結構化學分子對稱性省公開課金獎全國賽課一等獎微課獲獎課件

第四章分子對稱性第七章晶體結構對稱性1/672/673/67Symmetryisimportantinquantummechanicsfordeterminingmolecularstructureandforinterpretingspectroscopicinformation.Inadditionofbeingusedtosimplifycalculations,twopropertiesdirectlydependonsymmetry:opticalactivityanddipolemoments.Weconsiderequilibriumconfigurations,withtheatomsintheir

meanpositions.4/6755/67a)含有對稱中心b)沒有對稱中心6/677/67a)氨分子（NH3）三重軸b)水分子（H2O）二重軸8/679/67反應操作和鏡面10/6711/67σ垂直于主軸12/67σ經(jīng)過主軸13/67σ經(jīng)過主軸，平分兩副軸（C2軸）夾角14/67旋轉反應操作和映軸15/6716/67旋轉反演操作和反軸17/6718/67對稱元素組合-11、兩個旋轉軸組合交角為2pi/2n兩個C2軸，在其交點上必定出現(xiàn)一個垂直于這兩個C2軸Cn軸；而垂直于Cn軸經(jīng)過交點平面內(nèi)必有n個C2軸19/67對稱元素組合-22、兩個鏡面組合交角為2pi/2n兩個鏡面相交，則其交線必為n次軸Cn；Cn軸和經(jīng)過它鏡面組合，一定存在n個鏡面，相鄰鏡面夾角為2pi/2n20/67對稱元素組合-33、偶次旋轉軸和與它垂直鏡面組合一個偶次旋轉軸和與它垂直鏡面組合，必定在其交點上出現(xiàn)對稱中心---〉一個偶次旋轉軸和對稱中心組合，必有垂直于軸鏡面；，，，21/67群定義1、封閉性：2、主操作：3、逆操作：4、結合律：22/67群實例23/67群乘法表規(guī)則：先行后列，（列行）24/67分子點群分類Cn，Cnh，Cnv，Cni，Sn，Dn，Dnh，Dnd,

T,

Th,

Td,

O,

Oh,

I,

Ih25/6726PointgroupsPointgroupsareawayofclassifyingmoleculesintermsoftheirinternalsymmetry.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,acenterofsymmetryoran improperaxis.

ExampleofC2h:transdichloroethylene;ofC3h:B(OH)3.26/6727Dn: OnlyaCnandC2perpendiculartoit(propeller):Dnd: ACn,twoperpendicularC2andadihedralmirrorplanecolinear withtheprincipalaxis.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,isKh27/6728Decisiontree:28/67分子偶極矩和極化率偶極矩：dipole,μ=qr(庫侖米Cm)μ=4.8×10-18cmesu=4.8DDebye1D=3.336×10-30Cm只有屬于Cn和Cnv

這兩類點分子才可能含有永久偶極矩誘導偶極矩：在電場E中分子發(fā)生誘導極化而產(chǎn)生μ誘=αE，α分子極化率矢量標量29/67分子偶極矩和極化率μ誘=αE+βEE+γEEE+，α分子極化率，β分子第一超極化率，

γ分子第二超極化率

30/67分子手性和旋光性若分子含有反軸對稱性，一定沒有旋光性若分子沒有反軸對稱性，可能含有旋光性31/6732/67entrycatalystsolventhyield[%][b]ee[%][c]1HBC-1CH2Cl22474502HBC-2CH2Cl22489-703HBC-3CH2Cl22489724HBC-4CH2Cl22460625HBC-5CH2Cl22492826HBC-6CH2Cl22498807[d]HBC-6CH2Cl24085868[d]HBC-6CHCl31694869[d]HBC-6Et2O120838010[d]HBC-6CH3CN30838011[d]HBC-6CH3OH72946412[e]HBC-6CHCl36d929233/67Mod1(R1=R2=Me)34/67Symmetryoperationsobeythelawsofgrouptheory.Asymmetryoperationcanberepresentedbyamatrixoperatingonabasesetdescribingthemolecule.Differentbasissetscanbechosen,theyareconnectedbysimilaritytransformations.S-1AS

