Chapter 9 多原子的半经验方法 1. π- electron approximation

2. The free-electron MO method

3. The Huckel MO method

4. Conjugated Chain Molecules

5. Monocyclic Conjugated Polyenes

6. Polycyclic Conjugated Polyenes

7. Charges, Bond Orders and Free Valences

8. General Semi-empirical MO Methods

1. π-electron approximation

1 1

1core^ ^

i i j i ij

n n( i )H H r

21

2

core^

i V( i )H

(1)

(2)

由变分原理 , 极小化变分积

V(i): 第 i 个 π 电子在核与 σ 电子场中的势能 .

^*Min( d ) ,H E

2 .The free-electron MO method

V= , outside this region

如果 : 忽略 , 且 V=0, in a certain region 1

ijr

有 :

^

H E

ii

n

(3)

(4)

^

ii i( ) eH

core

i (5)

ii 1

neE

(6)

一维情况 , 有 2 2

i2 ii

, 1, 2,......8

h nem l

ne (7)

由 Pauli 原理 , 有 :

...

1

2

3

4...

1

2

3

4

HOMO-LUMO Excitation

n1 22

hcE e enhv

(8) n 21 22e

1 1( ) ( 1)

8e e nn

h

hc Cm l

对于 Polyenes:

设单键长 , l1, 双键长 , l2, 考虑 MOs 离域 , 两端分别增加

. 则电子运动的区域 :1 2

3 1

4 4l l

1 2 1 2 1 2

1 2 1 2

1 2

3 1 3( 1) ( )( )

2 2 21 1

(2 3)( ) (2 2 1)( )2 21

( 1)( )2 c

kl k l l l l l l

k l l k l l

n l l

代入 (8) 式有 :

1 21 22 ( ) ( 1) ( 1)(64.6 )e c cm ch l l n n nm

3 . The Huckel MO method

^ ^

( )eff

i

niH H

(11)

1

c

rii rr

nfc

^

( )eff

ii ii eH (12)

(13)

由线性变分法有 :

1

[( ) ] 0c

eff

rs rs i sis

ns e cH

r=1,2,3,….nc (14)

久期方程 :

det | | 0effrs rs iH S e (15)

积分 *( ) ( ) ( )eff eff

rr r r iH f i H i f i dv

^*

( ) ( ) ( )eff eff

irs r ri i i dvf fH H for Cr & Cs bonded

(16)

0eff

rsH for Cr & Cs not bonded together

*( ) ( ) irs rsr si i dvf fs

(17)

(18)

2r rf C p

归一化条件 (normalization condition )

(19)

21

c

rir s

nc

(20)

4. Conjugated Chain Molecules

1 2 3 4

...n

c c c c c (21)

本征方程 : 1

2

1

1 0 ... 0 0

1 1 ... 0 0

0 1 ... 0 00

... ... ... ... ... ...

... ... ... ... 1

... ... ... ... 1N

N

x

x

x

x

x

cc

cc

(22)

x

1 0 0

1 1 00

1

x

x

x

(23)

由行列式理论有 :

kk

2cos1k

k

Nx

k=1, 2, 3, …, N

(24)

2sin

1 1uk

ku

N Nc

k=1, 2, 3, …, N

u=1, 2, 3, …, N (25)

例子 : 丁二烯 1 2 3 4

2 2H C C C C H

12cos

5x

3

32cos

5x

2

22cos

5x

4

42cos

5x

(26)

i =1.618 , 0.618, -0.618 , -1.618

11.618 2

0.618

30.618 4

1.618

波函数 :

11

2sin 0.372

5 5c

21

2 2sin 0.602

5 5c

31

2 3sin 0.602

5 5c

41

2 4sin 0.372

5 5c

 

1 2 3 410.372 0.602 0.602 0.372f f f f

1 2 3 420.602 0.372 0.372 0.602f f f f

1 2 3 430.602 0.372 0.372 0.602f f f f

1 2 3 440.372 0.602 0.602 0.372f f f f

分子轨道节面与能级 (Nodal Planes and Energy Levels) :

4 +

+

+

+- -

- -

1.618 Ag

3 +

+

+-

- -

0.618 uB

+

+-

-

2

0.618 gA

+

-

1

0.618 gA

The ground state of Butadine in C2h symmetry

Ag1

HOMO LUMO A1g B1

u

E 1.236 一般情况 :

HOMO:

2ck n LUMO: 1 1

2ck n

1

( 2)2cos

2 2c

kc

nx n

2cos

2 2c

kc

nx n

1

( 2)2 cos cos

2 2 2 2

2 sin2 2

c ck k

c c

c

n nE

n n

n

5. Monocyclic Conjugated Polyenes.

HMO results :

k

22 cos

c

k

n k= 0, 1, …, nc-1

1 2 ( 1)exp

rkcc

i r kC nn

1

1 2 ( 1)exp

c

k rr cc

n i r k fnn

MO levels:

3cn

02

1 2

4cn 0

2

1

32

2

4n + 2 rule ( 占满成键轨道 )

C4H4 a triplet ground state

分子轨道 C6H6

k=0 1 1 2 3 4 5 6

1( )

6f f f f f f 2

2 4 5

3 3 3 32 1 2 3 4 5 6

1( )

6

i i i i

f f f f f fe e e e

k=1

2 4 5

3 3 3 33 1 2 3 4 5 6

1( )

