Chapter9多原子的半经验方法

Chapter9

多原子的半经验方法

1.π-electronapproximation2.Thefree-electronMOmethod3.TheHuckelMOmethod4.Conjugated

ChainMolecules

5.MonocyclicConjugatedPolyenes6.PolycyclicConjugatedPolyenes7.Charges,BondOrdersandFreeValences8.GeneralSemi-empiricalMOMethods1.π-electronapproximation(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.2.Thefree-electronMOmethod

V=,outsidethisregion如果:忽略,且V=0,inacertainregion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.TheHuckelMOmethod(11)

(12)

(13)

由线性变分法有:r=1,2,3,….nc

(14)

久期方程:(15)

积分

forCr&Csbonded(16)

forCr&Csnotbondedtogether(17)

(18)

归一化条件(normalizationcondition)(19)

(20)

4.Conjugated

ChainMolecules(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,-1.618波函数:

分子轨道节面与能级(NodalPlanesandEnergyLevels)

:

ThegroundstateofButadineinC2h

symmetry

HOMOLUMO

一般情况:HOMO:

LUMO:

5.MonocyclicConjugatedPolyenes.

HMOresults:

k=0,1,…,nc-1MOlevels:

4n+2rule(占满成键轨道)

C4H4atripletgroundstate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MOSimilarly

MOSymmetryspeciesThesymmetryspeciesofMOs

Thegroundstate:6.PolycyclicConjugatedPolyenesNaphthealene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=C8C9=C10根据Huckel近似:

令:x=-1,,,x=1,

7.p-ElectronCharges,BondOrders,FreeValence丁二烯:ElectronchargesWherethesumisovertheMOsFor1,3-butadiene,Theorem.Forthegroundstatesofaneutralalternanthydrocarbon,alltheHuckelelectronchargesqrare1.BondOrdersForbutadiene,P12P23

Thetotalbondordersare[bondcontributionincluded]CorrelationofBondlengthswithBondOrders

Rc-cvsPrsFree–ValenceIndex8.GeneralSemi-empiricalMOMethodsTheExtendedHuckelMethod.

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-stateionizationP