group theory - lecture notes complete 3rd ed

CourseLectureNotes 3rd Edition

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


Page48

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.


Page49

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.


Page50

SelectionRulesforVibrationsBornOppenheimerapproximation:electronsmovefastrelativetonuclearmotion.

Where:istheelectronicwavefunctionandisthenuclearwavefunction isthedipolemomentoperator

e e

Where:riistheradiusvectorfromtheorigintoachargeqianelectroninthiscaseeistheprotonchargeZisthenuclearchargeristheradiusvectorforanucleusIntegralsofthistypedefinetheoverlapofwavefunctions.Whentheaboveintegralisnotequalto0,avibrationaltransitionissaidtobeallowed.Thatis,thereexistssomedegreeofoverlapofthetwowavefunctionsallowingthetransitionfromonetotheother.In1800SirWilliamHerschelputathermometerinadispersedbeamoflight.Whenheputthethermometerintotheregionbeyondtheredlighthenotedthetemperatureincreasedevenmorethanwhenplacedinthevisiblelight.HehaddiscoveredinfraredIRlight.


Page51

SimilartoelectronictransitionswithvisibleandUVlight,IRcanstimulatetransitionsfrom12.Asimplifiedintegraldescribingthistransitionis

whichisallowedwhentheintegraldoesnotequalzero.Inthisintegralvibrationalgroundstatewavefunctionand isthethfundamentalvibrationallevelwavefunction.Whatthisallmeansisthatavibrationaltransitionintheinfraredregionisonlyallowedifthevibrationcausesachangeinthedipolemomentofthemolecule.DipolemomentstranslatejustliketheCartesiancoordinatevectorsx,yandz.Thereforeonlyvibrationsthathavethesamesymmetryasx,yorzareallowedtransitionsintheinfrared.SelectionRulesforRamanSpectroscopyInRamanspectroscopy,incidentradiationwithanelectricfieldvectormayinduceadipoleinamolecule.Theextentofwhichdependsonthepolarizabilityofthemoleculepolarizabilityoperator.

TransitionsinRamanspectroscopyareonlyallowedifthevibrationcausesachangeinpolarizability.Polarizabilitytransformslikethebinaryproducttermsxy,z2etcandthereforevibrationsthathavethesamesymmetryasthebinaryproductsareallowedtransitionsinRamanspectroscopy.Forwater,allthreevibrationsareIRandRamanactive.


Page52

NormalCoordinateAnalysisLetsfindallofthevibrationalmodesforwaterinasystematicway.Weexpect3N63vibrations.Asimplewaytodescribeallpossiblemotionsofamoleculeistoconsiderasetofthreeorthonormalcoordinatescenteredoneachatom.Forwater,thisresultsinasetof9vectorsshownbelow.Anypossiblemotionwillbethesumofallninecomponents.

Now,wewillwritethefourtransformationmatricesthatrepresentthetransformationoftheseninevectorsundertheoperationsoftheC2vpointgroupE,C2,vxz,vyz


Page53

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


Page54

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


Page55

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.


Page56

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.


Page57

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


Page58

Wewritethefollowing 3 4

However,3Ndescribesallpossiblemotion,includingtranslationandrotation.Inspectionofthecharactertablerevealsthat

Thisleavesthevibrationsas 2 2noticewepredicted6normalmodesandwehave6dimensionsrepresentedtwo11andtwo22.Nowwewillwritepicturesrepresentingwhatthevibrationslooklike.issymmetricwithrespecttoalloperations

symmetricstretchingsymmetricbending


Page59

TheEmodesaredegenerate.Thatistheyaremadeupofvibrationsthatareofequalenergy.asymmetricstretching

Thethirdpossiblewayofdrawinganasymmetricstretchisjustalinearcombinationofthetwoaboveaddthetwovibrations.asymmetricbending

Aswiththestretches,thethirdbendisformedfromalinearcombinationoftheothertwoandisnotuniquesubtractthetwobends.


Page60

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


Page61

3 and Subtractingthetranslationalandrotationalirreduciblerepresentationsweareleftwith 2 IRactivemodesarethe6modesin2 All9modesareRamanactiveCalculatedIRandRamanspectraforCH4areshownbelow2

2SpectracalculatedbyGAMESSusingHartreeFockmethods


Page62

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


Page63

4 5 and Subtractingthetranslationalandrotationalirreduciblerepresentationsweareleftwith 3 3All9modesareIRactiveAll9modesareRamanactiveCalculatedIRandRamanspectraforCH3Fareshownbelow


Page64

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.


