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Chemical Reactivity in Confined Systems: Theory,

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ChemicalReactivityinConfinedSystems

Theory,ModellingandApplications

Editedby

PratimKumarChattaraj

DepartmentofChemistry

IndianInstituteofTechnologyKharagpur

Kharagpur

India

DebduttaChakraborty

DepartmentofChemistry

KatholiekeUniversiteit

Leuven

Belgium

Thiseditionfirstpublished2021 ©2021JohnWiley&SonsLtd

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LibraryofCongressCataloging-in-PublicationData

Names:Chattaraj,PratimKumar,editor. | Chakraborty,Debdutta,editor.

Title:Chemicalreactivityinconfinedsystems:theory,modellingand applications/editedbyPratimKumarChattaraj,DebduttaChakraborty.

Description:Hoboken,NJ:Wiley,2021. | Includesbibliographical referencesandindex.

Identifiers:LCCN2021025002(print) | LCCN2021025003(ebook) | ISBN 9781119684022(cloth) | ISBN9781119683384(adobepdf) | ISBN 9781119683230(epub)

Subjects:LCSH:Reactivity(Chemistry)

Classification:LCCQD505.5.C4852021(print) | LCCQD505.5(ebook) | DDC541/.39–dc23

LCrecordavailableathttps://lccn.loc.gov/2021025002

LCebookrecordavailableathttps://lccn.loc.gov/2021025003

CoverDesign:Wiley

CoverImage:©DebduttaChakraborty

Setin9.5/12.5ptSTIXTwoTextbyStraive,Chennai,India

Contents

Preface xv

ListofContributors xvii

1EffectofConfinementontheTranslation-RotationMotionofMolecules:The InelasticNeutronScatteringSelectionRule 1

1.1Introduction 1

1.2DiatomicsinC60 :Entanglement,TRCoupling,Symmetry,BasisRepresentation,and EnergyLevelStructure 4

1.2.1EntanglementInducedSelectionRules 4

1.2.2H@C60 5

1.2.3H2 @C60 7

1.2.3.1Symmetry 7

1.2.3.2SphericalBasisandEigenstates 7

1.2.3.3EnergyLevelOrderingwithRespectto �� 8

1.2.4HX@C60 10

1.3INSSelectionRuleforSpherical(Kh )Symmetry 11

1.3.1InelasticNeutronScattering 11

1.3.2GroupTheoryDerivationoftheINSSelectionRule 12

1.3.2.1Group-Theory-BasedINSSelectionRuleforCylindrical(C∞�� )Environments 12

1.3.2.2Group-Theory-BasedINSSelectionRuleforSpherical(Kh )Environments 12

1.3.3SpecificSystems,QuantumNumbers,andBasisSets 13

1.3.3.1H@sphere 14

1.3.3.2H2 @sphere 14

1.3.3.3HX@sphere 15

1.3.4BeyondDiatomicMolecules 15

1.3.4.1H2 O@sphere 15

1.3.4.2CH4 @sphere 17

1.3.4.3AnyGuestMoleculeinanySpherical(Kh )Environment 18

1.4INSSelectionRulesforNon-SphericalStructures 18

1.5SummaryandConclusions 20

Acknowledgments 22

References 22

2Pressure-InducedPhaseTransitions 25 WojciechGrochala

2.1Pressure,APropertyofAllFlavours,andItsImportancefortheUniverse andLife 25

2.2Pressure:IsotropicandAnisotropic,PositiveandNegative 26

2.3ChangesoftheStateofMatter 27

2.4CompressionofSolids 30

2.4.1IsotropicorAnisotropicCompressibility 30

2.4.2NegativeLinearCompressibility 30

2.4.3NegativeAreaCompressibility 31

2.4.4AnomalousCompressibilityChangesatHighPressure 31

2.5StructuralSolid-SolidTransitions 32

2.5.1StructuralPhaseTransitionsAccompaniedbyVolumeCollapse 32

2.5.2EffectsofVolumeCollapseonFreeEnergy 33

2.5.3Structure-InfluencingFactorsatCompression 34

2.5.4ChangesintheNatureofChemicalBondinguponCompressionanduponPhase Transitions 35

