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 C v ifthereisnoaxisperpendiculartotheprincipalaxisD h 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 8 10 18cmesu 4 8DDebye 1D 3 336 10 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 1AS diagonalblockmatrix Fordifferentbasissetsthematricesdescribingthesymmetryoperationslookdifferent However theircharacter trace isthesame 群的表示 Matrixrepresentationsofsymmetryoperationscanoftenbereducedintoblockmatrices Similaritytransformationsmayhelptoreducerepresentationsfurther Thegoalistofindtheirreduciblerepresentation theonlyrepresentationthatcannotbereducedfurther Thesame type ofoperations rotations reflectionsetc belongtothesameclass FormallyRandR belongtothesameclassifthereisasymmetryoperationSsuchthatR S 1RS Symmetryoperationsofthesameclasswillalwayshavethesamecharacter 群的表示 C C C BlockMatrices A A AB B BC C 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 Here sasmallerone 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 Don tmistaketheoperationEfortheMullikensymbolE Irreduciblerepresentations ThedifferencebetweenAandBisthatthecharacterforarotationCnisalways1forAand 1forB Thesubscripts1 2 3etc arearbitrarylabels Subscriptsgandustandsforgeradeandungerade meaningsymmetricorantisymmetricwithrespecttoinversion Superscripts and denotessymmetryorantisymmetrywithrespecttoreflectionthroughahorizontalmirrorplane 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 Let suseanoldexamplefromtwoweeksago 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 Let saddthecharactertableoftheirreduciblerepresentation ByinspectionwefindGred 2A1 E ReducibletoIrreduciblerepresentation Thedecompositionofanyreduciblerepresentationintoirreducibleonesisuniqe soifyoufindcombinationthatworksitisright Ifdecompositionbyinspectiondoesnotworkwehavetouseresultsfromthegreatandlittleorthogonalitytheorems unlesswehaveaninfinitegroup ReducibletoIrreduciblerepresentation FromLOTwecanderivetheexpression seeEq4 6 2 whereaiisthenumberoftimestheirreduciblerepresentationGiappearsinGred htheorderofthegroup lanoperationofthegroup g c thenumberofsymmetryoperationsintheclassofl credthecharacteroftheoperationlinthereduciblerepresentationandcithecharacteroflintheirreduciblerepresentation ReducibletoIrreduciblerepresentation Let sgobacktoourexampleagain SoonceagainwefindGred 2A1 E ProjectionOperator Symmetry adaptedbasesTheprojectionoperatortakesnon symmetry adaptedbasisofarepresentationandandprojectsitalongnewdirectionssothatitbelongstoaspecificirreduciblerepresentationofthegroup wherePlistheprojectionoperatoroftheirreduciblerepresentationl c l isthecharacteroftheoperationRfortherepresentationlandRmeansapplicationofRtoouroriginalbasiscomponent Applications Canallofthisactuallybeuseful Yes inmanyareasforexamplewhenstudyingelectronicstructureofatomsandmolecules chemicalreactions crystallography stringtheory Lie algebra etc Let slookatonesimpleexampleconceringmolecularvibrations MartinJ nssonwilltellyoualotmoreinacoupleofweeks MolecularVibrations WaterMolecularvibrationscanalwaysbedecomposedintoquitesimplecomponentscallednormalmodes Waterhas9normalmodesofwhich3aretranslational 3arerotationaland3aretheactualvibrations Eachnormalmodeformsabasisforanirreduciblerepresentationofthemolecule MolecularVibrations Firstfindabasisforthemolecule Let stakethecartesiancoordinatesforeachatom 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