Page65

ProjectionOperatorTheprojectionoperatorallowsustofindthesymmetryadaptedlinearcombinationsweneedtovisualizethevibrationalmodesinCH4andCDH3.

WherethedimensionoftheirreduciblerepresentationitheorderofthegroupcharacterofoperationRresultofthesymmetryoperationRonabasisfunctionProjectouttheA1modeformethanerecall Td E 8C3 3C2 6S4 6dA1 1 1 1 1 1

Nowwemustlabelourbasisfunctionsandkeeptrackofwhathappenstothemundereachoperation.

124


Page66

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.


Page67

Inananalogousmannerthebendingmodesmaybeprojectedtorevealthefollowing:

Ebendingmodes T2bendingmodesAnimationsofthesevibrationscanbeseenonlinehere:http://www.molwave.com/software/3dnormalmodes/3dnormalmodes.htm


Page68

BondingTheoriesLewisBondingTheoryAtomsseektoobtainanoctetofelectronsintheoutershellduetforhydrogen.Asinglebondisformedwhentwoelectronsaresharedbetweentwoatomse.g.H:HAdoublebondisformedwhentwopairsofelectronsaresharede.g.OOLewistheoryworkswellforconnectivitybutdoesnotgivepredictionsaboutthreedimensionalshape.VSEPRValenceShellElectronPairRepulsionTheoryPredictsshapebyassumingthatatomsandloneelectronpairsseektomaximizethedistancebetweenotheratomsandlonepairs.WorkswellforshapesbutmustbecombinedwithLewistheoryforconnectivityandbondorder.e.g.Anatomwithfourthingsi.e.bondsorlonepairswouldadoptatetrahedralgeometrywithrespecttothethings.Themolecularshapeisdescribedrelativetothebondsonly.NH3hasthreebondsandonelonepairontheNatom,givingatetrahedralgeometry.However,theshapeofNH3isdescribedastrigonalpyramidalwhenonlythebondsareconsidered.


Page69

ValanceBondTheoryVBtheorywasdevelopedinthe1920sandwasthefirstquantummechanicaldescriptionofbonding.InVBtheorybondsareformedfromtheoverlapofatomicorbitalsbetweenadjacentatoms.Thistheorypredicteddifferentshapesforsingleandmultiplebonds.Considertheoverlapofanelectroninhydrogen1s1orbitalandanelectroninacarbon2p1orbital

Theresultingoverlapofwavefunctionsresultsincontinuouselectrondensitybetweentheatomswhichisclassifiedasabondinginteraction.Theresultingbondhascylindricalsymmetryrelativetothebondaxis.Doubleandtriplebondsformfromtheoverlapofadjacentparallelporbitals.Theresultisshownbelow.

Adoublebondconsistsofonebondandonebond.Atriplebondisabondandtwobonds.


Page70

HybridOrbitalTheoryVBtheorygetsusstartedbutweknowthatothergeometriesexistinmolecules.ConsidercarbonwithagroundstateelectronconfigurationofHe2s2sp2.Withtheadditionofasmallamountofenergyone2selectronispromotedtothe2pshellandthefollowingconfigurationresultsHe2s12px12py12pz1.Thisleavesfourhalffilledorbitalsthatcanoverlapwithotherorbitalsonadjacentatoms.Willthisresultindifferenttypesofbonds?Overlapofansorbitalandaporbitalwillnotbethesame.However,weknowthatinCH4allofthebondsareequivalent.Thesolutionistocreatehybridorbitalsthatareformedwhenwetakelinearcombinationsofthefouravailableatomicorbitals.Thefouruniquelinearcombinationsthatcanbeformedareasfollowsh1spxpypz h2spxpypzh3spxpypz h4spxpypzGraphicallytheresultsareshownbelow:


Page71

Theselinearcombinationgiverisetofourneworbitalsthathaveelectrondensitypointedalongtetrahedralangles.Theorbitalsarenamedsp3hybridorbitals.Otherhybridorbitalsarepossiblewithdifferinggeometry

Orbital Shapesp Linearsp2 Trigonalplanarsp3 Tetrahedralsp3d Trigonalpyramidalsp3d2 octahedral