2.6SelectedClassesofMagneticandElectronicTransitions 36

2.6.1HighSpin–LowSpinTransitions 36

2.6.2ElectronicCom-vsDisproportionation 37

2.6.3Metal-to-MetalChargeTransfer 37

2.6.4Neutral-to-IonicTransitions 37

2.6.5MetallizationofInsulators(andResistingIt) 38

2.6.6TurningMetalsintoInsulators 39

2.6.7SuperconductivityofElementsandCompounds 39

2.6.8TopologicalPhaseTransitions 41

2.7ModellingandPredictingHPPhaseTransitions 41 Acknowledgements 42 References 42

3ConceptualDFTandConfinement 49

PaulGeerlings,DavidJ.Tozer,andFrankDeProft

3.1IntroductionandReadingGuide 49

3.2ConceptualDFT 50

3.3ConfinementandConceptualDFT 52

3.3.1Atoms:GlobalDescriptors 52

3.3.2Molecules:GlobalandLocalDescriptors 56

3.3.2.1ElectronAffinities 57

3.3.2.2HardnessandElectronicFukuiFunction 59

3.3.3InclusionofPressureintheE = E[N,v]Functional 63

3.4Conclusions 65 Acknowledgements 65 References 66

4ElectronicStructureofSystemsConfinedbySeveralSpatialRestrictions 69 Juan-JoséGarcía-Miranda,JorgeGarza,IlichA.Ibarra,AnaMartínez,Michael-Adán Martínez-Sánchez,MarcosRivera-Almazo,andRubiceliaVargas

4.1Introduction 69

4.2ConfinementImposedbyImpenetrableWalls 69

4.3ConfinementImposedbySoftWalls 72

4.4BeyondConfinementModels 74

4.5Conclusions 77 References 77

5UnveilingtheMysteriousMechanismsofChemicalReactions 81

SoledadGutiérrez-Oliva,SilviaDíaz,andAlejandroToro-Labbé

5.1Introduction 81

5.1.1Context 81

5.1.2IdeasandMethods 82

5.1.3Application 82

5.2EnergyandReactionForce 83

5.2.1TheReactionForceAnalysis(RFA) 83

5.2.2RFA-BasedEnergyDecomposition 84

5.2.3MarcusPotentialEnergyFunction 85

5.2.4MarcusRFA 86

5.3ElectronicActivityAlongaReactionCoordinate 87

5.3.1ChemicalPotential,Hardness,andElectrophilicityIndex 87

5.3.2TheReactionElectronicFlux(REF) 88

5.3.2.1PhysicalDecompositionofREF 88

5.3.2.2ChemicalDecompositionofREF 89

5.4AnApplication:theFormationofAminoacetonitrile 90

5.4.1EnergeticAnalysis 91

5.4.2ReactionMechanisms 91

5.5Conclusions 94 Acknowledgments 95 References 95

6APerspectiveontheSo-CalledDualDescriptor 99 F.Guégan,L.Merzoud,H.Chermette,andC.Morell

6.1Introduction:ConceptualDFT 99

6.2TheDualDescriptor:FundamentalAspects 99

6.2.1InitialFormulation 99

6.2.2AlternativeFormulations 100

6.2.2.1DerivativeFormulations 100

6.2.2.2LinkwithFrontierMolecularOrbitalTheory 101

6.2.2.3State-SpecificDevelopment 101

6.2.2.4MODegeneracy 102

6.2.2.5QuasiDegeneracy 102

6.2.2.6SpinPolarization 103

6.2.2.7GrandCanonicalEnsembleDerivation 105

6.2.3Real-SpacePartitioning 105

6.2.4DualDescriptorandChemicalPrinciples 106

6.2.4.1PrincipleofMaximumHardness 106

6.2.4.2LocalHardnessDescriptors 106

6.2.4.3LocalElectrophilicityandNucleophilicity 106

6.2.4.4LocalChemicalPotentialandExcitedStatesReactivity 107

6.3Illustrations 108

6.3.1WoodwardHoffmannRulesinDiels-AlderReactions 108

6.3.2PerturbationalMOTheoryandDualDescriptor 109

6.3.3MarkovnikovRule 109

6.4Conclusions 110 References 111

7MolecularElectrostaticPotentials:SignificanceandApplications 113 PeterPolitzerandJaneS.Murray

