group theory - lecture notes complete 3rd ed
DESCRIPTION
group theory and chemistry.TRANSCRIPT
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CourseLectureNotes 3rd Edition
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IntroductiontotheChemicalApplicationsofGroupTheory
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AcknowledgmentsandWebResourcesTheselecturenoteshavebeenderivedfromseveralsourcesincludingGroupTheoryandChemistrybyDavidM.BishopISBN13:9780486673554andChemicalApplicationsofGroupTheorybyF.AlbertCottonISBN10:0471175706.PicturesofmolecularorbitalswerecalculatedusingFirefly.TheorbitalswereconvertedtocubeformatwithMoldenandrenderedwithPyMol.Forhelpwithsymmetryoperationsandsymmetryelementssee:http://www.molwave.com/software/3dmolsym/3dmolsym.htmAnimationsofmolecularvibrationscanbeseenhere:http://www.molwave.com/software/3dnormalmodes/3dnormalmodes.htm
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TableofContentsIntroduction.....................................................................................................................................6SymmetryElementsandSymmetryOperations..............................................................6IdentityE.......................................................................................................................................7RotationC.....................................................................................................................................7Reflection..................................................................................................................................8Inversioni.....................................................................................................................................8ImproperRotationS.................................................................................................................9ImmediateApplicationsofSymmetry...............................................................................10SymmetryOperations...............................................................................................................11AlgebraofOperators.................................................................................................................12Specialcaseoflinearoperators............................................................................................12Algebraoflinearoperators....................................................................................................13SumLaw.........................................................................................................................................13ProductLaw..................................................................................................................................13AssociativeLaw...........................................................................................................................13DistributiveLaw.........................................................................................................................13AlgebraofSymmetryOperators..........................................................................................14
AssociativeLaw:..................................................................................................................14DistributiveLaw:................................................................................................................14
DefinitionofaGroup.................................................................................................................16Summary........................................................................................................................................16ExampleGroups..........................................................................................................................17GroupMultiplicationTables..................................................................................................18RearrangementTheorem:......................................................................................................18Classes.............................................................................................................................................20SimilarityTransforms..............................................................................................................20
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PointGroups.................................................................................................................................22ClassificationofPointGroups...............................................................................................23SystematicMethodtoAssignPointGroups....................................................................24ClassesinSymmetryPointGroups.....................................................................................26PropertiesofMatrices..............................................................................................................28Matrixmathbasics.....................................................................................................................28
AdditionandSubtraction................................................................................................28Matrixmultiplication........................................................................................................29MatrixDivision....................................................................................................................29SpecialMatrices...................................................................................................................30
MatrixRepresentationsofSymmetryOperations........................................................31Identity....................................................................................................................................31Reflection...............................................................................................................................31Inversion.................................................................................................................................31Rotation..................................................................................................................................32ImproperRotations...........................................................................................................33
VectorsandScalarProducts..................................................................................................35RepresentationsofGroups.....................................................................................................36TheGreatOrthogonalityTheorem......................................................................................38IrreducibleRepresentations..................................................................................................39TheReductionFormula...........................................................................................................44CharacterTables.........................................................................................................................45RegionIMullikenSymbolsforIrreducibleRepresentations...............................45RegionIICharacters..............................................................................................................46RegionIIITranslationsandRotations...........................................................................46RegionIVBinaryProducts...................................................................................................47WritingChemicallyMeaningfulRepresentations.........................................................47
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Vibrations......................................................................................................................................48SelectionRulesforVibrations...............................................................................................50SelectionRulesforRamanSpectroscopy.........................................................................51
NormalCoordinateAnalysis..........................................................................................52IRandRamanSpectraofCH4andCH3F............................................................................60ProjectionOperator...................................................................................................................65BondingTheories.......................................................................................................................68LewisBondingTheory.............................................................................................................68VSEPRValenceShellElectronPairRepulsionTheory............................................68ValanceBondTheory................................................................................................................69HybridOrbitalTheory..............................................................................................................70MolecularOrbitalTheory........................................................................................................71
Quantummechanicaldescriptionoforbitals..........................................................73GroupTheoryandQuantumMechanics...........................................................................74LCAOApproximation................................................................................................................75electronApproximation......................................................................................................75HckelOrbitalMethod.........................................................................................................76HckelOrbitalsforNitrite......................................................................................................77HckelMOsforCyclobutadiene..........................................................................................84HckelMOsforBoronTrifluoride.....................................................................................90Onefinalexercise....................................................................................................................95
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IntroductionSymmetry:Relationshipbetweenpartsofanobjectwithrespecttosize,shapeandposition.Easytorecognizesymmetryinnature:Flowers,leaves,animalsetc.GroupTheorydevelopedinthelate1700s.Early1800svaristeGalois18111832inventedmuchofthefundamentalsofgrouptheory.Thiscoincidedwithdevelopmentsinmatrixmathematics.Chemistsuseasubsetofgrouptheorycalledrepresentationtheory.GroupcharacterswereprimarilytheworkofGeorgeFrobenious18491917EarlychemicalapplicationstoquantummechanicscamefromtheworkofHermannWeyl18851955andEugeneWigner19021995
SymmetryElementsandSymmetryOperationsAsymmetryelementisageometricentitypoint,lineorplaneAsymmetryoperatorperformsandactiononathreedimensionalobject
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Symmetryoperatorsaresimilartoothermathematicaloperators,,,log,cos,etcWewillbeuseonlyfivetypesofoperatorsinthissubjectOperator SymbolIdentity ERotation CMirrorplane Inversion iImproperrotation SAllsymmetryoperatorsleavetheshapemoleculeinanequivalentposition,i.e.itisindistinguishablebeforeandaftertheoperatorhasperformeditsaction.Identity(E)Thisoperatordoesnothingandisrequiredforcompleteness.Equivalenttomultiplyingby1oradding0inalgebra.Rotation(C)Rotateclockwisearoundanaxisby2/niftherotationbringstheshapemoleculeintoanequivalentposition.Thesymmetryelementiscalledtheaxisofsymmetry.Fora2\nrotationthereisannfoldaxisofsymmetry.ThisisdenotedasCn.Manymoleculeshavemorethanonesymmetryaxis.Theaxiswiththelargestniscalledtheprincipalaxis.
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ConsiderasquareplanarmoleculelikePtCl4.
possiblerotations. and WeclassifythisasE,2C4,C2.TherearealsotwootherC2axesalongthebondsandbetweenthebondsReflection()Theshapemoleculeisreflectedthroughaplane.spiegelisGermanformirrorIfaplaneistotheprincipalrotationaxisthenitiscalledhhorizontal.Ifitisalongtheprincipalaxisthenitiscalledvvertical.Theremaybemorethanonev.Iftheplanebisectsananglebetween3atomsthenitiscalledddihedral.Thereflectionplaneisthesymmetryelement.Inversion(i)Allpointsintheshapemoleculearereflectedthoughasinglepoint.Thepointisthesymmetryelementforinversion.Thisturnsthemoleculeinsideoutinasense.Thesymmetryelementisthepointthroughwhichtheshapeisinverted.
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ImproperRotation(S)Rotationby2/nfollowedbyreflection,totherotationaxis.SinceperformingtwotimesisthesameasdoingnothingE,Scanonlybeperformedanoddnumberoftime. ifkisodd ifkisevenkmustbeanoddvaluee.g. and Additionally ifnisodd ifnisevenThesymmetryelementforSistherotationaxis.
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ImmediateApplicationsofSymmetryDipoleMomentsIfamoleculehasadipolemomentthenthedipolemustliealongthesymmetryelementslines,planes.
Ifamoleculehasaxisofrotation,thennodipoleexists. Ifthereisa,thenthedipolemustliewithintheplane.Ifthereare
multiplethedipolemustlieattheintersectionoftheplanes. Ifamoleculehasaninversioncenterithennodipoleexists.
ExampleswithH2O,NH3,PtCl4.OpticalActivityIngeneral,ifamoleculehasanimproperrotationSn,thenitisopticallyinactive.Thisisbecause,amoleculewithanSnisalwayssuperimposableonitsmirrorimage.