Overlapofthehybridorbitalswithotherorbitalhybridoratomicdescribethebondingnetworkinmolecule.MultiplebondsareformedwithunusedporbitalsoverlappingtoformbondsasdescribedinVBtheory.MolecularOrbitalTheoryMOtheorywasalsodevelopedduringthe1920sandisaquantummechanicaldescriptionofbonding.Alloftheprecedingbondingmodelsaretermedlocalizedelectronbondingmodelsbecauseitisassumedthatabondisformedwhenelectronsaresharedbetweentwoatomsonly.MOtheoryallowsfordelocalizationofelectrons.Thatis,electronsmaybesharedbetweenmorethantwoatomsoverlongerdistances.SimilarlytoVBtheory,MOsareformedfromtheinteractionofatomicorbitals.Hereweformlinearcombinationsofatomicorbitals.WhenwecombinetwoAOswemustformtwoMOs


Page72

ConstructiveinterferenceleadstotheMOshowninthetopexampleandtheresultingMOissaidtobebonding.Destructiveinterferencegivesthesecondexampleistermedantibonding.ForO2theMOdiagramisshownbelow.ThiscorrectlypredictsthatO2hastwounpairedelectrons.Noneoftheprevioustheoriescouldaccommodateforthisfact.


Page73

QuantummechanicaldescriptionoforbitalsErwinSchrdingerproposedamethodtofindelectronwavefunctions.ThetimeindependentSchrdingerequationinonedimensionis

2

whereisthewavefunctionand

2 1.054 10

Vxisthepotentialenergyoftheelectronatpositionxandand

isthekineticenergyoftheelectron.Thisequationisoften

simplifiedas

where iscalledtheHamiltonianoperator.WritteninthisformweseethatthisisanEigenvalueequation.ThetotalenergyoftheelectronbecomestheEigenvalueoftheHamiltonianandistheEigenfunction.describesthedynamicinformationaboutagivenelectron.Theprobabilityoffindinganelectroninagivenvolumeofinfinitelysmallsizeis

||

Withregardtospecificallyidentifyingthedynamicinformationaboutanelectronwearelimitedbythefollowing:

12

ThisisknownastheHeisenberguncertaintyprincipleandstatesthatwecannotknowboththepositionandmomentumofaparticlewitharbitraryprecision.xistheerroruncertaintyinpositionandpistheerrorinmomentum.


Page74

GroupTheoryandQuantumMechanicsIfweexchangeanytwoparticlesinasystembycarryingoutasymmetryoperation,theHamiltonianmustremainunchangedbecauseweareinanequivalentstate.Inotherwords,theHamiltoniancommuteswithallRforagroup

TherearecasesinwhichmultipleEigenfunctionsgivethesameEigenvalue.

:

WesaythattheEigenvalueisdegenerateornfolddegenerate.InthesecasestheEigenfunctionsareasolutiontotheSchrdingerequationandalsoanylinearcombinationofthedegenerateEigenfunctions

WewillconstructtheEigenfunctionsandsubsequentlythelinearcombinationssothattheyareallorthonormaltoeachother.

ThesetoforthonormalEigenfunctionsforamoleculecanformthebasisofanirreduciblerepresentationofthegroup.ForanondegenerateEigenfunctionwehave

sothatRiisanEigenfunctionoftheHamiltonian.


Page75

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.


Page76

HckelOrbitalMethodWewillusearedefinedHamiltoniancalledtheeffectiveHamiltonian.

, 2

whereJandKaretheCoulombandexchangeintegralsrespectively.TheCoulombintegraltakesintoaccounttheelectronelectronrepulsionbetweentwoelectronsindifferentorbitals,andtheexchangeintegralrelatestotheenergywhenelectronsintwoorbitalsareexchangedwitheachother.3IntheHckelorbitalmethodweconstructnewMOsasfollows

whereNisthenumberofatomsintheorbitalsystem,sisapzorbitalonagivenatomandCsjisacoefficientdeterminedbyprojection.Hckeltheorymakesthefollowingapproximations:

,

, ifrandsarenearestneighbors0otherwise

istheCoulombicintegralwhichraisestheenergyofawavefunctionpositivevalueandistheresonanceintegralwhichlowerstheenergyofawavefunctionnegativevalue.Theseintegralscanbeevaluatednumericallybutarebeyondthescopeofwhatwehopetoaccomplishhere.3

3Formoreinformationonbondingtheoryconsultanappropriatetextonquantummechanics.


