南京大学结构化学双语课件CH6LEC2

0 c j cl c m c n 0
交替烃的非键轨道举例
a 0
-a 0
a
a 0
-2a 0 -a
a 0
-a 0 0 a -a 0 0 a 0 -a
a 0
0
6.3 Charge Densities and Bond Orders
•Charge Densities
•Bond Orders
参与共轭的原子可以被交替染色的共轭碳氢分子
*
*
* * *
*
*பைடு நூலகம்
*
*
*
*
*
*
? ?
Properties of alternant Hydrocarbons
Theorem For alternant hydrocarbons, the p energy levels +x and –x occurs in pairs (x). For the paired MOs, the atomic orbital coefficients for labeled atoms (*) are the same, while for unlabeled atoms, the only differences are that they are opposite in sign.
6.3 Charge Densities and Bond Orders
Free Valence nmax is calculated from the model compound (CH2)3C *
三亚甲基甲烷
(Trimethylenemethane, TMM)
n*=1.732
nmax=1.732
x2=0.618 x1=1.618
2 0 .6 0 2 1 0 .3 7 2 2 0 .3 7 2 3 0 .6 0 2 4
1 0.372 1 0.602 2 0.602 3 0.372 4
Charge Densities Energy level 1, x1=1.618, is occupied by two electrons, more probably appear on atom two and three.
xc i c r c s c t
( r,s,t stand for three unlabeled atoms connected to atom i, but it could be less than three ) For each unlabeled atom j,
xc j c l c m c n
xc i c k c l
Here, the kth and lth atom are connected to ith atom.
xc i c k c l
x
ck ci

cl ci
11 2
更复杂的体系
The is one eigenstate corresponding to x = 2, 1
Therefore, bond 1-2 is much stronger than bond 3-4, butadiene is often denoted as,
Bond Order vs. Bond Length
共轭CC键键长反比于其p键级
3. Free Valence
描述共轭分子中，不同原子的反应活性差异
(x) ( x)
i la b led

c i i c i i
j u n la b led

c j
j
i la b led

j u n la b led

c j
j
Proof For some alternant hydrocarbon molecule, suppose there exists one energy level x. For each labeled atom i,



c
r
r
1
Number of electrons Occupying orbital
2. Bond Orders
Bond Orders For connected atoms, the p-bond orders are defined as
p rs
n c
1
4. Molecular Graph
0.89 0.45
0.84
0.39 0.45 0.10 0.40
0.52
0.80
6.4 Aromaticity
• Aromaticity, Anti-aromaticity, 4n+2 rule
• Resonance Energy （共振能）
• The Eight-parameter Scheme
•Free Valences 自由价
•Molecular Graph 分子图
1. Charge Densities
Butadiene x4=-1.618 x3=-0.618
4 0 .3 7 2 1 0 .6 0 2 2 0 .6 0 2 3 0 .3 7 2 4
3 0 .6 0 2 1 0 .3 7 2 2 0 .3 7 2 3 0 .6 0 2 4

N ( 1 2 ... N )
x 3
The is one eigenstate corresponding to x = 3, 1

60 ( 1 2 ... 6 0 )
Alternant Hydrocarbons（交替烃）
Definition
For butadiene, F1 = F4 = 1.73-0.89 =0.84 F2 = F3 = 1.73-0.89-0.45 =0.39 1,4-addition is preferred to 1,2- or 2,3-additions
Naphthalene
a
b
Anthracene
g Fa>Fb a b Fg> Fa>Fb
Theorem
The p energy levels of the conjugated hydrocarbons
are bounded, satisfying
|x|3
|x| < 2
|x| 2
|x| < 3
|x| 3
PROOF：
One atom connects with no more than two other atoms
Unlabeled
Which means, wavefunction with {{ci}, {-cj}} as AO coefficients are also eigenstate, the corresponding eigenvalue(energy level) is –x.
交替烃的非键轨道
1 0.372 1 0.602 2 0.602 3 0.372 4
Probability of finding 1 electron (from MO 1) on atom 1 is calculated by
2 c1 ( 1 ) 2 0 .3 7 2 0 .2 7 7
H 2C
C H
C H
CH2
Equal Charge Density， Unequal Reactivity
Free Valence Definition
Fr=nmax-nr
nr : 原子 r与相邻原子间的共轭键p键级之和。
nmax : 碳原子所能取的最大 nr值。代表共轭分子中参 与共轭的碳原子的最大成键能力。 Fr : 自由价，代表第r个原子的剩余成键能力，即反应 活性。
c ( ) 2 c ( ) 2 2 (0 .3 7 2 2 0 .6 0 2 2 ) 1 .0 2 1 2 1 1
This could also be regarded as the p electron charge density of atom 1.
Secular Equation of Hexatriene
x 1 0 0 0 0 1 x 1 0 0 0 0 1 x 1 0 0 0 0 1 x 1 0 0 0 0 1 x 1 0 0 0 0 1 x c1 c 2 c3 0 c4 c 5 c6
6.2 More about Energy Levels and Molecular Orbitals
•The BOUND and SUM of energy levels （能级的 “和” 与 “界”）
•Alternant Hydrocarbons
（交替烃）
Bound of the Energy Levels
o cc
r
c s
This parameter reflects the strength of the corresponding p-bond, therefore can be correlated to the bond lengths
Bond Orders Butadiene
2 0 .6 0 2 1 0 .3 7 2 2 0 .3 7 2 3 0 .6 0 2 4
For A.H.s with odd-number of atoms, the AOs from the unlabeled atoms do not have contributions to the non-bonding MO.