7.1AQuickReviewofSomeClassicalPhysics 113

7.2MolecularElectrostaticPotentials 113

7.3TheElectronicDensityandtheElectrostaticPotential 114

7.4CharacterizationofMolecularElectrostaticPotentials 115

7.5MolecularReactivity 116

7.6SomeApplicationsofElectrostaticPotentialstoMolecularReactivity 118

7.6.1 σ-Holeand π-HoleInteractions 118

7.6.2HydrogenBonding:A σ-HoleInteraction 119

7.6.3InteractionEnergies 120

7.6.4CloseContactsandInteractionSites 121

7.6.5BiologicalRecognitionInteractions 124

7.6.6StatisticalPropertiesofMolecularSurfaceElectrostaticPotentials 125

7.7ElectrostaticPotentialsatNuclei 126

7.8DiscussionandSummary 127 References 127

8ChemicalReactivityWithintheSpin-PolarizedFrameworkofDensity FunctionalTheory 135 E.ChamorroandP.Pérez

8.1Introduction 135

8.2TheSpin-PolarizedDensityFunctionalTheoryasaSuitableFrameworktoDescribe BothChargeandSpinTransferProcesses 137

8.3PracticalApplicationsofSP-DFTIndicators 141

8.4ConcludingRemarksandPerspectives 145 Acknowledgements 147 References 147

9ChemicalBindingandReactivityParameters:AUnifiedCoarseGrained DensityFunctionalView 167 SwapanK.Ghosh

9.1Introduction 167

9.2Theory 169

9.2.1ConceptofElectronegativity,ChemicalHardness,andChemicalBinding 169

9.2.1.1ElectronegativityandHardness 169

9.2.1.2InteratomicChargeTransferinMolecularSystems 169

9.2.1.3ConceptofChemicalPotentialandHardnessfortheBondRegion 170

9.2.1.4Spin-PolarizedGeneralizationofChemicalPotentialandHardness 171

9.2.1.5ChargeEquilibriationMethods:SplitChargeModelsandModelswithCorrect DissociationLimits 172

9.2.1.6DensityFunctionalPerturbationApproach:ACoarseGrainingProcedure 173

9.2.1.7AtomicChargeDipoleModelforInteratomicPerturbationandResponse Properties 174

9.2.1.8ForceFieldGenerationinMolecularDynamicsSimulation 174

9.3PerspectiveonModelBuildingforChemicalBindingandReactivity 175

9.4ConcludingRemarks 175 Acknowledgements 175 References 175

10SoftnessKernelandNonlinearElectronicResponses 179 PatrickSenet

10.1Introduction 179

10.2LinearandNonlinearElectronicResponses 181

10.2.1LinearResponseTheory 181

10.2.1.1Ground-State 181

10.2.1.2LinearResponses[1] 181

10.2.2NonlinearResponsesandtheSoftnessKernel 182

10.2.3EigenmodesofReactivity 184

10.3One-DimensionalConfinedQuantumGas:AnalyticalResultsfromaModel Functional 185

10.4Conclusion 188 References 188

11ConceptualDensityFunctionalTheoryintheGrandCanonical Ensemble 191 JoséL.Gázquez,MarcoFranco-Pérez,PaulW.Ayers,andAlbertoVela

11.1Introduction 191

11.2FundamentalEquationsforChemicalReactivity 192

11.3Temperature-DependentResponseFunctions 195

11.4LocalCounterpartofaGlobalDescriptorandNon-LocalCounterpartofaLocal Descriptor 200

11.5ConcludingRemarks 203

Acknowledgements 204

References 204

12EffectofConfinementontheOpticalResponsePropertiesofMolecules 213 WojciechBartkowiak,MartaChołuj,andJustynaKozłowska

12.1Introduction 213

12.2ElectronicContributionstoLongitudinalElectric-DipolePropertiesofAtomicand MolecularSystemsEmbeddedinHarmonicOscillatorPotential 215

12.3VibrationalContributionstoLongitudinalElectric-DipolePropertiesofSpatially ConfinedMolecularSystems 218

