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1、6.2MoreaboutEnergyLevelsandMolecularOrbitalsTheBOUNDandSUMofenergylevels（能级的“和”与“界”）AlternantHydrocarbons（交替烃）BoundoftheEnergyLevelsTheoremThepenergylevelsoftheconjugatedhydrocarbonsarebounded,satisfying
2、x
3、3
4、x
5、<2
6、x
7、<3
8、x
9、3
10、x
11、2PROOF：OneatomconnectswithnomorethantwootheratomsSecularEquationo
12、fHexatrieneEachlinecontainsnomorethanthreenon-zerotermsSuppose,thereexistsaneigenstate,withthelargestcontributionfromtheithorbital.Thentheithlinereads,Here,thekthandlthatomareconnectedtoithatom.更复杂的体系Theisoneeigenstatecorrespondingtox=2,Theisoneeigenstatecorrespondingtox=3,AlternantHydrocarb
13、ons（交替烃）参与共轭的原子可以被交替染色的共轭碳氢分子Definition*************??PropertiesofalternantHydrocarbonsForalternanthydrocarbons,thepenergylevels+xand–xoccursinpairs(x).ForthepairedMOs,theatomicorbitalcoefficientsforlabeledatoms(*)arethesame,whileforunlabeledatoms,theonlydifferencesarethattheyareoppositeins
14、ign.TheoremProofForsomealternanthydrocarbonmolecule,supposethereexistsoneenergylevelx.Foreachlabeledatomi,(r,s,tstandforthreeunlabeledatomsconnectedtoatomi,butitcouldbelessthanthree)Foreachunlabeledatomj,(l,m,nstandforthreelabeledatomsconnectedtoatomi,butitcouldbelessthanthree)Obviously,Unla
15、beledUnlabeledWhichmeans,wavefunctionwith{{ci},{-cj}}asAOcoefficientsarealsoeigenstate,thecorrespondingeigenvalue(energylevel)is–x.交替烃的非键轨道ForA.H.swithodd-numberofatoms,theAOsfromtheunlabeledatomsdonothavecontributionstothenon-bondingMO.Whilethesummationofthecoefficientsfromalltheatomsconnec
16、ttooneunlabeledatomsiszero交替烃的非键轨道举例a-a-aa-aa0000000a-aa-2a000a-aa006.3ChargeDensitiesandBondOrdersChargeDensitiesBondOrdersFreeValences自由价MolecularGraph分子图1.ChargeDensitiesButadienex1=1.618x2=0.618x3=-0.618x4=-1.618ChargeDensitiesEnergylevel1,x1=1.618,isoccupiedbytwoelectrons,moreprobablyap
17、pearonatomtwoandthree.Probabilityoffinding1electron(fromMOy1)onatom1iscalculatedbyProbabilityoffinding1electron(fromMOy2)onatom1iscalculatedbyChargeDensitiesSincey1andy2aretheonlytwoMOsoccupiedbyelectrons,theprobabilityoffinding1electron(fromalloft
