chapter 9 多原子的半经验方法 1. π- electron approximation 2. the free-electron mo method...
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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.