 3 0:9510x4 ị

ẳ 0:6015x 1 þ 0:3718x2 À0:3718x3 À0:6015x 4



ð3:20Þ



the second bonding MO (HOMO, one nodal plane);

8

!

4

X

>

3p

3p

6p

9p

12p

>

>

>

x m sinm ẳ C x1 sin ỵx 2 sin ỵx 3 sin ỵx4 sin

f3 ẳ C

>

>

5

5

5

5

5

>

mẳ1

<

>

>

>

>

>

>

>

:



ẳ C0:9510x1 0:5878x2 0:5878x 3 ỵ0:9510x 4 ị

ẳ 0:6015x 1 0:3718x 2 0:3718x3 ỵ0:6015x4



3:21ị



the first antibonding MO (LUMO, two nodal planes);

0

1

8

4

X

>

4p

4p

8p

12p

16p

>

>

A

> f4 ẳ C

xm sinm ẳ C@x 1 sin ỵx 2 sin ỵx 3 sin

ỵx 4 sin

>

>

5

5

5

5

5

>

mẳ1

<

>

>

>

>

>

>

>

:



ẳ C0:5878x1 0:9510x2 ỵ0:9510x3 0:5878x 4 ị

ẳ 0:3718x1 0:6015x2 ỵ0:6015x 3 0:3718x4



3:22ị



the last antibonding MO, highest in energy (three nodal planes). These

MOs coincide with those given in Equations (2.272) of Chapter 2, and

whose shapes are sketched in Figure 2.27.



3.2



THE CLOSED POLYENE CHAIN



Next, we want to find the general solution for the system of homogeneous

linear equations for the closed polyene chain with N atoms yielding the

Nth degree secular equation



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