1/2
(6) A
△ E(6) (4)
S
△ E(4)
△ E(1) = △ E(6) = │1/√6│2*0.5 = 1/12
1/2√3 1/√3
AS
(2) △ E(2)
△ E(2) = △ E(4) = │1/√3│2*0.5 = 1/6
(1) △ E(1)
1/√6
21
精确的HMO计算
x+0.5 1 0 0 0 1
2.3 利用分子对称性对Hückel 计算简化 ◆基本原理：
若体系的骨架对某一镜面具有对称性，则对这一 镜面而言，其MO-波函数只能是对称或反对称
若有简并轨道存在，则总能找到其线性组合方式， 对镜面一个为对称，另一个则为反对称
例如：乙烯、苯、吡啶、环丁四烯等
1
H H
H
H
Antisymmetric
9
若考虑存在的两个对称面:很快得到四个对 称性不同的分子轨道
10
苯：考虑两个对称面
1 2
SS
SS
SA
SA
AS
AA
ψ3
1
6
ψ1
1
23
1
3
ψ4
1
6
1
6
ψ6
1
23
1
3
ψ2
ψ5
11
The π molecular orbitals for benzene
12
2.4 PMO Method ---Perturbation Theory
= +;
1
1
S121'[1∫'=1∫11d1'+1'∫d1=02∫d2+(∫12+012d)+
2 (1+2)d ∫22d]
1
= 1;
同理：H1'2' = H2'1' = ; H2'2' = +; S1'2' = S2'1' = 0; S2'2' = 1
7
8
同样方法可得到其 antisymmetric分子轨道。 请尝试计算
*
n
2p
O
H3C
H
O
O
H3C
HO
25
2.5 Applications
2.5.1 Energy Level Diagram
LUMO
HOMO
Hg
Hg
PE spectrum Ekin = h -
UV spectrum h =
Molecule
Molecule
H = L (LUMO - Hg) G = H - TS = E0F E0 = G/F = 1/F [L(LUMO - Hg) - TS] 26
a) 杂原子取代
H
H
CC
H
H
H CO
H
N
b) Sigma-诱导基团取代
CF3
13
c) Mesomeric Interaction with n-orbital
O H
N HH
14
d) The -overlap Changes Through -Bond Rotation
O H
HN H
15
16
2.4.1 The -MO Changes through Heteroatom Substitution and -inductive Interactions
CHFClBr 1-chloro-3-fluorobenzene trans-1,2-dichlorocyclobutane trans-1,2-dichloroethylene H2O NH3 Ethylene Methane H-F
Point Group
C1 Cs C2 C2h C2v C3v D2h Td C∞v
△Hll = △Vll = △l = <lV l> (l: 取代的位置) △Hmm = △Vmm = 0 (m ≠ l) △Hmn = △Vmn = 0
一级微扰得到
1）(i) = 0(i) 2) (i) = <0(i)│V│0(i)> = │cl(i)│2△ l
17
乙烯 甲醛
18
乙烯 甲醛
H
23
0.17
0.06精确HMO源自△ (1) = 0.21
△ (2) = 0.07
△ (3) = 0.15
24
2.4.3 Purtubation through Mesomeric Interactions
练习：丙酮（ = 280 nm, = 15）,乙酸（ = 204 nm, = 50） 如何解释这一hyposochromic effect (blue shift).
Symmetric Antisymmetric
Symmetric
2
对称操作、重要的对称元素
1. The symmetry plane, labled
2. The rotational axis, labeld Cn 3. A center of symmetry i
3
常见分子的点群及对称元素
Molecule
N
1 x 1000
0 1x 100 =0
0 01 x10
0 00 1x1
1 00 01x
请比较
E(1) E(2) 一级微扰 2.0833 1.1667 HMO 2.1074 1.1672
E(3) E(4) E(5) 1 -0.8333 -1 1 -0.8410 -1
E(6) -1.9167 -1.9337
Symmetry Elements
None C2 C2, h, i C2, 2v C2, 3v 3C2, 3, i 4C3, 3S4, 3C2, 6 C∞, ∞Cv
4
轨道对称性对HMO计算简化举例
Cyclobutadiene
一般计算的基轨道为4个参 与成键的p轨道：
4
3
1
2
5
按一个对称面1导出的基轨道：
Symmetric AO's:
1 = 12（1 + 2）
1
2 = 2(3 + 4）
1'
'
Antisymmetric AO's:
3
=
1 2
（1
-
2）
4 = 12（3 - 4）
'
'
1
6
Hückel矩阵元素
H1'1'= ∫1' Ĥ 1'd = ∫
1
2 (1+2)Ĥ
1 2
(1+2)d
1
= 2 [∫1 Ĥ 1d + ∫1 Ĥ 2d +∫2 Ĥ 1d + ∫2 Ĥ 2d]
22
2.4.2 Bond Twist
若 键l-m中扭曲了 角度:
△Hlm = △Vlm = < lV m> = △ lm = (cos 1) △E(i) = 2 △ cl(i)cm(i) = 2 (cos 1) cl(i)cm(i)
3O H
例如
N
H1 2 H
= 45; cos45 = 0.707
H
H
H
H 1 2 1 = 0
O 2 = O C = b
H
OZcore = 1, hO = 1
2 =
12
1
2 =
2
2
12
1
1 =
2 =
2
2
19
实例：2. 有简并 苯 吡啶
20
m
N: Z-core = 1, △ N = N m 0.5; △E(3) = △E(5) = 0
(5)
(3)