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1、6Grouptheory6.1IntroductionGroup TheoryisoneofthemostpowerfulmathematicaltoolsusedinQuantumChemistryandSpectroscopy.Itallowstheusertopredict,interpret,rationalize,andoftensimplifycomplextheoryanddata.Group theorycanbeconsideredthestudyofsymmetry.Group theoryisabasicstructureofmodernalgebra,consistin
2、gofasetofelementsandanoperation.Group theoryisthesubjectofintensestudywithinmathematics,andisusedinmanyscientificfields.e.g.,groupsareusedinchemistrytodescribethesymmetriesofmolecules,andtheLorentzgroupisacentralpartofspecialrelativity.Also,thetheoryofgroupsplaysacentralroleinparticlephysics,whereit
3、hasledtothediscoveryofnewelementaryparticles.CurlKrotoSmalley1985Fullerenes1990,KratcshmerTheinvolvementofsymmetryinchemistryhasalonghistory;in540BCthesocietyofPythagorasheldthattheearthhadbeenproducedfromthecube,firefromthetetrahedron,airfromtheoctahedron,waterfromtheicosahedron,andtheheavenlyspher
4、efromtheregulardodecahedron.Symmetryexistsallaroundusandmanypeopleseeitasbeingathingofbeauty.Symmetry,is related to equivalence, mutuallycorrespondingarrangementofvariouspartsofabody,producingaproportionate,balancedform.AtitsheartisthefactthattheSet of OperationsassociatedwiththeSymmetry Elementsofa
5、moleculeconstituteamathematicalsetcalledaGroup.ThisallowstheapplicationofthemathematicaltheoremsassociatedwithsuchgroupstotheSymmetry Operations.6.2 Symmetry elements and operationsSymmetryoperationsAsymmetryoperationisdefinedas:movementofamoleculetoaneworientationinwhicheverypointinthemoleculeiscoi
6、ncidentwithanequivalentpoint(orthesamepoint)ofthemoleculeinitsoriginalorientation.SymmetryElementsAsymmetryelement isageometricalentity(aline,planeorpoint)withrespecttowhichoneormoresymmetryoperationsmaybecarriedout.SymmetryelementsandoperationsSymmetry ElementSymmetry ElementSymmetry OperationSymme
7、try Operation SymbolSymbol n n-Foldsymmetryaxis-FoldsymmetryaxisMirrorPlaneMirrorPlaneCenterofInversionCenterofInversionn n-FoldImproperRotation-FoldImproperRotationAxisAxisIdentityIdentityRotationby(2Rotationby(2 / /n n)radians)radiansReflectionReflectionInversionInversionRotationby(2Rotationby(2 /
8、 /n n)radians)radiansfollowedbyreflectionfollowedbyreflectionperpendiculartorotationaxisperpendiculartorotationaxisE EC Cn n i iS Sn n1. Types of symmetry operation(a)Inversion,i (x,y,z)-(-x,-y,-z)in (x,y,z)-(-1)nx,(-1)ny,(-1)nz)Ni(CN)42-C2H4benzeneMatrix representationofainversion:(c)Properrotation
9、s,CCnisarotationabouttheaxisby2/nThus,C2isarotationby180,whileC3isarotationby120.(b)Identity,E,nochangeatalli2n=E, n=integerin=iforoddnPrinciple axisisalwaysdefinedastheaxiswiththe highest order.Matrix representationofaproperrotation:Cnmisarotationabouttheaxisbym2/nNote:Cnn=E=Cn2n=Cn3nCnaxisgenerate
10、snoperations:Cn,Cn2,Cn3Cnn(d)Reflections,v:inaplanewhichcontainstheprincipleaxis(suffixvfor“vertical”).h:inaplaneprincipleaxis(suffixhfor“horizontal”).d:inaplanecontainingprincipleaxisandbisectinglowerorderaxes(suffixdfor“dihedral”or“diagonal”).(xy):(x,y,z)-(x,y,-z)(e)Improperrotations,SSn=CnhN3S2PC
11、l4O2Sn=hCn=Cnh(Cnandhalwayscommute).(Notethatingeneral,R1R2doesnotequalR2R1)2. Operator multiplicationAswasimplicitabove,theconsecutiveapplicationoftwosymmetryoperationsmayberepresentedalgebraicallybytheproductoftheindividualoperations.TheproductoftwooperatorsisdefinedbyTheidentity operator doesnoth
12、ing(ormultipliesbyE)Theassociative lawholdsforoperatorsThecommutative lawdoesnotgenerallyholdforoperators.Ingeneral,e.g.orderC2C2=C22=Ev(yz)v(yz)= Ev(xz)v(xz)= Ev(xz)C2=v(yz)Multiplication tableC C2v2vECEC2 2 v v( (xzxz) ) v v( (yzyz) ) EEC C2 2 v v( (xzxz) ) v v( (yzyz) )ECEC2 2 v v( (xzxz) ) v v(
13、(yzyz) ) CC2 2 EE v v( (yzyz) ) v v( (xzxz) ) v v( (xzxz) ) v v( (yzyz) )ECEC2 2 v v( (yzyz) ) v v( (xzxz) )CC2 2 EEOrder:,6.3 Mathematical groupsAbstractGroupTheoryConsiderasetofobjectsGandaproductruledenotedthatallowsustocombinethem.DenotedFG,whereF,GG .Gcanbeobjectssuchasnumbersorvariables,oroper
14、ators.ExamplesTheintegersandanyofthebinaryoperationsofarithmetic:=+:1+5=6(1)=-:1-5=-45-1(2)(12-3)-7=312-(3-7)=16(3)=:123=4312(notevenaninteger)(4)Notethatsofartherearenorequirementsthatshouldobeycertainrules,suchascommutativityorclosure.Translationsorrotationsofaphysicalobjectintwoorthreedimensions.