6

i i i i

f f f f f fe e e e

k=5

2 4 2 4

3 3 3 34 1 2 3 4 5 6

1( )

6

i i i i

f f f f f fe e e e

k=2 2 4 2 4

3 3 3 35 1 2 3 4 5 6

1( )

6

i i i i

f f f f f fe e e e

k=4

6 1 2 3 4 5 6

1( )

6f f f f f f k=3 2

简并轨道的线性组合 实 MO

'

2 2 3

1( )

2

'

3 2 3

1( )

2i

'

2 1 2 3 4 5 6

1(2 2 )

12f f f f f f

2'

3 2 3 5 6

1( )

2f f f f

Similarly '

4 1 2 3 4 5 6

1(2 2 )

12f f f f f f

2'

5 2 3 5 6

1( )

2f f f f

 

1

2 3

4

5

6

2aa 1ge1ge 2ue 2ue

2g b

MO

Symmetry

species

The symmetry species of MOs

The ground state: 42

2 11 1

u ga e

6. Polycyclic Conjugated Polyenes

Naphthealene Z

y

1 2

34

5

67

8 9

10

久期方程 :

10

( ) 0eff

rs rs i sis

S CH r = 1, 2, …, 10

久期行列式 :

0eff

rs rs iSH

利用 , 对波函数进行分类 :zσ yσ

z yS S yzS A z ySA z yA A

原子轨道等价组 : 1 2 3 4, , ,

5 6 7 8, , , 9 10

,

z yS S

C1 = C2 = C3 = C4 '

1 1 2 3 4

1

2

C5 = C6 = C7 = C8 '

2 5 6 7 8

1

2

C9 = C10 '

3 9 10

1

2

' ' '

1 2 31 2 3C C C

Z

y

1 2

34

5

67

8 9

10

' ' '' ' '

11 12 1311 12 13

''

33 33

... ... ... 0

... ...

k k k

k

S S SH H H

SH

根据 Huckel 近似 :

^

i iH ^ ( 1)

0i j

j iH

| 1i i

j| 0( )

ij i

2

0 0

2 0

k

k

k

令 :

kx

1 2

1 1 0 0

2 0 1

x

x

x

x = -1 , 1 1

132 2

, ,

2.3028 1.3028

1 2 3 4

5 8 6 7

9 10

yz

c c c cS c c c cA

c c

'

1 1 2 3 4

1

2 '

2 5 8 7 6

1

2

'

3 9 10

1

2

' ' '

1 2 31 2 3C C C

2

0 0

2 0

Z

y

1 2

34

5

67

8 9

10

1 2

1 1 0 0

2 0 1

x

x

x

x =1 ,

1 113

2 2

2.3028 1.3028

1 2 3 4

5 8 6 7

9 100

z y

c c c cS c c c cA

c c

Z

y

1 2

34

5

67

8 9

10

10

1 1

x

x

1 5

2x

1.618 0.618

1 2 3 4

5 6 8 7

9 100

z y

c c c cc c c cA A

c c

Z

y

1 2

34

5

67

8 9

10

10

1 1

x

x

1 5

2x

0.618 1.618

7. Electron Charges, Bond Orders, Free Valence

丁二烯 : 1 2 3 4

C C C C

1 1 2 3 40.372 0.602 0.602 0.372f f f f

2 1 2 3 40.602 0.372 0.372 0.602f f f f

3 1 2 3 40.602 0.372 0.372 0.602f f f f

4 1 2 3 40.372 0.602 0.602 0.372f f f f

Electron charges 2

rq ri

ii Cn

Where the sum is over the MOs

For 1,3-butadiene ,

2 2 2 2

11 1212 2 2 0.372 2 0.602 1.000q C C

2 3 41.000q q q

Theorem. For the ground states of a neutral alternant hydrocarbon, all the Huckel electron charges qr are 1 .

Bond Orders sirs

p Crii

iCn

For butadiene ,

P12 =2(0.372)(0.602)+2(0.602)(0.372)=0.894P23 =2(0.602)(0.602)+2(0.372)(0.372)=0.447

The total bond orders are

CH2 CH2CH1.894 1.447 1.894

CH

[ bond contribution included ] σ 1pσ

Correlation of Bond lengths with Bond Orders

150

140

0.0 0.5 1.0 bond order

Rc-c vs Prs

Free – Valence Index

3s rss

F P 1 4

3 0.89 1.73 0.89 0.84F F

2 33 0.89 0.45 0.39F F

C C CC0.89 0.45 0.89

0.84 0.39 0.39 0.84

8. General Semi-empirical MO Methods

The Extended Huckel Method.

价电子 Hamilton 看作单电子 Hamiltonian 的求和

^ ^

H effvali

iH

分子轨道看作价原子轨道的线性组合

rii rr

fC

价单电子方程 ^

ii iieffH e

vali

iE e

应用变分法有 : ( ) 0

eff

rs i rs sis

e S CH 由 Koopmans’ 定理引入 VSIP 参数化VSIP = Valence-state ionization Potential

对角元 : eff

eff

rr rrr rf fH H I

如 20.8eff IsIs Is

eVC CH I

非对角元 :

( )eff eff effrs rr ss rsH k H H S

The CNDO , INDO , and NDDO Methods

CNDO: The complete neglect of differential overlap method. INDO: The Intemediate neglect of differential overlap method.