12.4Two-PhotonAbsorptioninSpatialConfinement 219

12.5Conclusions 220

References 221

13ADensityFunctionalTheoryStudyofConfinedNobleGasDimersin FullereneMolecules 225 DongboZhao,MengLi,XinHe,BinWang,ChunyingRong,PratimK.Chattaraj, andShubinLiu

13.1Introduction 225

13.2ComputationalDetails 226

13.3ResultsandDiscussion 227

13.3.1ChangesinStructure 227

13.3.2ChangesinInteractionEnergy 227

13.3.3ChangesinBondingEnergy 228

13.3.4ChangesinEnergyComponents 228

13.3.5ChangesinNoncovalentInteractions 229

13.3.6ChangesinInformation-TheoreticQuantities 231

13.3.7ChangesinSpectroscopy 232

13.3.8ChangesinReactivity 233

13.4Conclusions 236

Acknowledgments 236 References 236

14ConfinementInducedChemicalBonding:CaseofNobleGases 239 SudipPan,GabrielMerino,andLiliZhao

14.1Introduction 239

14.2ComputationalDetailsandTheoreticalBackground 241

14.3TheBondinginHe@C10 H16 :ADebate 243

14.4ConfinementInducingChemicalBondBetweenTwoNgs 244

14.5XNgYInsertionMolecule:ConfinementinOneDirection 251

14.6Conclusions 254

Acknowledgements 255

References 255

15EffectofBothStructuralandElectronicConfinementsonInteraction, ChemicalReactivityandProperties 263

MaheshKumarRavva,RavinderPawar,ShyamVinodKumarPanneer,VenkataSuryaKumar Choutipalli,andVenkatesanSubramanian

15.1Introduction 263

15.2GeometricalChangesinSmallMoleculesUnderSphericalandCylindrical Confinement 264

15.3HydrogenBondingInteractionofSmallMoleculesintheSphericalandCylindrical Confinement 265

15.4SphericalandCylindricalConfinementandChemicalReactivity 267

15.5ConcludingRemarks 268 References 270

16EffectofConfinementonGasStoragePotentialandCatalyticActivity 273 DebduttaChakraborty,SukantaMondal,RanjitaDas,andPratimKumarChattaraj

16.1Introduction 273

16.2EndohedralGasAdsorptionInsideClathrateHydrates 274

16.3HydrogenHydrates 276

16.4MethaneHydrates 278

16.5NobleGasHydrates 279

16.6ConfinementInducedCatalysisofSomeChemicalReactions 280

16.7Outlook 285 Acknowledgements 285 References 286

17EngineeringtheConfinedSpaceofMOFsforHeterogeneousCatalysisof OrganicTransformations 293 TapanK.Pal,DineshDe,andParimalK.Bharadwaj

17.1Introduction 293

17.2CatalysisattheOpenMetalSites 293

17.2.1MOFsEndowedwithOpenMetalSite(s) 294

17.2.2RemovalofVolatileMoleculesFromMetalNodestoPerformCatalysis 297

17.2.3CatalysisattheMetalnodePostTransmetalation 299

17.3FunctionalizationintheMOFtoFurnishCatalyticSite 301

17.3.1AttachingtheCatalyticallyActiveMoietiestotheMetalNodes(SBU) 301

17.3.2PreconceivedCatalyticSiteintotheLinker 301

17.3.3PostSyntheticModificationoftheLinker 304

17.3.4MOFswithLinkersHavingCoordinatedMetalIons(Metalloligands) 306

17.4MOFsasBifunctionalCatalyst 310

17.5Impregnation/EncapsulationofNanoparticlesintheMOFCavityforCatalysis 317

17.6EngineeringHomochiralMOFsforEnantioselectiveCatalysis 320

17.7Conclusion 325

Acknowledgements 326 References 326

18ControllingExcitedStateChemistryofOrganicMoleculesinWaterThrough Incarceration 335 V.Ramamurthy

18.1Introduction 335

18.2ComplexationPropertiesofOA 337

18.3PropertiesofOAcapsule 339

18.4DynamicsofEncapsulatedGuests 340

18.5DynamicsofHost-GuestComplex 346

18.6RoomTemperaturePhosphorescenceofEncapsulatedOrganicMolecules 353

18.7ConsequenceofConfinementonthePhotophysicsofAnthracene 356

18.8SelectivePhoto-OxidationofCycloalkenes 358

18.9RemoteActivationofEncapsulatedGuests:ElectronTransferAcrossanOrganic Wall 360