diagonalblockmatrixFordifferentbasissetsthematricesdescribingthesymmetryoperationslookdifferent.However,theircharacter(trace)isthesame!群表示35/67Matrixrepresentationsofsymmetryoperationscanoftenbereducedintoblockmatrices.Similaritytransformationsmayhelptoreducerepresentationsfurther.Thegoalistofindtheirreduciblerepresentation,theonlyrepresentationthatcannotbereducedfurther.Thesame”type”ofoperations(rotations,reflectionsetc)belongtothesameclass.FormallyRandR’belongtothesameclassifthereisasymmetryoperationSsuchthatR’=S-1RS.Symmetryoperationsofthesameclasswillalwayshavethesamecharacter.群表示36/67C’C’’CBlockMatricesA’A’’=AB’B’’=BC’C’’=CBlockmatricesaregood37/67BlockMatricesIfamatrixrepresentingasymmetryoperationistransformedintoblockdiagonalformtheneachlittleblockisalsoarepresentationoftheoperationsincetheyobeythesamemultiplicationlaws.Whenamatrixcannotbereducedfurtherwehavereachedtheirreduciblerepresentation.Thenumberofreduciblerepresentationsofsymmetryoperationsisinfinitebutthereisasmallfinitenumberofirreduciblerepresentations.Thenumberofirreduciblerepresentationsisalwaysequaltothenumberofclassesofthesymmetrypointgroup.38/67GroupTheoryIIAsstatedbeforeallrepresentationsofacertainsymmetryoperationhavethesamecharacterandwewillworkwiththemratherthanthematricesthemselves.Thecharactersofdifferentirreduciblerepresentationsofpointgroupsarefoundincharactertables.Charactertablescaneasilybefoundintextbooks.Reducingbigmatricestoblockdiagonalformisalwayspossiblebutnoteasy.Fortunatelywedonothavetodothisourselves.39/67CharacterTablesTheC3vcharactertableIrreduciblerepresentationsSymmetryoperationsTheorderhis6Thereare3classes40/67CharacterTablesOperationsbelongingtothesameclasswillhavethesamecharactersowecanwrite:Irreduciblerepresentations(symmetryspecies)Classes41/67TheGreatOrthogonalityTheorem”Consideragroupoforderh,andletD(l)(R)betherepresentativeoftheoperationRinadl-dimensionalirreduciblerepresentationofsymmetryspeciesG(l)ofthegroup.Then

”Readmoreaboutitinsection4.6.3.42/67Here’sasmallerone,wherec(l)(R)isthecharacteroftheoperation(R).Orevenmoresimpleifthenumberofsymmetryoperationsinaclasscisg(c).Thensincealloperationsbelongingtothesameclasshavethesamecharacter.TheLittleOrthogonalityTheorem

43/67

characterTablesThereisanumberofusefulpropertiesofcharactertables:ThesumofthesquaresofthedimensionalityofalltheirreduciblerepresentationsisequaltotheorderofthegroupThesumofthesquaresoftheabsolutevaluesofcharactersofanyirreduciblerepresentationisequaltotheorderofthegroup.Thesumoftheproductsofthecorrespondingcharactersofanytwodifferentirreduciblerepresentationsofthesamegroupiszero.Thecharactersofallmatricesbelongingtotheoperationsinthesameclassareidenticalinagivenirreduciblerepresentation.Thenumberofirreduciblerepresentationsinagroupisequaltothenumberofclassesofthatgroup.44/67IrreduciblerepresentationsEachirreduciblerepresentationofagrouphasalabelcalledasymmetryspecies,generallynotedG.WhenthetypeofirreduciblerepresentationisdetermineditisassignedaMullikensymbol:One-dimensionalirreduciblerepresentationsarecalledAorB.Two-dimensionalirreduciblerepresentationsarecalledE.Three-dimensionalirreduciblerepresentationsarecalledT(F).Thebasisforanirreduciblerepresentationissaidtospantheirreduciblerepresentation.Don’tmistaketheoperationEfortheMullikensymbolE!45/67IrreduciblerepresentationsThedifferencebetweenAandBisthatthecharacterforarotationCnisalways1forAand-1forB.Thesubscripts1,2,3etc.arearbitrarylabels.Subscriptsgandustandsforgeradeandungerade,meaningsymmetricorantisymmetricwithrespecttoinversion.Superscripts’and’’denotessymmetryorantisymmetrywithrespecttoreflectionthroughahorizontalmirrorplane.46/67characterTablesExample:ThecompleteC4vcharactertableThesearebasisfunctionsfortheirreduciblerepresentations.Theyhavethesamesymmetrypropertiesastheatomicorbitalswiththesamenames.47/67characterTablesExample:ThecompleteC4vcharactertableA1transformslikez.Edoesnothing,C4rotates90oaboutthez-axis,C2rotates180oaboutthez-axis,svreflectsinverticalplaneandsdinadiagonalplane.48/67characterTablesA2transformslikearotationaroundz.E+RzC4+RzC2+Rzsv-Rzsd-Rz49/67ReducibletoIrreduciblerepresentationGivenageneralsetofbasisfunctionsdescribingamolecule,howdowefindthesymmetryspeciesoftheirreduciblerepresentationstheyspan?50/67ReducibletoIrreduciblerepresentationIfwehaveaninterestingmoleculethereisoftenanaturalchoiceofbasis.Itcouldbecartesiancoordinatesorsomethingmoreclever.Fromthebasiswecanconstructthematrixrepresentationsofthesymmetryoperationsofthepointgroupofthemoleculeandcalculatethecharactersoftherepresentations.51/67ReducibletoIrreduciblerepresentationHowdowefindtheirreduciblerepresentation?Let’suseanoldexamplefromtwoweeksago:123NC3vinthebasis(Sn,S1,S2,S3)Tofindthecharactersofthesymmetryoperationswelookathowmanybasiselements”fallontothemselves”(ortheirnegativeself)afterthesymmetryoperation.E:c=4C3:c=1sv:c=252/67ReducibletoIrreduciblerepresentation123NSoC3vinthebasis(Sn,S1,S2,S3)willhavethefollowingcharactersforthedifferentsymmetryoperations.53/67ReducibletoIrreduciblerepresentation123NSoC3v

