1、Chapter 9 多原子的半经验方法,1. - electron approximation 2. The free-electron MO method 3. The Huckel MO method 4. Conjugated Chain Molecules 5. Monocyclic Conjugated Polyenes6. Polycyclic Conjugated Polyenes7. Charges, Bond Orders and Free Valences8. General Semi-empirical MO Methods,1. -electron approximat
2、ion,(1),(2),由变分原理, 极小化变分积,V(i):第i个电子在核与电子场中的势能.,2 .The free-electron MO method,V= , outside this region,如果: 忽略, 且 V=0, in a certain region,有:,(3),(4),(5),(6),一维情况,有,(7),由Pauli 原理,有:,(8),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:,代入(8)式有:,3 . The Huckel MO method,(11),(12),(13),由线性变分法
3、有:,r=1,2,3,.nc,(14),久期方程:,(15),积分,for Cr & Cs bonded,(16),for Cr & Cs not bonded together,(17),(18),归一化条件(normalization condition ),(19),(20),4. Conjugated Chain Molecules,(21),本征方程:,(22),(23),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),ci =1.618 , 0.618, -0.618 , -1.
4、618,波函数:,分子轨道节面与能级 (Nodal Planes and Energy Levels),:,The ground state of Butadine in C2h symmetry,HOMO LUMO,一般情况: HOMO:,LUMO:,5. Monocyclic Conjugated Polyenes.,HMO results :,k= 0, 1, , nc-1,MO levels:,4n + 2 rule (占满成键轨道),C4H4 a triplet ground state,分子轨道 C6H6,k=0,k=1,k=5,k=2,k=4,k=3,简并轨道的线性组合 实 MO
5、,Similarly,The symmetry species of MOs,The ground state:,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:,原子轨道等价组:,C1 = C2 = C3 = C4,C5 = C6 = C7 = C8,C9 = C10,根据Huckel近似:,令:,x = -1 , ,x =1 ,7. p- Electron Charges, Bond Orders, Free Valence,丁二烯:,Electron char
6、ges,Where the sum is over the MOs,For 1,3-butadiene ,Theorem. For the ground states of a neutral alternant hydrocarbon, all the Huckel electron charges qr are 1 .,Bond Orders,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
7、, bond contribution included ,Correlation of Bond lengths with Bond Orders,Rc-c vs Prs,Free Valence Index,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,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.,