18.10Summary 362

Acknowledgements 363 References 363

19EffectofConfinementonthePhysicochemicalPropertiesofChromophoric Dyes/DrugswithCucurbit[n]uril:ProspectiveApplications 371 J.Mohanty,N.Barooah,andA.C.Bhasikuttan

19.1Introduction 371

19.1.1ConfinementofDyes/DrugsinMacrocyclicHosts 372

19.1.1.1Cyclodextrins 372

19.1.1.2Calixarenes 373

19.1.1.3Cucurbiturils 373

19.2ConfinementinCucurbiturilHosts:EffectsonthePhysicochemicalPropertiesofGuest Molecules–AdvantagesforVariousTechnologicalApplications 374

19.2.1EnhancedPhotostabilityandSolubilityofRhodamineDyes 375

19.2.1.1Water-BasedDyeLaser 376

19.2.2EnhancedLuminescenceandPhotostabilityofCH3 NH3 PbBr3 Perovskite Nanoparticles 377

19.2.3EnhancedAntibacterialActivityandExtendedShelf-LifeofFluoroquinoloneDrugs withCucurbit[7]uril 377

19.2.4EffectofConfinementonthePrototropicEquilibrium 379

19.2.4.1Salt-InducedpKa TuningandGuestRelocation 379

19.2.5ConfinementinCucurbit[7]uril-MediatedBSA:Stimuli-ResponsiveUptakeand ReleaseofDoxorubicin 380

19.2.6EffectofConfinementontheFluorescenceBehaviorofChromophoricDyeswith Cucurbiturils 380

19.2.6.1FluorescenceBehaviorofChromophoricDyeswithCucurbit[7]uril 381

19.2.6.2FluorescenceBehaviorofChromophoricDyeswithCucurbit[8]uril 383

19.2.7EffectofConfinementontheCatalyticPerformancewithinCucurbiturils 386

19.3Conclusion 388 Acknowledgement 389 References 389

20Box-ShapedHosts:EvaluationoftheInteractionNatureandHost CharacteristicsofExBoxDerivativesinHost-GuestComplexesfrom ComputationalMethods 395

GiovanniF.CaramoriandAlvaroMuñoz-Castro

20.1Introduction 395

20.2NoncovalentInteractionsThroughEnergyDecompositionAnalysis 396

20.3Ex0 Box4+ (CBPQT4+ ) 398

20.4ExBox4+ andEx2 Box4+ 400

20.5LargerBoxes 406

20.6NMRFeatures 408

20.7AllCarbonCounterpart 409

20.8Conclusions 409

Acknowledgments 410 References 411

Index 417

Preface

Confinedsystemsoftenexhibitunusualbehaviorregardingtheirstructure,stability,reactivity, bonding,interactionsanddynamics.Quantizationisadirectconsequenceofconfinement. Confinementmodifiestheelectronicenergylevels,orbitals,electronicshellfilling,etc.,ofa systemtherebyaffectingitsreactivityaswellasvariousresponsepropertiesascomparedtothe correspondingunconfinedsystem.Confinementmayenforcetworaregasatomstoformapartly covalentbond.Gasstorageisfacilitatedthroughconfinementandunprecedentedoptoelectronic propertiesareobservedincertaincases.Someslowreactionsgethighlyacceleratedinan appropriateconfinedenvironment.

Analyzingthereactivityofatomsandmolecules,presentinaconfinedenvironment,byutilizingvarioustheoreticalandcomputationalmethodscanunravelnumerousnewparadigms vis-à-vis physicochemicalpropertiesofthesystemsunderconsideration.Therefore,confinedquantumsystemshavebeenextensivelyanalyzedfrombothepistemologicalandappliedpointsofview.The crucialpointwhileanalyzingquantumconfinedsystemsistobeabletoconstructanaccuratetheoreticalmodelthattakesintoaccountchangesintheelectronicwavefunctionduetotheeffectof confinement.Tothisend,modeltheoreticalcalculationscouldbeconceivedbysuitablechoiceof theboundarycondition.Host–guestcomplexesprovideanideal‘real’platformwherethechanges ofreactivityandresponsepropertiesofvarioussystemscouldbeunderstood.Tothisend,densityfunctionaltheory(DFT)basedcalculationprovidesacost-effectiveandreasonablyaccurate method.