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SymmetryOperationsIdentifyingallsymmetryelementsandoperationsinmolecules.CyclopropaneD3h
E,2C3,3C2,h,2S3,3vThereisanandanalsocalled EthanestaggeredD3d
E,2C3,3C2,i,2S6,3d1,3,5trihydroxybenzeneplanarC3h
E,2C3,h2S3Thereisanandanalsocalled
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AlgebraofOperatorsAnoperatorisasymbolfortheoperationrotation,reflection,etce.g.C3istheoperatorfortheoperationofclockwiserotationby2/3Operatorscanoperateonfunctionsfxtogeneratenewfunctionse.g.Omultiplyby3fx23x2ThenOfx69x2Ocanbedefinedanywaywelike,d/dx,2,etcSpecialcaseoflinearoperatorsLinearoperatorshavethefollowingpropertyOf1f2Of1Of2AndOkf1kOf1wherekisaconstantDifferentiationclearlyisalinearoperator
2 1or 2 1
and 3
3 6
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Algebraoflinearoperators1. SumLaw2. ProductLaw3. AssociativeLaw4. DistributiveLaw
SumLaw
ProductLaw
O2operatesfirsttoproduceanewfunction,thenO1operatestoproduceanothernewfunction.Note:theorderofoperationsisimportanthere.O1O2maynotbethesameasO2O1,i.e.operatorsdonotnecessarilycommutewitheachother.AssociativeLaw
2nd1st2nd1st
DistributiveLaw
and
Dosymmetryoperatorsobeytheselaws?theydo
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consideracetoneE,C2,v1,v2SumLaw:thereisnoprocesstoaddsymmetryoperatorsAlgebraofSymmetryOperatorsProductLaw:Wedefinetheproductofsymmetryoperatorsas:dooneoperationfollowedbyanother:e.g.PQfmeansapplyQtofandthenapplyPtotheresultwherePandQaresomesymmetryoperation.Or,alternativelyPQRwhereRisalsoasymmetryoperation.C2v1fC2v1fC2v1v2andC2v1fresultsinthesameconfigurationasv2fAssociativeLaw:C2v1v2f C2v1v2fv1v2C2 C2v1v2C2C2E v2v2EGenericallyPQRPQRDistributiveLaw:Thereisnoprocesstoaddsymmetryoperators
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ConsidertheammoniamoleculeC3v
E,2C3,3vNoteherethat Iftwooperatorscombinetogivetheidentity,wesaythattheyareinversetoeachother.
Itisalsotruethat orgenericallyPQQPEi.e.symmetryoperatorsthatareinversetooneanothercommute.hhEiiEmirrorplanesarealwaysinversetothemselves,likewiseinversionisalwaysinversetoitself.GenericallywewritePQ1P1Q1
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DefinitionofaGroupTherearefourdefiningrulesforgroups.
1. Thecombinationofanytwoelementsaswellasthesquareofeachelementmustbeinthegroup.
Combiningrulecanbedefinedasanythingmultiplication,differentiation,onefollowedbyanother,etcPQR;RmustbeinthegroupThecommutativelawmaynotholdABBA
2. Oneelementmustcommutewithallotherelementsandleavethemunchanged.Thatis,anidentityelementmustbepresent.
ERRER;Emustbeinthegroup3. Theassociativelawmusthold.
PQRPQR;forallelements4. Everyelementmusthaveaninversewhichisalsointhegroup.
RR1R1RE;R1mustbeinthegroupSummaryDefinitionofagroupPQR
RmustbeinthegroupERRER
EmustbeinthegroupPQRPQR
forallelementsRR1R1RE
R1mustbeinthegroup
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ExampleGroupsWithacombiningruleofaddition,allintegersformagroup.Theidentityelementis0,andtheinverseofeachelementisthenegativevalue.Thisisanexampleofaninfinitegroup.Withacombiningruleofmultiplication,wecanformafinitegroupwiththefollowingseti,i,1,1Theidentityelementis1inthiscase.Asetofmatricescanalsoformafinitegroupwiththecombiningruleofmatrixmultiplication.
1 0 0 00 1 0 00 0 1 00 0 0 1
0 1 0 01 0 0 00 0 0 10 0 1 0
0 0 0 10 0 1 00 1 0 01 0 0 0
0 0 1 00 0 0 11 0 0 00 1 0 0
Theidentitymatrixis1 0 0 00 1 0 00 0 1 00 0 0 1
e.g.
0 1 0 01 0 0 00 0 0 10 0 1 0
0 0 0 10 0 1 00 1 0 01 0 0 0
0 0 1 00 0 0 11 0 0 00 1 0 0
aikelementintheithrowandkthcolumn
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Lastly,thesetofsymmetryoperatorsnotsymmetryelementspresentforagivenmolecularshapeformsagroupwiththecombiningruleofonefollowedbyanother.Thesetypesofgroupsarecalledpointgroups.
GroupMultiplicationTablesThenumberofelementssymmetryoperatorsinthegroupiscalledtheorderofthegrouphRearrangementTheorem:Inagroupmultiplicationtable,eachrowandcolumnlistseachelementinthegrouponceandonlyonce.Notworowsortwocolumnsmaybeidentical.Consideragroupoforder3G3 E A BE E A BA A ? ?B B ? ?TherearetwooptionsforfillingoutthetableAABorAAEIfAAEthenthetablebecomesG3 E A BE E A BA A E BB B A EThisviolatestherearrangementtheoremasthelasttwocolumnshaveelementsthatappearmorethanonce.
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TheonlysolutionforgroupG3isG3 E A BE E A BA A B EB B E ANote:ThegroupG3isamemberofasetofgroupscalledcyclicgroups.CyclicgroupshavethepropertyofbeingAbelian,thatisallelementscommutewitheachother.Acyclicgroupisonewhicheveryelementcanbegeneratedbyasingleelementanditspowers.InthiscaseAAandAAA2BandAAAA3E.Therearetwopossiblegroupsoforder4G4 E A B CE E A B CA A B C EB B C E AC C E A B
G4 E A B CE E A B CA A E C BB B C E AC C B A E
InthesecondcaseofG4thereisasubgroupoforder2present.G2 E AE E AA A ETheorderofasubgroupgmustbeadivisoroftheorderofthemaingrouph,thatish/gk,wherekisaninteger.
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ClassesGroupscanfurtherbedividedintosmallersetscalledclasses.SimilarityTransformsIfA,BandXareinagroupandX1AXBwesaythatBissimilaritytransformofAbyX.WealsocansaythatAandBareconjugateofeachother.Conjugateelementshavethefollowingproperties
1 AllelementsareconjugatewiththemselvesAX1AXforsomeX
2 IfAisconjugatetoB,thenBisconjugatetoAAX1BXandBY1AYwithX,Yinthegroup
3 IfAisconjugatetoBandCthenBandCarealsoconjugatesofeachother.
Thecompletesetofelementsoperationsthatareconjugatetoeachotheriscalledaclass.FindtheclassesinG6G6 E A B C D FE E A B C D FA A E D F B CB B F E D C AC C D F E A BD D C A B F EF F B C A E DEisinaclassbyitselfoforder1A1EAEetc..
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OtherclassesinG6E1AEAA1AAAB1ABCC1ACBD1ADBF1AFCWeseeherethattheelementsA,BandCareallconjugatetoeachotherandformaclassoforder3.E1DEDA1DAFB1DBFC1DCFD1DDDF1DFDWeseeherethattheelementsDandFareconjugatetoeachotherandformaclassoforder2.
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PointGroupsConsiderallofthesymmetryoperationsinNH3
EvvvC3NH3 E v v v C3 E E v v v C3 v v E C3 v vv v E C3 v vv v C3 E v vC3 C3 v v v E v v v E C3NotethatalloftherulesofagroupareobeyedforthesetofallowedsymmetryoperationsinNH3.G6 E A B C D FE E A B C D FA A E D F B CB B F E D C AC C D F E A BD D C A B F EF F B C A E DComparethemultiplicationtableofNH3tothatofG6.