15、Heredenotessuccessivetransformations.ClosureRequirethatifF,GG,thenFGGandGFG.NotethatthisdoesnotimplyFG=GF.Forexample,theintegersareclosedunderaddition,multiplication,andsubtraction,butnotunderdivision.SuccessiverotationsandtranslationsinMdimensionsareclosed.AssociativityRequirethatif F,G,HG,wehave(F
16、G)H=F(GH).Forexample,theadditionandmultiplicationofintegersisassociative,whereassubtractionisnot.Successivetranslationsandrotationsareassociative.Identity elementRequirethatinGthereisanelement,theidentity,suchthatE G=G E=G.Fortheintegers,theidentityforadditionis0,formultiplicationitis1;thereisnoiden
17、tityfordivision.Fortranslationstheidentityisthenulloperation,forrotationsitistheidentityrotationwhichisgiveninmatrixformbyaunitmatrix.InverseForeveryelementGGthereexistsanelementdenoted G-1suchthatG-1G=G .G-1=E.Fortheintegers,theinverseofkis-k.Thereisnoinverseundermultiplicationingeneral.Underdivisi
18、oneveryelementmayappeartobeitsowninverse,butthisisnotso,since1isnot theidentity.Foratranslationtheinverseis-1timestheoriginaltranslation.Forarotationtheinverseisthesamerotationintheoppositesense(matrixinverse).CommutativityIfthesetGhasthepropertythatforanytwoelementsF,GGwehave F G -GF=0,thentheeleme
19、ntsofGcommute.Integeradditioniscommutative,andsoisintegermultiplication;integersubtractionisnot.Translationsarecommutative,andsoaresuccessiverotationsaroundthesameaxis.GroupsTheelementsofasetGtogetherwithaproductruleformagroup G if:G,HG,GHG(closure).F,G,HG,F(GH)=(FG )H(associativity).AnelementEG exi
20、stssuchthatEG=GE=GGG(identity).ForeachGGthereexistsanelementG-1G suchthatG-1G=GG-1= E (inverse).IfinadditionGH-HG G,HG,GisAbelian.Multiplication tableC C2v2vECEC2 2 v v( (xzxz) ) v v( (yzyz) ) EEC C2 2 v v( (xzxz) ) v v( (yzyz) )ECEC2 2 v v( (xzxz) ) v v( (yzyz) ) C C2 2 EE v v( (yzyz) ) v v( (xzxz)
21、 ) v v( (xzxz) ) v v( (yzyz) )ECEC2 2 v v( (yzyz) ) v v( (xzxz) )CC2 2 E EG =E,C2,v(xz),v(yz)Wecanseethatsomeelementsmultiplyamongthemselvesonly,formingasubgroup.Theorderofasubgroupmustbeadivisoroftheorderofthegroup(Lagrange).SubgroupsIf HG,and GGbutGH,GHisaleft coset andHGisaright coset.CosetsClassesIfthereisatleastoneXGsuchthatH=XGX-1,G,HGHisconjugate toG.Clearly,ifHisconjugatetoG ,Gisconjugateto H:theyaremutuallyconjugate.AsubsetoftheelementsofGinwhichalltheelementsaremutuallyconjugateiscalledaconjugacy class,orsimplyclass.