Ontheotherhand,experimentalstudieshavehelpedtoshedlightonmanyfascinatingaspectsof confinement.Theutilityofconfinedsystemscouldbegaugedinvariousdisciplinessuchascatalysis,gasstorage,designingsuperioroptoelectronicandmagneticmaterials,etc.

Inthisbook,severalexperts,whohavemadeseminalcontributionsinunderstandingthebehaviorofconfinedsystems,havewrittenauthoritativeaccountsonstate-ofthe-artresearchtopics encompassingboththeoreticalaswellasexperimentalendeavors.Hopefully,thisbookwillbebeneficialforgraduatestudentsinchemistry,materialsscienceandphysicsinunderstandingtherecent developmentsinthisfield.

ListofContributors

PaulW.Ayers

McMasterUniversity

Canada

NilotpalBarooah BhabhaAtomicResearchCentre

Mumbai India

WojciechBartkowiak WroclawUniversityofScienceandTechnology

Poland

ParimalKantiBharadwaj IndianInstituteofTechnologyKanpur

India

A.C.Bhasikuttan

BhabhaAtomicResearchCentre

Mumbai India

GiovanniF.Caramori FederalUniversityofSantaCatarina–UFSC Brazil

EduardoChamorro UniversidadAndrésBello

Chile

DebduttaChakraborty KatholiekeUniversiteit

Leuven

Belgium

PratimKumarChattaraj IndianInstituteofTechnologyKharagpur Kharagpur India

HenryChermette UniversitédeLyon1 France

MartaChołuj WroclawUniversityofScienceandTechnology Poland

SilviaDíaz PontificiaUniversidadcatólicadeChile Chile

FrankDeProft DurhamUniversity Durham UK

MarcoFranco-Pérez UniversidadNacionalAutónomadeMéxico Mexico

JorgeGarzaOlguín UAM-Iztapalapa Mexico

JoséLuisGázquez UniversidadAutónoma Metropolitana-Iztapalapa Mexico

ListofContributors

PaulGeerlings VrijeUniversiteitBrussel Belgium

FrédéricGuégan UniversitédePoitiers France

SoledadGutiérrez-Oliva PontificiaUniversidadCatólicadeChile Chile

SwapanKumarGhosh UniversityofMumbai India

WojciechGrochala UniversityofWarsaw Poland

JustynaKozłowska WroclawUniversityofScienceandTechnology Poland

ShubinLiu UniversityofNorthCarolina USA

GabrielMerino Cinvestav Mexico

LyndaMerzoud UniversitédeLyon1 France

JyotirmayeeMohanty BhabhaAtomicResearchCentre Mumbai India

ChristopheMorell UniversitédeLyon1 France

AlvaroMuñozCastro UniversidadAutónomadeChile Chile

SudipPan Philipps-UniversitätMarburg Germany

PatriciaPerez UniversidadAndrésBello Chile

L.WilliamPoirier TexasTechUniversity USA

PeterPolitzer UniversityofNewOrleans USA

VaidhyanathanRamamurthy UniversityofMiami USA

ChunyingRong HunanNormalUniversity China

PatrickSenet UniversitédeBourgogneFranche-Comté (UBFC) France

VenkatesanSubramanian CSIR-CentralLeatherResearchInstitute India

AlejandroToro-Labbe PontificiaUniversidadCatolicadeChile Chile

DavidJ.Tozer DurhamUniversity Durham UK

RubiceliaVargas UniversidadAutónomaMetropolitana Iztapalapa

AlbertoVela Cinvestav Mexico

DongboZhao YunnanUniversity China

LiliZhao NanjingTechUniversity China

EffectofConfinementontheTranslation-RotationMotionof Molecules:TheInelasticNeutronScatteringSelectionRule