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Thereisa1:1correspondencebetweentheelementsineachgroupEEvAvBvCC3DCFGroupsthathavea1:1correspondencearesaidtobeisomorphictoeachother.Ifthereisamorethan1:1correspondencebetweentwogroups,theyaresaidtobehomomorphictoeachother.AllgroupsarehomomorphicwiththegroupE.i.e.AE,BE,CEetcClassificationofPointGroupsSchoenfliesNotation
GroupName
EssentialSymmetryElements*
Cs oneCi oneiCn oneCnDn oneCn plusnC2 toCnCnv oneCn plusnvCnh oneCn plushDnh thoseofDn plushDnd thoseofDn plusdSnevenn oneSnTd tetrahedron
SpecialGroupsOh octahedronIh icosahedronsHh sphere
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SystematicMethodtoAssignPointGroups
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Assignthepointgroupstothefollowingmolecules
ionlyCi
CnC3
C3h C3v
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ClassesinSymmetryPointGroupsYoucantestallpossiblesimilaritytransformstofindtheconjugateelements.X1AXB,howeverthisistediousandwithsymmetryelementsitismucheasiertosortclasses.Twooperationsbelongtothesameclassifonemaybereplacedbyanotherinanewcoordinatesystemwhichisaccessiblebyanallowedsymmetryoperationinthegroup.ConsiderthefollowingforaD4hgroup
C4x,yy,xandx,yy,xReflectthecoordinatesystembyd
x,yy,xandx,yy,xBychangingthecoordinatesystemwehavesimplyreplacetherolesthatC4andplay.Thatis and
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Moregenerallywecanstatethefollowing
1. E,iandharealwaysinaclassbythemselves.
2. and areinthesameclassforeachvalueofkaslongasthereisaplaneofsymmetryalongtheaxisoraC2to.Ifnotthenand areinclassesbythemselves.Likewiseforand .
3. andareinthesameclassifthereisanoperationwhichmovesoneplaneintotheother.Likewiseforandthatarealongdifferentaxes.
ConsidertheelementsofD4hsquareplaneThereare10classesinthisgroupwithorder14
E and ihvandvdanddC2alongzandxyandx,yand
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PropertiesofMatricesMatrix:rectangulararrayofnumbersorelements
aijithrowandjthcolumn
Avectorisaonedimensionalmatrix
ThiscouldbeasetofCartesiancoordinatesx,y,z
MatrixmathbasicsAdditionandSubtractionMatricesmustbethesamesizeCijAijBij addorsubtractthecorrespondingelementsineachmatrixMultiplicationbyascalarkkaijkaij everyelementismultipliedbytheconstantk
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Matrixmultiplication
aikelementintheithrowandkthcolumn
Where c11a11b11a12b21 c12a11b12a12b22 etcmatrixmultiplicationisnotcommutativeABBAMatrixDivisionDivisionisdefinedasmultiplyingbytheinverseofamatrix.Onlysquarematricesmayhaveaninverse.TheinverseofamatrixisdefinedasAA1ij ijKroneckerdelta
ij1ifijotherwiseij0
1 0 00 1 00 0 1
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SpecialMatricesBlockdiagonalmatrixmultiplication
1 0 0 0 0 01 2 0 0 0 00 0 3 0 0 00 0 0 1 3 20 0 0 1 2 20 0 0 4 0 1
4 1 0 0 0 02 3 0 0 0 00 0 1 0 0 00 0 0 0 1 20 0 0 3 0 20 0 0 2 1 1
4 1 0 0 0 08 7 0 0 0 00 0 3 0 0 00 0 0 13 3 100 0 0 10 3 80 0 0 2 5 9
Eachblockismultipliedindependentlyi.e.1 01 2
4 12 3
4 18 7
31 3
1 3 21 2 24 0 1
0 1 23 0 22 1 1
13 3 1010 3 82 5 9
SquareMatrices Thisisthesumofthediagonalelementsofamatrixtrace.AiscalledthecharacterofmatrixApropertiesof ifCABandDBAthenCDconjugatematriceshaveidenticalPB1PBthenRBTherefore,operationsthatareinthesameclasshavethesamecharacter.
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MatrixRepresentationsofSymmetryOperationsWewillnowusematricestorepresentsymmetryoperations.Considerhowanx,y,zvectoristransformedinspaceIdentityE
1 0 00 1 00 0 1
Reflectionxy
1 0 00 1 00 0 1
xz
1 0 00 1 00 0 1
Inversioni
1 0 00 1 00 0 1
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RotationCnaboutthezaxis
? ? 0? ? 00 0 1
Thezcoordinateremainsunchanged.
Consideracounterclockwiserotationbyaboutthezaxis
Fromtrigonometryweknowthat cos sin and sin cos Representedinmatrixformthisgives:cos sin sin cos
Foraclockwiserotationwefind
cos sin sin cos
recallcos cos sin sin
Thetransformationmatrixforaclockwiserotationbyis:
cos sin 0 sin cos 0
0 0 1
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ImproperRotationsSnBecauseanimproperrotationmaybeexpressedasxyCnwecanwritethefollowingsincematricesalsofollowtheassociativelaw.
1 0 00 1 00 0 1
cos sin 0 sin cos 0
0 0 1
cos sin 0 sin cos 0
0 0 1
Thesetofmatricesthatwehavegeneratedthattransformasetofx,y,zorthogonalcoordinatesarecalledorthogonalmatrices.Theinverseofthesematricesarefoundbyexchangingrowsintocolumnstakingthetransposeofthematrix.ConsideraC3rotationaboutthezaxis.
0
00 0 1
exchangingrowsintocolumnsgives
0
00 0 1
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Multiplyingthesetwomatricesgivestheidentitymatrix
0
00 0 1
0
00 0 1
1 0 00 1 00 0 1
Wealreadyknowfromsymmetrythat Hereweseethatandareinverseandorthogonaltoeachother.IngeneralwecanwriteasetofhomomorphicmatricesthatformarepresentationofagivenpointgroupForexample,considerthewatermoleculewhichbelongstotheC2vgroup.C2vcontainsE,C2,xz,yzThesetoffourmatricesbelowtransformandmultiplyexactlylikethesymmetryoperationsinC2v.Thatis,theyarehomomorphictothesymmetryoperations.
1 0 00 1 00 0 1
,1 0 00 1 00 0 1
,1 0 00 1 00 0 1
,1 0 00 1 00 0 1
EC2xzyzShowthatC2xzyz
1 0 00 1 00 0 1
1 0 00 1 00 0 1
1 0 00 1 00 0 1
Thealgebraofmatrixmultiplicationhasbeensubstitutedforthegeometryofapplyingsymmetryoperations.
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VectorsandScalarProductsConsidertwovectorsin2Dspace
ThescalarordotproductresultsinascalarornumberDefinedasthelengthofeachvectortimeseachothertimesthecosoftheanglebetweenthem:ABABcos
If90thenAB0If0thenABAB
Wecanwritethefollowing:angletothexaxisforAgreaterangletothexaxisforB
ProjectionsAxAcosAyAsinBxBcosByBsin
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UsingatrigidentitywecanwriteABABcoscossinsinRearrangetoAcosBcosAsinBsinSubstitutefromaboveABAxBxAyByMoregenerally
forndimensionalspace
Orthogonalvectorsaredefinedasthoseforwhichthefollowingistrue
0
RepresentationsofGroupsThefollowingmatricesformarepresentationoftheC2vpointgroup
1 0 00 1 00 0 1
,1 0 00 1 00 0 1
,1 0 00 1 00 0 1
,1 0 00 1 00 0 1
EC2xzyzGroupMultiplicationTableforC2vC2v E C2 xz yzE E C2 xz yzC2 C2 E yz xzxz xz yz E C2yz yz xz C2 E
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HowmanyotherrepresentationsexistfortheC2vpointgroup?A:AsmanyaswecanthinkupThesetofnumbers1,1,1,1transformlikeC2vetcHowever,thereareonlyafewrepresentationsthatareoffundamentalimportance.ConsiderthematricesE,A,B,C,andweperformasimilaritytransformwithQEQ1EQAQ1AQBQ1BQEtcForexampleAQ1AQ
AQ1AQ
ThesimilaritytransformofAbyQwillblockdiagonalizeallofthematricesAlloftheresultingsubsetsformrepresentationsofthegroupaswelle.g. , , .WesaythatE,A,B,Carereduciblematricesthatformasetofreduciblerepresentations.IfQdoesnotexistwhichwillblockdiagonalizeallofthematrixrepresentationsthenwehaveanirreduciblerepresentation.
