结构化学 分子对称性.ppt

,第四章分子的对称性第七章晶体结构的对称性,Symmetryisimportantinquantummechanicsfordeterminingmolecularstructureandforinterpretingspectroscopicinformation.Inadditionofbeingusedtosimplifycalculations,twopropertiesdirectlydependonsymmetry:opticalactivityanddipolemoments.Weconsiderequilibriumconfigurations,withtheatomsintheirmeanpositions.,5,a)具有对称中心的,b)没有对称中心的,a)氨分子（NH3）的三重轴,b)水分子（H2O）的二重轴,反映操作和镜面,垂直于主轴,通过主轴,通过主轴，平分两副轴（C2轴）的夹角,旋转反映操作和映轴,旋转反演操作和反轴,对称元素的组合-11、两个旋转轴的组合交角为2pi/2n的两个C2轴，在其交点上必定出现一个垂直于这两个C2轴的Cn轴；而垂直于Cn轴通过交点的平面内必有n个C2轴,对称元素的组合-22、两个镜面的组合交角为2pi/2n的两个镜面相交，则其交线必为n次轴Cn；Cn轴和通过它的镜面组合，一定存在n个镜面，相邻镜面的夹角为2pi/2n,对称元素的组合-33、偶次旋转轴和与它垂直的镜面的组合一个偶次旋转轴和与它垂直的镜面的组合，必定在其交点上出现对称中心-一个偶次旋转轴和对称中心组合，必有垂直于轴的镜面；，,群的定义,1、封闭性：2、主操作：3、逆操作：4、结合律：,群的实例,群的乘法表,规则：先行后列，（列行）,分子点群的分类,Cn，Cnh，Cnv，Cni，Sn，Dn，Dnh，Dnd,T,Th,Td,O,Oh,I,Ih,26,PointgroupsPointgroupsareawayofclassifyingmoleculesintermsoftheirinternalsymmetry.Moleculescanhavemanysymmetryoperationsthatresultintoindistinguishableconfigurations.Differentcollectionsofsymmetryoperationsareorganizedintogroups.These11groupsweredevelopedbySchoenflies.C1:onlyidentity.Example:CHBrClFCs:onlyareflectionplane.Example:CH2BrClCi:onlyacenterofsymmetry.Example:staggered1,2-dibromo-1,2-dichloroethane.Cn:onlyaCncenterofsymmetry.ExampleofC2:hydrogenperoxide(notcoplanar)Cnv:onlyn-foldaxisandnvertical(ordihedral)mirrorplanes.ExampleofC2v:water;ofC3v:ammoniaCnh:onlyn-foldaxis,ahorizontalmirrorplane,acenterofsymmetryoranimproperaxis.ExampleofC2h:transdichloroethylene;ofC3h:B(OH)3.,27,Dn:OnlyaCnandC2perpendiculartoit(propeller):Dnd:ACn,twoperpendicularC2andadihedralmirrorplanecolinearwiththeprincipalaxis.D2dAllene:H2C=C=CH2.Dnh:ACn,andahorizontalmirrorplaneperpendiculartoCn.D6hbenzeneSn:ASnaxis.S41,3,5,7-tetramethylcyclooctatetraeneSpecial:Linearmolecules:Cv:ifthereisnoaxisperpendiculartotheprincipalaxisDh:ifthereisanaxisperpendiculartotheprincipalaxisTetrahedralmolecules:Td(acubeisTh)Octahedralmolecules:OhIcosahedronanddodecahedronmolecules:IhAsphere,likeanatom,isKh,28,Decisiontree:,分子的偶极矩和极化率,偶极矩：dipole,=qr(库仑米Cm),=4.810-18cmesu=4.8DDebye,1D=3.33610-30Cm,只有属于Cn和Cnv这两类点的分子才可能具有永久偶极矩,诱导偶极矩：在电场E中分子发生诱导极化而产生的,诱=E，分子的极化率,矢量,标量,分子的偶极矩和极化率,诱=E+EE+EEE+，分子的极化率,，分子的第一超极化率，分子的第二超极化率,分子的手性和旋光性,若分子具有反轴对称性，一定没有旋光性,若分子没有反轴对称性，可能具有旋光性,Mod1(R1=R2=Me),Symmetryoperationsobeythelawsofgrouptheory.,Asymmetryoperationcanberepresentedbyamatrixoperatingonabasesetdescribingthemolecule.,Differentbasissetscanbechosen,theyareconnectedbysimilaritytransformations.S-1ASdiagonalblockmatrix,Fordifferentbasissetsthematricesdescribingthesymmetryoperationslookdifferent.However,theircharacter(trace)isthesame!,群的表示,Matrixrepresentationsofsymmetryoperationscanoftenbereducedintoblockmatrices.Similaritytransformationsmayhelptoreducerepresentationsfurther.Thegoalistofindtheirreduciblerepresentation,theonlyrepresentationthatcannotbereducedfurther.,Thesame”type”ofoperations(rotations,reflectionsetc)belongtothesameclass.FormallyRandRbelongtothesameclassifthereisasymmetryoperationSsuchthatR=S-1RS.Symmetryoperationsofthesameclasswillalwayshavethesamecharacter.,群的表示,C,C,C,BlockMatrices,AA=ABB=BCC=C,Blockmatricesaregood,BlockMatrices,Ifamatrixrepresentingasymmetryoperationistransformedintoblockdiagonalformtheneachlittleblockisalsoarepresentationoftheoperationsincetheyobeythesamemultiplicationlaws.,Whenamatrixcannotbereducedfurtherwehavereachedtheirreduciblerepresentation.Thenumberofreduciblerepresentationsofsymmetryoperationsisinfinitebutthereisasmallfinitenumberofirreduciblerepresentations.,Thenumberofirreduciblerepresentationsisalwaysequaltothenumberofclassesofthesymmetrypointgroup.,GroupTheoryII,Asstatedbeforeallrepresentationsofacertainsymmetryoperationhavethesamecharacterandwewillworkwiththemratherthanthematricesthemselves.Thecharactersofdifferentirreduciblerepresentationsofpointgroupsarefoundincharactertables.Charactertablescaneasilybefoundintextbooks.,Reducingbigmatricestoblockdiagonalformisalwayspossiblebutnoteasy.Fortunatelywedonothavetodothisourselves.,CharacterTables,TheC3vcharactertable,Irreduciblerepresentations,Symmetryoperations,Theorderhis6Thereare3classes,CharacterTables,Operationsbelongingtothesameclasswillhavethesamecharactersowecanwrite:,Irreduciblerepresentations(symmetryspecies),Classes,TheGreatOrthogonalityTheorem,”Consideragroupoforderh,andletD(l)(R)betherepresentativeoftheoperationRinadl-dimensionalirreduciblerepresentationofsymmetryspeciesG(l)ofthegroup.Then”Readmoreaboutitinsection4.6.3.,Heresasmallerone,wherec(l)(R)isthecharacteroftheoperation(R).Orevenmoresimpleifthenumberofsymmetryoperationsinaclasscisg(c).Thensincealloperationsbelongingtothesameclasshavethesamecharacter.,TheLittleOrthogonalityTheorem,characterTables,Thereisanumberofusefulpropertiesofcharactertables:,Thesumofthesquaresofthedimensionalityofalltheirreduciblerepresentationsisequaltotheorderofthegroup,Thesumofthesquaresoftheabsolutevaluesofcharactersofanyirreduciblerepresentationisequaltotheorderofthegroup.,Thesumoftheproductsofthecorrespondingcharactersofanytwodifferentirreduciblerepresentationsofthesamegroupiszero.,Thecharactersofallmatricesbelongingtotheoperationsinthesameclassareidenticalinagivenirreduciblerepresentation.,Thenumberofirreduciblerepresentationsinagroupisequaltothenumberofclassesofthatgroup.,Irreduciblerepresentations,Eachirreduciblerepresentationofagrouphasalabelcalledasymmetryspecies,generallynotedG.WhenthetypeofirreduciblerepresentationisdetermineditisassignedaMullikensymbol:One-dimensionalirreduciblerepresentationsarecalledAorB.Two-dimensionalirreduciblerepresentationsarecalledE.Three-dimensionalirreduciblerepresentationsarecalledT(F).Thebasisforanirreduciblerepresentationissaidtospantheirreduciblerepresentation.DontmistaketheoperationEfortheMullikensymbolE!,Irreduciblerepresentations,ThedifferencebetweenAandBisthatthecharacterforarotationCnisalways1forAand-1forB.,Thesubscripts1,2,3etc.arearbitrarylabels.,Subscriptsgandustandsforgeradeandungerade,meaningsymmetricorantisymmetricwithrespecttoinversion.,Superscriptsanddenotessymmetryorantisymmetrywithrespecttoreflectionthroughahorizontalmirrorplane.,characterTables,Example:ThecompleteC4vcharactertable,Thesearebasisfunctionsfortheirreduciblerepresentations.Theyhavethesamesymmetrypropertiesastheatomicorbitalswiththesamenames.,characterTables,Example:ThecompleteC4vcharactertable,A1transformslikez.Edoesnothing,C4rotates90oaboutthez-axis,C2rotates180oaboutthez-axis,svreflectsinverticalplaneandsdinadiagonalplane.,characterTables