Page77

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


Page78

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


Page79

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


Page80

Therefore;1

1

Theintegralintheaboveexpressionmustevaluateto1sincethebasisfunctionsjareorthonormalwavefunctions.SolvingforthenormalizationfactorNigives;

1

Applyingthisto2abovewefind,

and

andthenormalizedwavefunctionbecomes;

B1symmetry

Similarlywefindthenormalized3tobe

B1symmetry


Page81

Ifweevaluate,foreachneworbitalwefindthat

, 22

, 22

DistributingtheiacrosstheHamiltonianandseparatingtheintegralsweareleftwiththefollowingsum

22 , 22

22

, 22 22

, 22 22

, 22

TheseintegralsevaluateaccordingtoHckelapproximationsas

, 12 0 0

12

Simplifyingleavesuswith:

, 12

Asimilaranalysisfor2and3yields:,

,

Inorderofincreasingenergywefind213

Recallthatisanegativetermandlowerstheenergyoftheorbital.


Page82

Drawingpicturesoftheorbitalshelpstovisualizetheresults

B1antibondingorbital

43A2nonbondingorbital

B1bondingorbital

43


Page83

Nitriteorbitalscalculatedusingabinitiomethods.

B1antibondingorbital

A2nonbondingorbital

B1bondingorbital


Page84

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


Page85

Nowwemustprojectoutthesymmetryadaptedlinearcombinationsandthennormalizetheresultingfunctions.

18

14

18

14

28 2 2 12

BecauseEistwodimensional,wemustprojectanadditionalorbital.

28 2 2 12

ThetwoEMOswillbethelinearcombinationsumanddifferenceofthetwoprojectionswehavejustmade.

12

12


Page86

NormalizingtheMOsresultsinthefollowingorthonormalsetoffunctions A2symmetry

B2symmetry

Esymmetry

EsymmetryToputtheorbitalinthecorrectorderenergetically,weevaluate,foreachMO., 2

, 2

,

, BecausetheEorbitalisdegenerate,bothorbitalsmusthavethesameenergy.Wefindtheinorderofincreasingenergy,13,42.LastlywecanlookathowtheeachoftheseorbitalstransformsintheD4hpointgroupandassignthegorusubscript.Forexample,theB2orbitalcouldbeB2uorB2g.Underinversionitheorbitalgoesinto1ofitselfsoitmustbeB2uthecharacterunderiforB2gis1and1forB2u.AnalysisoftheremainingorbitsgivesA2uandEg.


Page87

B2u antibonding

2

Egbonding

A2ubonding

2molecularorbitalsascalculatedbyabinitiomethods

B2u

Eg

A2u


Page88

Anoteofcaution.WemightbetemptedtoviewcyclobutadieneasanaromaticringsystembasedupontheappearanceoftheA2uorbital.However,examinationoftheelectrondensitycolormappedwiththeelectrostaticpotentialrednegative,bluepositiverevealsanonuniformelectrondensitydistribution.Thisindicatesthatcyclobutadieneisnotaromaticbutratheralternatingdoubleandsinglebonds.

Ifthestructureisgeometryoptimized,weseethatthesymmetryisnolongerD4h,butratherthatofarectangle,D2h.Theelectrostaticpotentialmapplacesextraelectrondensityalongtheshorterdoublebondswewouldpredict.TheHckelruleforaromaticityrequires4n2electronsandherewehaveonly4electrons,sothelackofaromaticityisexpected.

Ifthesymmetrychanges,thesymmetryofthemolecularorbitalsalsochanges.Calculationsoftheneworbitalsareshownonthefollowingpage.


Page89

Abinitiocalculationsfortheorbitalsingeometryoptimizedcyclobutadiene.

Au

B3g

B2g

B1u

NoticethatthegeneralshapeoftheMOsissimilartowhatwecalculatedforthemoleculeunderD4hsymmetry,buttheEgorbitalhassplitintothenondegenerateB2gandB3gorbitalsshownabove.


Page90

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


Page91

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.


Page92

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


Page93

Step5.ConstructtheMOdiagramanddrawpicturesoftheMOs

A2

E

A2

Thesesketchesareatopdownview.Thesignofthewavefunctionisoppositeonthebottomside.


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AbinitiocalculationsgivethefollowingMOpicturesandthecorrespondingenergyinHartree.

A20.0418Ha

E0.9073Ha

A20.9411Ha


Page95

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


Page96

Abinitiocalculationsfortheorbitalsofbenzeneareshownbelow

B2g0.3480Ha

E2u0.1322Ha

E1g0.3396Ha

A2u0.5073Ha

group theory - lecture notes complete 3rd ed

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