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TheGreatOrthogonalityTheoremThetheoremstates
Where;horderofthegroup#ofsymmetryoperatorsithrepresentation dimensionofe.g.33,3RgenericsymbolforanoperatortheelementinthemthrowandnthcolumnofanoperatorRinrepresentationcomplexconjugateoftheelementinthemthrowandnthcolumnofanoperatorRinrepresentationWhatdoesthisallmean?Foranytwoirreduciblerepresentations, Anycorrespondingmatrixelementsonefromeachmatrixbehaveascomponentsofavectorinhdimensionalspace,suchthatallvectorsareorthonormal.Thatis,orthogonalandofunitlength.
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ExaminethetheoremundervariousconditionsIfvectorsarefromdifferentrepresentationsthentheyareorthogonal
0ifij
Ifvectorsarefromthesamerepresentationbutaredifferentsetsofelementsthentheyareorthogonal
0ifmm'ornn'
Thesquareofthelengthofanyvectorish/li
IrreducibleRepresentationsTherearefiveimportantrulesconcerningirreduciblerepresentations
1 Thesumofthesquaresofthedimensionsoftheirreduciblerepresentationsofagroupisequaltotheorderofthegroup
2 Thesumofthesquaresofthecharactersinanirreduciblerepresentationisequaltotheorderofthegroup
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3 Vectorswhosecomponentsarethecharactersoftwoirreduciblerepresentationsareorthogonal
0whenij
4 Inagivenrepresentationreducibleorirreduciblethecharactersofallmatricesbelongingtothesameclassareidentical
5 Thenumberofirreduciblerepresentationsofagroupisequaltothe
numberofclassesinthegroup.Letslookatasimplegroup,C2vE,C2,v,vTherearefourelementseachinaseparateclass.Byrule5,theremustbe4irreduciblerepresentations.Byrule1,thesumofthesquaresofthedimensionsmustbeequaltoh4.
4Theonlysolutionis 1ThereforetheC2vpointgroupmusthavefouronedimensionalirreduciblerepresentations.C2v E C2 v v1 1 1 1 1Allotherrepresentationsmustsatisfy 4Thiscanonlyworkfori1.Andforeachoftheremainingtobeorthogonalto1theremustbetwo1andtwo1.
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Therefore,theremainingmustbeEisalways1C2v E C2 v v1 1 1 1 12 1 1 1 13 1 1 1 14 1 1 1 1Takeanytwoandverifythattheyareorthogonal12111111110ThesearethefourirreduciblerepresentationofthepointgroupC2vConsidertheC3vgroupE,2C3,3vTherearethreeclassessotheremustbethreeirreduciblerepresentations
6Theonlyvalueswhichworkare 1, 1, 2Thatis,twoonedimensionalrepresentationsandonetwodimensionalrepresentation.Sofor1wecanchooseC3v E 2C3 3v1 1 1 1For2weneedtochoose1tokeeporthogonalityC3v E 2C3 3v1 1 1 12 1 1 112112113110
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C3v E 2C3 3v1 1 1 12 1 1 13 2 Tofind3wemustsolvethefollowing
12 21 31 0
12 21 31 0
Solvingthissetoftwolinearequationandtwounknownsgives 1and 0
ThereforethecompletesetofirreduciblerepresentationsisC3v E 2C3 3v1 1 1 12 1 1 13 2 1 0WehavederivedthecharactertablesforC2vandC3vcheckthebookappendixC2v E C2 v v C3v E 2C3 3vA1 1 1 1 1 A1 1 1 1A2 1 1 1 1 A2 1 1 1B1 1 1 1 1 E 2 1 0B2 1 1 1 1
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Wenowknowthatthereisasimilaritytransformthatmayblockdiagonalizeareduciblerepresentation.Duringasimilaritytransformthecharacterofarepresentationisleftunchanged.
WhereRisthecharacterofthematrixforoperationRandajisthenumberoftimesthatthejthirreduciblerepresentationappearsalongthediagonal.ThegoodnewsisthatwedonotneedtofindthematrixQtoperformthesimilaritytransformandblockdiagonalizethematrixrepresentations.Becausethecharactersareleftintact,wecanworkwiththecharactersalone.WewillmultiplytheabovebyiRandsumoveralloperations.
and
Recallthat
Foreachsumoverjwehave
Thecharactersforiandjformorthogonalvectorswecanonlyhavenonzerovalueswhenij
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TheReductionFormulaTheaboveleadstotheimportantresultcalledTheReductionFormula
1
Whereaiisthenumberoftimestheithirreduciblerepresentationappearsinthereduciblerepresentation.C3v E 2C3 3v1 1 1 12 1 1 13 2 1 0a 5 2 1b 7 1 3ApplythereductionformulatoaandbFora
16 115 212 311 1
16 115 212 311 2
16 125 212 301 1
Forb
16 117 211 313 0
16 117 211 313 3
16 127 211 303 2
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SumthecolumnsForaC3v E 2C3 3v1 1 1 12 1 1 12 1 1 13 2 1 0a 5 2 1 ForbC3v E 2C3 3v2 1 1 12 1 1 12 1 1 13 2 1 03 2 1 0b 7 1 3
CharacterTablesForC3vwefindthefollowingcharactertablewithfourregions,IIV.C3v E 2C3 3v A1 1 1 1 z x2y2, z2A2 1 1 1 RzE 2 1 0 x,yRx,Ry x2y2,xyxz,yzI II III IV
RegionIMullikenSymbolsforIrreducibleRepresentations
1 All11representationsareAorB,22areEand33areT
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2 11whicharesymmetricwithrespecttorotationby2/nabouttheprincipleCnaxisareAi.e.thecharacteris1underCn.ThosethatareantisymmetricarelabeledBthecharacteris1underCn.
3 Subscripts1or2areaddedtoAandBtodesignatethosethataresymmetric1orantisymmetric2toaC2toCnorifnoC2ispresentthentoav.
4 andareattachedtothosethataresymmetricorantisymmetricrelativetoah.
5 Ingroupswithaninversioncenteri,subscriptgGermanforgeradeorevenisaddedforthosethataresymmetricwithrespecttoiorasubscriptuGermanforungeradeorunevenisaddedforthoseantisymmetricwithrespecttoi.
6 LabelsforEandTrepresentationsaremorecomplicatedbutfollowthesamegeneralrules.
RegionIICharactersThisregionlistthecharactersoftheirreduciblerepresentationsforallsymmetryoperationsineachgroup.RegionIIITranslationsandRotationsTheregionassignstranslationsinx,yandzandrotationsRx,Ry,Rztoirreduciblerepresentations.E.g.,inthegroupabovex,yislistedinthesamerowastheEirreduciblerepresentation.Thismeansthatifoneformedamatrixrepresentationbasedonxandycoordinates,itwouldtransformthatishavethesamecharactersasidenticallyasE.RecallthatpreviouslywelookedataC3transformationmatrixforasetofCartesiancoordinates
0
00 0 1
Noticethatthismatrixisblockdiagonalized.Ifwebreakthisintoblocksweareleftwith
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and1
ComparethecharactersofthesematricestothecharactersunderC3inthetableabove.Noticethatforx,y1andforz1.Ifyoucomparedthecharactersforalloftheothertransformationmatricesyouwillseethatx,yEandzA1asshowninregionIIIofthetable.Similaranalysiscanbemadewithrespecttorotationsaboutx,yandz.RegionIVBinaryProductsThisregionlistvariousbinaryproductsandtowhichirreduciblerepresentationthattheybelong.Thedorbitalshavethesamesymmetryasthebinaryproducts.Forexamplethedxyorbitaltransformsthesameasthexybinaryproduct.