TexasTechUniversity,USA

1.1Introduction

Oneofthecoretheoreticalideasusedtounderstandthedynamicsoffreemoleculesisthe simplifyingnotionthattheoverall(i.e.,center-of-mass)translationalmotioncanbecleanlyseparatedfrominternalvibrationsandrotations.Indeed,thisseparationissouniversallyapplied–and thetranslationalmotionsoeasilydealtwith–thatitcanbeeasyto“forget”thatthelattereven exists!Ontheotherhand,ifyouplacethatsamemoleculeinananoconfinedenvironment,the situationcanbevastlydifferent.Firstandforemost,thecontinuumofstatesthatcharacterizes translationalmotionforfreemoleculesnecessarilybecomes quantized intheconfinedcontext. Thequantumtranslationalstatesforthetrapped“guest”moleculecanbe“particle-in-a-boxlike” ormorecomplicated,dependingonthenatureoftheexternalfieldprovidedbythecagestructure. Ofcourse,largercagesgiverisetosmallertranslationallevelspacings–whichareinanyevent generallysmallerthantherotationallevelspacings(and much smallerthanthevibrationallevel spacings).Iftheconfinementisverysevere,however,thenthetranslationalandrotationallevel spacingscanbecomecomparabletoeachother–andevenstronglycoupled.

ThisisthesituationforsmallmoleculestrappedinsideC60 fullerenecages–e.g.,H2 @C60 [1–16],HD@C60 [4,7,12,15,17,18],HF@C60 [14–16,19],H2 O@C60 [14–16,20,21],and CH4 @C60 [15,22],allofwhichwillbeconsideredinthischapter.Basedonthephysicalsizeof thefullerene(diameter ≈ 7Å),onemightwellimaginethatanumberofguestmoleculescould becrammedintoasingleC60 cage.Inreality,theguestmoleculesaretrappedvialong-range vanderWaalsinteractionsthatpreventthemfromgettingcloserthanafewÅfromthecage wall.Consequently,the effective cagesizeismuchsmaller–ontheorderofthesizeoftheguest moleculeitself.Thisimpliesthat:(a)onlyoneguestmoleculecanfitinsideasingleC60 cage; (b)thecorrespondingtranslationallevelspacingiscomparabletotherotationallevelspacing.It ishardlysurprising,then,toalsofindthattranslationandrotationareindeedstronglycoupled inthesesystems.Thiscomplicateslifefromatheoretical/computationalstandpoint,forwhich onemustadoptanexact,coupledquantumdynamicaltreatment,encompassingallrelevant translation-rotation(TR)guestmoleculedegreesoffreedom[3–5,15,16,19,23–25].Thatsaid,the combinedTRstatesthatresultarenotnecessarilyentirelydevoidofstructureeither–itisjustnot thatofthestandardformthatoneexpectsintermsofTRseparability.

*Email:bill.poirier@ttu.edu

ChemicalReactivityinConfinedSystems:Theory,ModellingandApplications, FirstEdition.EditedbyPratimKumarChattarajandDebduttaChakraborty. ©2021JohnWiley&SonsLtd.Published2021byJohnWiley&SonsLtd.

1EffectofConfinementontheTranslation-RotationMotionofMolecules

Inadditiontoprovidingafundamentallydifferentspectroscopicpicture,thestrongconfinementinducedTRcouplingalsogivesrisetoaremarkablephysicaleffectthatwasoncethoughttobe impossible– selectionrules forinelasticneutronscattering(INS)[26–28].INSisanexperimental techniqueinwhichabeamofneutronsisscatteredthroughnuclearforceinteractionswiththe nucleiofthetargetsample.Itisanextremelyusefultoolforprobingnanoconfinedhydrogen,in largepartbecausetheHatomnucleusprovidesthe largest neutronscatteringcrosssectionacross theentireperiodictable.Eventhatofthe“secondplace”contender–i.e.,deuterium–ismorethan anorderofmagnitudesmaller.INSoffersotheradvantagesaswell,suchastheabilitytoexamine transitionsbetweenindividualquantumstates–includingortho-paranuclearspintransitionsin H2 ,whichwouldbeforbidden,e.g.,inanelectricdipoleorevenRamanfar-infrared(IR)spectrum. However,whereassuchopticallyforbiddenspectroscopictransitionscertainlyareallowedinINS, thisisnoguaranteethat allother transitionsarealsonecessarilyallowed.