,A2transformslikearotationaroundz.,ReducibletoIrreduciblerepresentation,Givenageneralsetofbasisfunctionsdescribingamolecule,howdowefindthesymmetryspeciesoftheirreduciblerepresentationstheyspan?,ReducibletoIrreduciblerepresentation,Ifwehaveaninterestingmoleculethereisoftenanaturalchoiceofbasis.Itcouldbecartesiancoordinatesorsomethingmoreclever.,Fromthebasiswecanconstructthematrixrepresentationsofthesymmetryoperationsofthepointgroupofthemoleculeandcalculatethecharactersoftherepresentations.,ReducibletoIrreduciblerepresentation,Howdowefindtheirreduciblerepresentation?Letsuseanoldexamplefromtwoweeksago:,C3vinthebasis(Sn,S1,S2,S3),Tofindthecharactersofthesymmetryoperationswelookathowmanybasiselements”fallontothemselves”(ortheirnegativeself)afterthesymmetryoperation.,E:c=4,C3:c=1,sv:c=2,ReducibletoIrreduciblerepresentation,SoC3vinthebasis(Sn,S1,S2,S3)willhavethefollowingcharactersforthedifferentsymmetryoperations.,ReducibletoIrreduciblerepresentation,SoC3vinthebasis(Sn,S1,S2,S3)willhavethefollowingcharactersforthedifferentsymmetryoperations.,Letsaddthecharactertableoftheirreduciblerepresentation,ByinspectionwefindGred=2A1+E,ReducibletoIrreduciblerepresentation,Thedecompositionofanyreduciblerepresentationintoirreducibleonesisuniqe,soifyoufindcombinationthatworksitisright.,Ifdecompositionbyinspectiondoesnotworkwehavetouseresultsfromthegreatandlittleorthogonalitytheorems(unlesswehaveaninfinitegroup).,ReducibletoIrreduciblerepresentation,FromLOTwecanderivetheexpression(seeEq4.6.2)whereaiisthenumberoftimestheirreduciblerepresentationGiappearsinGred,htheorderofthegroup,lanoperationofthegroup,g(c)thenumberofsymmetryoperationsintheclassofl,credthecharacteroftheoperationlinthereduciblerepresentationandcithecharacteroflintheirreduciblerepresentation.,ReducibletoIrreduciblerepresentation,Letsgobacktoourexampleagain.,SoonceagainwefindGred=2A1+E,ProjectionOperator,Symmetry-adaptedbasesTheprojectionoperatortakesnon-symmetry-adaptedbasisofarepresentationandandprojectsitalongnewdirectionssothatitbelongstoaspecificirreduciblerepresentationofthegroup.,wherePlistheprojectionoperatoroftheirreduciblerepresentationl,c(l)isthecharacteroftheoperationRfortherepresentationlandRmeansapplicationofRtoouroriginalbasiscomponent.,Applications?,Canallofthisactuallybeuseful?Yes,inmanyareasforexamplewhenstudyingelectronicstructureofatomsandmolecules,chemicalreactions,crystallography,stringtheory(Lie-algebra)etc,Letslookatonesimpleexampleconceringmolecularvibrations.MartinJnssonwilltellyoualotmoreinacoupleofweeks.,MolecularVibrations,WaterMolecularvibrationscanalwaysbedecomposedintoquitesimplecomponentscallednormalmodes.,Waterhas9normalmodesofwhich3aretranslational,3arerotationaland3aretheactualvibrations.,Eachnormalmodeformsabasisforanirreduciblerepresentationofthemolecule.,MolecularVibrations,Firstfindabasisforthemolecule.Letstakethecartesiancoordinatesforeachatom.,WaterbelongstotheC2vgroupwhichcontainstheoperationsE,C2,sv(xz)andsv(yz).,TherepresentationbecomesEC2sv(xz)sv(yz)Gred,9,-1,1,3,MolecularVibrations,CharactertableforC2v.,NowreduceGredtoasumofirreduciblerepresentations.Useinspectionortheformula.,MolecularVibrations,TherepresentationreducestoGred=3A1+A2+2B1+3B2,Gtrans=A1+B1+B2,Grot=A2+B1+B2,Gvib=2A