WritingChemicallyMeaningfulRepresentationsWewillbeginbyconsideringthesymmetryofmolecularvibrations.Toagoodapproximation,molecularmotioncanbeseparatedintotranslational,rotationalandvibrationalcomponents.Eachatominamoleculehasthreedegreesoffreedommotionpossible.Anentiremoleculethereforehas3NdegreesoffreedomforNatoms.3DOFarefortranslationinx,yandz.3DOFareforrotationinx,yandznote:linearmoleculescanonlyrotatein2dimensionsTheremainDOFarevibrationalinnatureAmoleculewillhave 3N6possiblevibrations
3N5forlinearmolecules
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Usingthetoolsofclassicalmechanicsitispossibletosolvefortheenergiesofallvibrationsthinkballsandspringsmodelforamolecule.Thecalculationsaretediousandcomplicatedandasearlyasthe1960scomputershavebeenusedtodothecalculations.1VibrationsWecanusethetoolsofgrouptheorytodeducethequalitativeappearanceofthenormalmodesofvibration.WellstartwithasimplemoleculelikeH2O.Forwaterweexpect3N63normalmodesofvibration.Waterissimpleenoughthatwecanguessthemodes.
symmetricstretchingantisymmetricstretchingbendingAssignthesethreevibrationstoirreduciblerepresentationsintheC2vpointgroup.C2v E C2 xz 'yz A1 1 1 1 1 zA2 1 1 1 1 RzB1 1 1 1 1 x,RyB2 1 1 1 1 y,RxConsiderthedisplacementvectorsredarrowsforeachmodeandwritewhathappensundereachsymmetryoperation.1Formoreinformationonthesecalculations,lookupFandGmatricesinagrouptheoryorphysicalchemistrytext.Insummarythismethodsumsupandsolvesallofthepotentialenergiesbasedontheforceconstants(bondstrength)anddisplacementvectors(vibrations). . where,fikistheforceconstantandsiandskaredisplacements(stretchingorbending).Thetermfiisi2representsthepotentialenergyofapurestretchorbendwhilethecrosstermsrepresentinteractionsbetweenthevibrationalmodes.
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Symmetricstretching1E1,C21,xz1,'yz1AntiSymmetricstretching2E1,C21,xz1,'yz1Bending3E1,C21,xz1,'yz1C2v E C2 xz 'yzA1 1 1 1 1 zA2 1 1 1 1 RzB1 1 1 1 1 x,RyB2 1 1 1 1 y,Rx1 1 1 1 12 1 1 1 13 1 1 1 1Inamorecomplicatedcasewewouldapplythereductionformulatofindtheirrwhichcomprise.However,inthiscaseweseebyinspectionthat1A12B13A1Amoregeneralizedapproachtofindingwillbediscussedlater.
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SelectionRulesforVibrationsBornOppenheimerapproximation:electronsmovefastrelativetonuclearmotion.
Where:istheelectronicwavefunctionandisthenuclearwavefunction isthedipolemomentoperator
e e
Where:riistheradiusvectorfromtheorigintoachargeqianelectroninthiscaseeistheprotonchargeZisthenuclearchargeristheradiusvectorforanucleusIntegralsofthistypedefinetheoverlapofwavefunctions.Whentheaboveintegralisnotequalto0,avibrationaltransitionissaidtobeallowed.Thatis,thereexistssomedegreeofoverlapofthetwowavefunctionsallowingthetransitionfromonetotheother.In1800SirWilliamHerschelputathermometerinadispersedbeamoflight.Whenheputthethermometerintotheregionbeyondtheredlighthenotedthetemperatureincreasedevenmorethanwhenplacedinthevisiblelight.HehaddiscoveredinfraredIRlight.
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SimilartoelectronictransitionswithvisibleandUVlight,IRcanstimulatetransitionsfrom12.Asimplifiedintegraldescribingthistransitionis
whichisallowedwhentheintegraldoesnotequalzero.Inthisintegralvibrationalgroundstatewavefunctionand isthethfundamentalvibrationallevelwavefunction.Whatthisallmeansisthatavibrationaltransitionintheinfraredregionisonlyallowedifthevibrationcausesachangeinthedipolemomentofthemolecule.DipolemomentstranslatejustliketheCartesiancoordinatevectorsx,yandz.Thereforeonlyvibrationsthathavethesamesymmetryasx,yorzareallowedtransitionsintheinfrared.SelectionRulesforRamanSpectroscopyInRamanspectroscopy,incidentradiationwithanelectricfieldvectormayinduceadipoleinamolecule.Theextentofwhichdependsonthepolarizabilityofthemoleculepolarizabilityoperator.
TransitionsinRamanspectroscopyareonlyallowedifthevibrationcausesachangeinpolarizability.Polarizabilitytransformslikethebinaryproducttermsxy,z2etcandthereforevibrationsthathavethesamesymmetryasthebinaryproductsareallowedtransitionsinRamanspectroscopy.Forwater,allthreevibrationsareIRandRamanactive.
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NormalCoordinateAnalysisLetsfindallofthevibrationalmodesforwaterinasystematicway.Weexpect3N63vibrations.Asimplewaytodescribeallpossiblemotionsofamoleculeistoconsiderasetofthreeorthonormalcoordinatescenteredoneachatom.Forwater,thisresultsinasetof9vectorsshownbelow.Anypossiblemotionwillbethesumofallninecomponents.
Now,wewillwritethefourtransformationmatricesthatrepresentthetransformationoftheseninevectorsundertheoperationsoftheC2vpointgroupE,C2,vxz,vyz
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1 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 00 0 0 1 0 0 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 1 0 0 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 1
9
0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 10 0 0 1 0 0 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 1 0 0 01 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0
1
1 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 00 0 0 1 0 0 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 1 0 0 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 1
3
0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 10 0 0 1 0 0 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 1 0 0 01 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0
1
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Thereduciblerepresentationthatwehavejustformedisdenoted3N,toindicatewehaveused3orthonormalvectorsoneachatom.Nowthatwehavethetransformationmatriceswecanwritethecharactersfor3NC2v E C2 xz 'yz A1 1 1 1 1 zA2 1 1 1 1 RzB1 1 1 1 1 x,RyB2 1 1 1 1 y,Rx3N 9 1 3 1 Thenextstepistofindthelinearcombinationofirreduciblerepresentationsthecomprise3N.Wedothisbyapplicationofthereductionformulato3N.
a 14 91 11 31 11 3
a 14 91 11 31 11 1
a 14 91 11 31 11 3
a 14 91 11 31 11 2
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LetsfindallofthevibrationalmodesforNH3.Weexpect3N66vibrations.Asimplewaytodescribeallpossiblemotionsofamoleculeistoconsiderasetofthreeorthonormalcoordinatescenteredoneachatom.ForNH3,thisresultsinasetof12vectors.Anymotionwillbethesumofalltwelvecomponents.
Asperformedpreviouslyforasetofthreex,yandzvectorswecanwriteatransformationmatrixthatdescribeswhathappenstoeachofthevectorsforeachsymmetryoperationinthegroup.Wenowneedtofindthecharactersof3NC3v E 2C3 3v A1 1 1 1 z x2y2,z2A2 1 1 1 RzE 2 1 0 x,yRx,Ry x2y2,xyxz,yz3N ? ? ?Thetransformationmatriceswillbe1212.However,weareonlyinterestedinthecharactersofeachmatrix.ForEthecharacterwillbe12sinceallelementsremainunchanged.
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ConsideraC3rotation:OnlyvectorsontheNatomwillgointothemselves.Fromourpreviousresultsweknowthatx,yandztransformlike
1232 0 0 0 0 0 0 0 0 0 0
32 12 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0Allothercomponentsareoffdiagonalanddonotcontributetothecharacterofthematrix.Here,0forC3.