inthebasis(Sn,S1,S2,S3)willhavethefollowingcharactersforthedifferentsymmetryoperations.Let’saddthecharactertableoftheirreduciblerepresentationByinspectionwefindGred=2A1+E54/67ReducibletoIrreduciblerepresentationThedecompositionofanyreduciblerepresentationintoirreducibleonesisuniqe,soifyoufindcombinationthatworksitisright.Ifdecompositionbyinspectiondoesnotworkwehavetouseresultsfromthegreatandlittleorthogonalitytheorems

(unlesswehaveaninfinitegroup).55/67ReducibletoIrreduciblerepresentationFromLOTwecanderivetheexpression(seeEq

4.6.2)whereaiisthenumberoftimestheirreduciblerepresentationGiappearsinGred,htheorderofthegroup,lanoperationofthegroup,g(c)thenumberofsymmetryoperationsintheclassofl,credthecharacteroftheoperationlinthereduciblerepresentationandcithecharacteroflintheirreduciblerepresentation.56/67ReducibletoIrreduciblerepresentationLet’sgobacktoourexampleagain.SoonceagainwefindGred=2A1+E

57/67ProjectionOperatorSymmetry-adaptedbasesTheprojectionoperatortakesnon-symmetry-adaptedbasisofarepresentationandandprojectsitalongnewdirectionssothatitbelongstoaspecificirreduciblerepresentationofthegroup.wherePlistheprojectionoperatoroftheirreduciblerepresentationl,c(l)isthecharacteroftheoperationRfortherepresentationlandRmeansapplicationofRtoouroriginalbasiscomponent.^^58/67Applications?Canallofthisactuallybeuseful?Yes,inmanyareasforexamplewhenstudyingelectronicstructureofatomsandmolecules,chemicalreactions,crystallography,stringtheory(Lie-algebra)etc…Let’slookatonesimpleexampleconceringmolecularvibrations.MartinJ?nssonwilltellyoualotmoreinacoupleofweeks.59/67MolecularVibrationsWaterMolecularvibrationscanalwaysbedecomposedintoquitesimplecomponentscallednormalmodes.Waterhas9normalmodesofwhich3aretranslational,3arerotationaland3aretheactualvibrations.Eachnormalmodeformsabasisforanirreduciblerepresentationofthemolecule.60/67MolecularVibrationsFirstfindabasisforthemolecule.Let’stakethecartesiancoordinatesforeachatom.x1x3x2y1y2y3z1z2z3WaterbelongstotheC2vgroupwhichcontainstheoperationsE,C2,sv(xz)andsv’(yz).Therepresentationbecomes

E C2

sv(xz) sv’(yz)Gred

9-11361/67MolecularVibrationsCharactertableforC2v.NowreduceGredtoasumofirreduciblerepresentations.Useinspectionortheformula.62/67MolecularVibrationsTherep