1. 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

2. 1. π-electron approximation (1) (2) 由变分原理, 极小化变分积 V(i): 第 i 个 π 电子在核与  电子场中的势能.

3. 2.The free-electron MO method V=, outside this region 如果 : 忽略, 且 V=0, in a certain region 有:有: (3) (4) (5)

4. (6) 一维情况, 有 (7) 由 Pauli 原理, 有 :

5. (8) 对于 Polyenes: 设单键长, l 1, 双键长, l 2, 考虑 MOs 离域, 两端分别增加. 则电子运动的区域 :

6. 代入 (8) 式有 : 3. The Huckel MO method (11) (12) (13) 由线性变分法有 :

7. r=1,2,3,….n c (14) 久期方程 : (15) 积分 for C r & C s bonded (16) for C r & C s not bonded together (17) (18)

8. 归一化条件 (normalization condition ) (19) (20) 4. Conjugated Chain Molecules (21)

9. 本征方程 : (22) (23)

10. 由行列式理论有 : k=1, 2, 3, …, N (24) k=1, 2, 3, …, N u=1, 2, 3, …, N (25) 例子 : 丁二烯 (26)

11.  i =1.618, 0.618, -0.618, -1.618 波函数 :

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

13. The ground state of Butadine in C 2h symmetry

14. HOMO LUMO 一般情况 : HOMO: LUMO:

15. 5. Monocyclic Conjugated Polyenes. HMO results : k= 0, 1, …, n c -1 MO levels:

16. 4n + 2 rule ( 占满成键轨道 ) C 4 H 4 a triplet ground state

17. 分子轨道 C 6 H 6 k=0 k=1 k=5 k=2 k=4 k=3

18. 简并轨道的线性组合 实 MO Similarly

19. MO Symmetry species The symmetry species of MOs The ground state:

20. 6. Polycyclic Conjugated Polyenes Naphthealene 久期方程 : r = 1, 2, …, 10 久期行列式 : 利用, 对波函数进行分类 :

21. 原子轨道等价组 : C 1 = C 2 = C 3 = C 4 C 5 = C 6 = C 7 = C 8 C 9 = C 10

22. 根据 Huckel 近似 :

23. 令:令: x = -1,,

25. x =1,

27. 7.  Electron Charges, Bond Orders, Free Valence 丁二烯 : Electron charges Where the sum is over the MOs

28. For 1,3-butadiene, Theorem. For the ground states of a neutral alternant hydrocarbon, all the Huckel electron charges q r are 1. Bond Orders For butadiene, P 12 =2(0.372)(0.602)+2(0.602)(0.372)=0.894 P 23 =2(0.602)(0.602)+2(0.372)(0.372)=0.447

29. The total bond orders are [ bond contribution included ] Correlation of Bond lengths with Bond Orders Rc-c vs Prs

30. Free – Valence Index

31. 8. General Semi-empirical MO Methods The Extended Huckel Method. 价电子 Hamilton 看作单电子 Hamiltonian 的求和 分子轨道看作价原子轨道的线性组合 价单电子方程

32. 应用变分法有 : 由 Koopmans’ 定理引入 VSIP 参数化 VSIP = Valence-state ionization Potential 对角元 : 如 非对角元 : The CNDO, INDO, and NDDO Methods CNDO: The complete neglect of differential overlap method. INDO: The Intemediate neglect of differential overlap method.