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Fortheverticalmirrorplane,vFourvectorsremainunchangedonNandHandtwogointo1ofthemselvesonNandH.Theother6ontheoutofplaneHatomsallbecomeoffdiagonalelements.2111111000000
Nowwecanwrite3NC3v E 2C3 3v A1 1 1 1 z x2y2,z2A2 1 1 1 RzE 2 1 0 x,yRx,Ry x2y2,xyxz,yz3N 12 0 2Applythereductionformulatofindwhatirrcomprise3N
1
a 16 1112 210 312 3
a 16 1112 210 312 1
a 16 1212 210 302 4
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Wewritethefollowing 3 4
However,3Ndescribesallpossiblemotion,includingtranslationandrotation.Inspectionofthecharactertablerevealsthat
Thisleavesthevibrationsas 2 2noticewepredicted6normalmodesandwehave6dimensionsrepresentedtwo11andtwo22.Nowwewillwritepicturesrepresentingwhatthevibrationslooklike.issymmetricwithrespecttoalloperations
symmetricstretchingsymmetricbending
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TheEmodesaredegenerate.Thatistheyaremadeupofvibrationsthatareofequalenergy.asymmetricstretching
Thethirdpossiblewayofdrawinganasymmetricstretchisjustalinearcombinationofthetwoaboveaddthetwovibrations.asymmetricbending
Aswiththestretches,thethirdbendisformedfromalinearcombinationoftheothertwoandisnotuniquesubtractthetwobends.
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IRandRamanSpectraofCH4andCH3FMethaneCH4belongstotheTdpointgroupPerformthenormalcoordinateanalysisformethaneanddeterminethenumberandsymmetryofallIRandRamanactivebandsTherewillbe3569normalmodesofvibrationinbothmoleculesNormalCoordinateAnalysisofCH41stwritethecharactersfor3NTd E 8C3 3C2 6S4 6d A1 1 1 1 1 1 x2y2z2A2 1 1 1 1 1 E 2 1 2 0 0 2z2x2y2T1 3 0 1 1 1 Rx,Ry,Rz x2y2T2 3 0 1 1 1 x,y,z xy,xz,yz3N 15 0 1 1 3 Reduce3Nintoitsirreduciblerepresentations.
124 15 0 3 6 18 1
124 15 0 3 6 18 0
124 30 0 6 0 0 1
124 45 0 3 6 18 1
124 45 0 3 6 18 3
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3 and Subtractingthetranslationalandrotationalirreduciblerepresentationsweareleftwith 2 IRactivemodesarethe6modesin2 All9modesareRamanactiveCalculatedIRandRamanspectraforCH4areshownbelow2
2SpectracalculatedbyGAMESSusingHartreeFockmethods
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Raman Spectrum
1400160018002000220024002600280030003200Energy (cm-1)
0
1
Inte
nsity
NormalCoordinateAnalysisofCH3FNowweperformtheidenticalanalysisforCH3FwhichbelongstotheC3vpointgroup.1stwritethecharactersfor3NC3v E 2C3 3v A1 1 1 1 z x2y2,z2A2 1 1 1 RzE 2 1 0 x,yRx,Ry x2y2,xyxz,yz3N 15 0 3 Reduce3Nintoitsirreduciblerepresentations.
16 15 0 9 4
16 15 0 9 1
16 30 0 0 5
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4 5 and Subtractingthetranslationalandrotationalirreduciblerepresentationsweareleftwith 3 3All9modesareIRactiveAll9modesareRamanactiveCalculatedIRandRamanspectraforCH3Fareshownbelow
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AlternateBasisSetsTohelpdrawpicturesofthevibrationalmodeswecanuseabasissetthatrelatesmoredirectlytovibrations,calledinternalcoordinates.ForCH4wecanuseCHbondstretchesasabasisandHCHbendsasabasis.Td E 8C3 3C2 6S4 6dCH 4 1 0 0 2 StretchesHCH 6 0 2 0 2 bends Ifweaddthisupwefindthatthisis10normalmodesbutweexpectonly9.LookingatthebendingmodesweseeanA1representation.SincethereisnowaytoincreaseallthebondanglesatonceinCH4thismustbediscounted.Inordertovisualizethevibrationsbemustmakelinearcombinationsofourbasissetelementsthatareorthonormalsymmetryadapted.
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ProjectionOperatorTheprojectionoperatorallowsustofindthesymmetryadaptedlinearcombinationsweneedtovisualizethevibrationalmodesinCH4andCDH3.
WherethedimensionoftheirreduciblerepresentationitheorderofthegroupcharacterofoperationRresultofthesymmetryoperationRonabasisfunctionProjectouttheA1modeformethanerecall Td E 8C3 3C2 6S4 6dA1 1 1 1 1 1
Nowwemustlabelourbasisfunctionsandkeeptrackofwhathappenstothemundereachoperation.
124
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6 6 6 6
Asyouwouldpredictthisisthetotallysymmetricstretch
A1stretchingmodeDothesamefortheT2modeTd E 8C3 3C2 6S4 6dT2 3 0 1 1 1 324 3 0
324 6 2 2 2 14 3
Thisisanasymmetricstretchingmodethatistriplydegenerate.
T2stretchingmodesOnemustprojectouttwootherbasisfunctionstofindtheothertwomodes.Theywilllookidenticalbutberotatedrelativetothemodedrawnhere.Noticethe1functionisdisplaced3timesmorethantheother3basisfunctions.
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Inananalogousmannerthebendingmodesmaybeprojectedtorevealthefollowing:
Ebendingmodes T2bendingmodesAnimationsofthesevibrationscanbeseenonlinehere:http://www.molwave.com/software/3dnormalmodes/3dnormalmodes.htm
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BondingTheoriesLewisBondingTheoryAtomsseektoobtainanoctetofelectronsintheoutershellduetforhydrogen.Asinglebondisformedwhentwoelectronsaresharedbetweentwoatomse.g.H:HAdoublebondisformedwhentwopairsofelectronsaresharede.g.OOLewistheoryworkswellforconnectivitybutdoesnotgivepredictionsaboutthreedimensionalshape.VSEPRValenceShellElectronPairRepulsionTheoryPredictsshapebyassumingthatatomsandloneelectronpairsseektomaximizethedistancebetweenotheratomsandlonepairs.WorkswellforshapesbutmustbecombinedwithLewistheoryforconnectivityandbondorder.e.g.Anatomwithfourthingsi.e.bondsorlonepairswouldadoptatetrahedralgeometrywithrespecttothethings.Themolecularshapeisdescribedrelativetothebondsonly.NH3hasthreebondsandonelonepairontheNatom,givingatetrahedralgeometry.However,theshapeofNH3isdescribedastrigonalpyramidalwhenonlythebondsareconsidered.
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ValanceBondTheoryVBtheorywasdevelopedinthe1920sandwasthefirstquantummechanicaldescriptionofbonding.InVBtheorybondsareformedfromtheoverlapofatomicorbitalsbetweenadjacentatoms.Thistheorypredicteddifferentshapesforsingleandmultiplebonds.Considertheoverlapofanelectroninhydrogen1s1orbitalandanelectroninacarbon2p1orbital
Theresultingoverlapofwavefunctionsresultsincontinuouselectrondensitybetweentheatomswhichisclassifiedasabondinginteraction.Theresultingbondhascylindricalsymmetryrelativetothebondaxis.Doubleandtriplebondsformfromtheoverlapofadjacentparallelporbitals.Theresultisshownbelow.
Adoublebondconsistsofonebondandonebond.Atriplebondisabondandtwobonds.
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HybridOrbitalTheoryVBtheorygetsusstartedbutweknowthatothergeometriesexistinmolecules.ConsidercarbonwithagroundstateelectronconfigurationofHe2s2sp2.Withtheadditionofasmallamountofenergyone2selectronispromotedtothe2pshellandthefollowingconfigurationresultsHe2s12px12py12pz1.Thisleavesfourhalffilledorbitalsthatcanoverlapwithotherorbitalsonadjacentatoms.Willthisresultindifferenttypesofbonds?Overlapofansorbitalandaporbitalwillnotbethesame.However,weknowthatinCH4allofthebondsareequivalent.Thesolutionistocreatehybridorbitalsthatareformedwhenwetakelinearcombinationsofthefouravailableatomicorbitals.Thefouruniquelinearcombinationsthatcanbeformedareasfollowsh1spxpypz h2spxpypzh3spxpypz h4spxpypzGraphicallytheresultsareshownbelow:
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Theselinearcombinationgiverisetofourneworbitalsthathaveelectrondensitypointedalongtetrahedralangles.Theorbitalsarenamedsp3hybridorbitals.Otherhybridorbitalsarepossiblewithdifferinggeometry
Orbital Shapesp Linearsp2 Trigonalplanarsp3 Tetrahedralsp3d Trigonalpyramidalsp3d2 octahedral
Overlapofthehybridorbitalswithotherorbitalhybridoratomicdescribethebondingnetworkinmolecule.MultiplebondsareformedwithunusedporbitalsoverlappingtoformbondsasdescribedinVBtheory.MolecularOrbitalTheoryMOtheorywasalsodevelopedduringthe1920sandisaquantummechanicaldescriptionofbonding.Alloftheprecedingbondingmodelsaretermedlocalizedelectronbondingmodelsbecauseitisassumedthatabondisformedwhenelectronsaresharedbetweentwoatomsonly.MOtheoryallowsfordelocalizationofelectrons.Thatis,electronsmaybesharedbetweenmorethantwoatomsoverlongerdistances.SimilarlytoVBtheory,MOsareformedfromtheinteractionofatomicorbitals.Hereweformlinearcombinationsofatomicorbitals.WhenwecombinetwoAOswemustformtwoMOs
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ConstructiveinterferenceleadstotheMOshowninthetopexampleandtheresultingMOissaidtobebonding.Destructiveinterferencegivesthesecondexampleistermedantibonding.ForO2theMOdiagramisshownbelow.ThiscorrectlypredictsthatO2hastwounpairedelectrons.Noneoftheprevioustheoriescouldaccommodateforthisfact.
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QuantummechanicaldescriptionoforbitalsErwinSchrdingerproposedamethodtofindelectronwavefunctions.ThetimeindependentSchrdingerequationinonedimensionis
2
whereisthewavefunctionand
2 1.054 10
Vxisthepotentialenergyoftheelectronatpositionxandand
isthekineticenergyoftheelectron.Thisequationisoften
simplifiedas
where iscalledtheHamiltonianoperator.WritteninthisformweseethatthisisanEigenvalueequation.ThetotalenergyoftheelectronbecomestheEigenvalueoftheHamiltonianandistheEigenfunction.describesthedynamicinformationaboutagivenelectron.Theprobabilityoffindinganelectroninagivenvolumeofinfinitelysmallsizeis
||
Withregardtospecificallyidentifyingthedynamicinformationaboutanelectronwearelimitedbythefollowing:
12
ThisisknownastheHeisenberguncertaintyprincipleandstatesthatwecannotknowboththepositionandmomentumofaparticlewitharbitraryprecision.xistheerroruncertaintyinpositionandpistheerrorinmomentum.
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GroupTheoryandQuantumMechanicsIfweexchangeanytwoparticlesinasystembycarryingoutasymmetryoperation,theHamiltonianmustremainunchangedbecauseweareinanequivalentstate.Inotherwords,theHamiltoniancommuteswithallRforagroup
TherearecasesinwhichmultipleEigenfunctionsgivethesameEigenvalue.
:
WesaythattheEigenvalueisdegenerateornfolddegenerate.InthesecasestheEigenfunctionsareasolutiontotheSchrdingerequationandalsoanylinearcombinationofthedegenerateEigenfunctions
WewillconstructtheEigenfunctionsandsubsequentlythelinearcombinationssothattheyareallorthonormaltoeachother.
ThesetoforthonormalEigenfunctionsforamoleculecanformthebasisofanirreduciblerepresentationofthegroup.ForanondegenerateEigenfunctionwehave
sothatRiisanEigenfunctionoftheHamiltonian.
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BecauseiisnormalizedRi1i.ThereforeifweapplyallRinagrouptoanondegeneratei,wegetarepresentationwereeachmatrixelement,iRwillbe1.Aonedimensionalmatrixisbydefinitionirreducible.AsimilaranalysisfornfolddegenerateEigenfunctionswillresultinanndimensionalirreduciblerepresentation.HowdowefindthelinearcombinationofwavefunctionsthatresultinasetoforthonormalEigenfunctions?Theprojectionoperatordiscussedpreviouslyforvibrationalanalysisisusefulagainhere.
Constructionofbasissetstoprojectisthesubjectoftheoreticalchemistryandphysics.BecausewecannotsolvetheSchrdingerequationdirectlywemustmakeapproximations.HartreeFockApproximationwriteMOsforeachelectronindependentlyoftheothers.Theerrorthatisintroducedhereisthattheelectronpositiondependsonthepositionofalloftheotherelectronselectronelectronrepulsion.Acorrectionfactormustbeappliedaftersolvingtheproblemtoaccountforthis.Thisiscalledthecorrelationenergy.RulesforMolecularOrbitals
1 Wavefunctionscannotdistinguishbetweenelectrons2 Ifelectronsexchangepositions,thesignofthewavefunctionmust
change.LCAOApproximationMolecularOrbitalsarelinearcombinationsofatomicorbitals.electronApproximationassumethatandbondsareindependentofeachother.Thatis,bondsarelocalized,whilebondsmaybedelocalized.
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HckelOrbitalMethodWewillusearedefinedHamiltoniancalledtheeffectiveHamiltonian.
, 2
whereJandKaretheCoulombandexchangeintegralsrespectively.TheCoulombintegraltakesintoaccounttheelectronelectronrepulsionbetweentwoelectronsindifferentorbitals,andtheexchangeintegralrelatestotheenergywhenelectronsintwoorbitalsareexchangedwitheachother.3IntheHckelorbitalmethodweconstructnewMOsasfollows
whereNisthenumberofatomsintheorbitalsystem,sisapzorbitalonagivenatomandCsjisacoefficientdeterminedbyprojection.Hckeltheorymakesthefollowingapproximations:
,
, ifrandsarenearestneighbors0otherwise
istheCoulombicintegralwhichraisestheenergyofawavefunctionpositivevalueandistheresonanceintegralwhichlowerstheenergyofawavefunctionnegativevalue.Theseintegralscanbeevaluatednumericallybutarebeyondthescopeofwhatwehopetoaccomplishhere.3
3Formoreinformationonbondingtheoryconsultanappropriatetextonquantummechanics.
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Asitstandssofar,wehaveamethodforfindingtheenergiesofNorbitalsforNatoms.TheresultisanNdimensionalpolynomial.Problemsofthistypecanbesolvedbutapplicationofsymmetrytothesystemgreatlyreducestheamountofworktobedone.HckelOrbitalsforNitriteConsidertheorbitalsystemforthenitriteanion
ThismoleculebelongstotheC2vpointgroup.Wewilluseapzorbitaloneachatomtoconstructtheorbitalsystemfornitrite.Thisformsthebasisforareduciblerepresentation,AO
WemustwritethecharactersforAOinananalogousmanneraswewrotethecharactersfor3N.Keepinmindthesignofthewavefunctionswhenperformingthesymmetryoperations.
C2v E C2 xz 'yzA1 1 1 1 1A2 1 1 1 1B1 1 1 1 1B2 1 1 1 1AO 3 1 1 3
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NextwereduceAOtofindtheirreduciblecomponents.
14 13 11 11 13 0
14 13 11 11 13 1
14 13 11 11 13 2
14 13 11 11 13 0
2Nextweprojectthebasisfunctionsoutoftheirreduciblerepresentations.Because1and3areequivalentitdoesntmatterwhichonewepick.However,2isuniqueandmustalsobeprojectedeachtime.
14 12
14 0i.e.nocontribution
14 12
14
TherearetwoorbitalswithB1symmetry.Sowewilltakelinearcombinationsofthetwoprojectedorbitalstofindtheorthogonalresults
12
12 12
12 12
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Lastlywemustnormalizetheorbitalssinceitisassumedthat
ThereisanormalizationfactorNthatisfoundasfollowsfor1
12 12 1
14
1
14
1
Notethatthesumsheregiveijfortheindividualijterms.Therearefourintegralsinthiscaseinvolvingoverlapof1,1,1,3,3,1,3,3.Thesegive1001respectively.
14
2 1
solvingforNwefindN2ournormalized1isthen A2symmetry
Moregenerallyforanormalizedwavefunctionhastheform
andweknownormalizedwavefunctionsfollow;
1
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Therefore;1
1
Theintegralintheaboveexpressionmustevaluateto1sincethebasisfunctionsjareorthonormalwavefunctions.SolvingforthenormalizationfactorNigives;
1
Applyingthisto2abovewefind,
and
andthenormalizedwavefunctionbecomes;
B1symmetry
Similarlywefindthenormalized3tobe
B1symmetry
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Ifweevaluate,foreachneworbitalwefindthat
, 22
, 22
DistributingtheiacrosstheHamiltonianandseparatingtheintegralsweareleftwiththefollowingsum
22 , 22
22
, 22 22
, 22 22
, 22
TheseintegralsevaluateaccordingtoHckelapproximationsas
, 12 0 0
12
Simplifyingleavesuswith:
, 12
Asimilaranalysisfor2and3yields:,
,
Inorderofincreasingenergywefind213
Recallthatisanegativetermandlowerstheenergyoftheorbital.
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Drawingpicturesoftheorbitalshelpstovisualizetheresults
B1antibondingorbital
43A2nonbondingorbital
B1bondingorbital
43
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Nitriteorbitalscalculatedusingabinitiomethods.
B1antibondingorbital
A2nonbondingorbital
B1bondingorbital
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HckelMOsforCyclobutadieneFindtheorbitalsforcyclobutadieneusingtheHckelorbitalmethod.CyclobutadienebelongstotheD4hpointgroup.Thisgrouphasanorderof16.Toreducethework,wecanuseasubgroupofD4h,D4whichhasanorderof8.Indoingsoweloseinformationabouttheoddorevennatureoftheorbitalgoru.However,oncetheMOsareconstructedwecaneasilydetermineanMOsgorustatusthroughexaminationofsymmetryoperationsonthenewMOs.Thebasissetwillbethefourpzorbitalsperpendiculartothemolecularplane.
WritingAOforD4wefind;
D4 E 2C4 C2 2C2 2C2A1 1 1 1 1 1A2 1 1 1 1 1B1 1 1 1 1 1B2 1 1 1 1 1E 2 0 2 0 0AO 4 0 0 2 0
ReducingAOresultsinAO
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Nowwemustprojectoutthesymmetryadaptedlinearcombinationsandthennormalizetheresultingfunctions.
18
14
18
14
28 2 2 12
BecauseEistwodimensional,wemustprojectanadditionalorbital.
28 2 2 12
ThetwoEMOswillbethelinearcombinationsumanddifferenceofthetwoprojectionswehavejustmade.
12
12
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NormalizingtheMOsresultsinthefollowingorthonormalsetoffunctions A2symmetry
B2symmetry
Esymmetry
EsymmetryToputtheorbitalinthecorrectorderenergetically,weevaluate,foreachMO., 2
, 2
,
, BecausetheEorbitalisdegenerate,bothorbitalsmusthavethesameenergy.Wefindtheinorderofincreasingenergy,13,42.LastlywecanlookathowtheeachoftheseorbitalstransformsintheD4hpointgroupandassignthegorusubscript.Forexample,theB2orbitalcouldbeB2uorB2g.Underinversionitheorbitalgoesinto1ofitselfsoitmustbeB2uthecharacterunderiforB2gis1and1forB2u.AnalysisoftheremainingorbitsgivesA2uandEg.
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B2u antibonding
2
Egbonding
A2ubonding
2molecularorbitalsascalculatedbyabinitiomethods
B2u
Eg
A2u
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Anoteofcaution.WemightbetemptedtoviewcyclobutadieneasanaromaticringsystembasedupontheappearanceoftheA2uorbital.However,examinationoftheelectrondensitycolormappedwiththeelectrostaticpotentialrednegative,bluepositiverevealsanonuniformelectrondensitydistribution.Thisindicatesthatcyclobutadieneisnotaromaticbutratheralternatingdoubleandsinglebonds.
Ifthestructureisgeometryoptimized,weseethatthesymmetryisnolongerD4h,butratherthatofarectangle,D2h.Theelectrostaticpotentialmapplacesextraelectrondensityalongtheshorterdoublebondswewouldpredict.TheHckelruleforaromaticityrequires4n2electronsandherewehaveonly4electrons,sothelackofaromaticityisexpected.
Ifthesymmetrychanges,thesymmetryofthemolecularorbitalsalsochanges.Calculationsoftheneworbitalsareshownonthefollowingpage.
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Abinitiocalculationsfortheorbitalsingeometryoptimizedcyclobutadiene.
Au
B3g
B2g
B1u
NoticethatthegeneralshapeoftheMOsissimilartowhatwecalculatedforthemoleculeunderD4hsymmetry,buttheEgorbitalhassplitintothenondegenerateB2gandB3gorbitalsshownabove.
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HckelMOsforBoronTrifluoride
D3h E 2C3 3C2 h 2S3 3vA1 1 1 1 1 1 1A2 1 1 1 1 1 1E 2 1 0 2 1 0A1 1 1 1 1 1 1A2 1 1 1 1 1 1E 2 1 0 2 1 0
ConstructtheHckelmolecularorbitalsforcarbonateandcomparetotheresultsforsemiempiricalcalculationsofallmolecularorbitalsforcarbonate.Step1.FindAOD3h E 2C3 3C2 h 2S3 3vAO 4 1 2 4 1 2Step2.Applythereductionformulatofindtheirreduciblerepresentations.
112 4 2 6 4 2 6 0
112 4 2 6 4 2 6 0
112 8 2 0 8 2 0 0
112 4 2 6 4 2 6 0
112 4 2 6 4 2 6 2
112 8 2 0 8 2 0 1
2
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Step3.ProjectthenewMOsBecause1isonthecenteratomwemustproject1andoneoftheothersandmakelinearcombinationstobesuretohavecompleteorbitals.
112
112 13
Takingthelinearcombinationsofthesetwoweobtainthefollowing:
13 13
13
13 13
13
Normalizingtheseresultsgives:
32 13
13
13
32 13
13
13
NowprojecttheEorbitals 112 2 2 0
Because1doesnotcontributetotheEorbitalwemustprojecttwootherbasisfunctionsandtaketheirlinearcombinations.
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212 2 2 16 2
212 2 2 16 2
16 2
16 3 3
Afternormalizationweareleftwith
16 2
12
Step4.DeterminetheenergyofeachorbitalbyevaluatingHeff,.
, 34
13
13
13
19
13
19
13
19
13
32
, 34
13
13
13
19
13
19
13
19
13
32
, 16 4
, 12
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Step5.ConstructtheMOdiagramanddrawpicturesoftheMOs
A2
E
A2
Thesesketchesareatopdownview.Thesignofthewavefunctionisoppositeonthebottomside.
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AbinitiocalculationsgivethefollowingMOpicturesandthecorrespondingenergyinHartree.
A20.0418Ha
E0.9073Ha
A20.9411Ha
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Onefinalexercise
1. ConstructtheorthonormalHckelmolecularorbitalsforbenzeneusingthesubgroupD6givenbelowandabasissetofthesixpzorbitalsthatlieperpendiculartotheplaneofthering.
2. DeterminethecorrectorderingoftheMOsenergeticallyandconstruct
theMOdiagramforthesystem.RefertothefullD6hpointgroupcharactertabletoassigngandudesignationstotheorbitals.
3. Sketcheachofthesixorbitals.Compareyourresultstotheabinitio
calculationsonthefollowingpage.D6 E 2C6 2C3 C2 3C2 3C2A1 1 1 1 1 1 1A2 1 1 1 1 1 1B1 1 1 1 1 1 1B2 1 1 1 1 1 1E1 2 1 1 2 0 0E2 2 1 1 2 0 0AO
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Abinitiocalculationsfortheorbitalsofbenzeneareshownbelow
B2g0.3480Ha
E2u0.1322Ha
E1g0.3396Ha
A2u0.5073Ha