ThatINSspectroscopydoesindeedhaveforbiddentransitions,wasfirstproposedandexperimentallyverifiedforH2 @C60 ,inastunningsetofpapersbyBa ˇ ci ´ c,Horsewill,andcoworkers [9,10,29,30].Inthisearlierwork,inadditiontocomputingtheTRquantumeigenstatesthemselves (energylevelsandwavefunctions)[3–5],anexplicitnumericalsimulationoftheexperimentalINS spectrumwasalsoperformed[9,12,29,30].Theserepresentheroiccalculations,appliedtoeachTR transitionindividually,whichalsotakevariousexperimentalcircumstancesintoproperaccount. TheendresultisareasonablyaccuratepredictionofbothINStransitionenergiesandintensities(both“stickspectra”andmoreexperimentally-relevantconvolvedspectracanbeobtained). Inthismanner,itwasdiscoveredthatsometransitionintensitiesforH2 @C60 arevastlysmaller thanothers–byfourormoreordersofmagnitude.Thisprovidedexcellentnumericalevidence foraselectionrule–which,indeed,wassubsequentlyconfirmedthroughactualINSexperiments [7,8,10].

Asimpressiveandunexpectedasthisdiscoveryprovedtobe,oneofthedrawbacksoftheabove approachisthatonemustinferageneralpatternfortheselectionrule,fromamongstanecessarily limitedsetofspecifictransitions.Indeed,thisledBa ˇ ci ´ candcoworkerstoinitiallyassumeaselection ruleforH2 @C60 ofthefollowingform:

restrictedINSselectionruleforH2 @C60 (fromp-H2 groundstate):Transitionsare forbiddentoallstatesforwhich (j + l ��)= 1[note:alltermswillbeexplained].

Infact,thecorrectrulefortransitionsstartingfromtheH2 @C60 groundstateismoregeneral:

correctINSselectionruleforH2 @C60 (fromp-H2 groundstate):Transitionsareforbiddentoallstatesforwhich (j + l ��)= odd.

Totheauthor’sknowledge,thelatterformabovewasfirstproposedbytheauthorhimself,ata scientificmeetinginMay,2015.SinceconfirmedassignmentsfortheexperimentalH2 @C60 data didnotexistbeyond (j + l ��)= 2,thetheoreticalsimulationswerenotperformedbeyondthis pointeither,andsotheavailableinformationatthetimewasconsistentwith either ofthetworules above.Thisstateofaffairsmotivatedthepresentauthortodevelopa grouptheoretical derivationof thegeneralINSselectionrule[11].Inparallelwiththiseffort,Ba ˇ ci ´ candcoworkersextendedthe analyticalpartoftheircalculationsofthetransitionintegrals[12],toencompassallpossibleinitial andfinalstates.Bothapproachesthengaverisetothefollowing,

mostgeneralINSselectionruleforH2 @C60 :Transitionsareforbiddenbetweenstates forwhich (j + l ��) changesfromeventoodd(orvice-versa),andatleastone �� = 0.

Aswillbedescribedinthischapter,thegrouptheoryapproachprovidesphysicalunderstanding, andalsohasthegreatadvantagethatitleadstothecorrectandcompletelygeneralINSselection rulefortheappropriatesymmetrygroup,“allatonce.”However,thisapproachhassomelimitations andissues,aswell,whichwillalsobeaddressed.Tobeginwith,itdoesnotprovideanyintensity informationforallowedtransitions;forthis,itisnecessarytocalculatetransitionintegralsexplicitly,asperBa ˇ ci ´ cet.al.Secondlythereisthequestionofchoosingthemostcorrectsymmetrygroup toworkwith,intermsofexperimentalrelevance.ThisisaparticularlyimportantquestionforINS spectroscopy,forwhichthereisapreferreddirectionororientation–i.e.,thatofthe momentum transfervector , ⃗ k,ofthescatteredneutronbeam.Thirdly,therearesomegrouptheoreticalcomplicationsthatarise,duetothefactthattheINSinteractionoperatoritselfisincommensuratewithmost ofthestandardmolecularpointgroups–akeydifference,e.g.,fromopticalspectroscopy.Finally, alltruegroup-theory-basedselectionrulesareexpressedintermsofthe irreduciblerepresentations (irreps)oftheappropriatesymmetrygroup.Whilethisformofaselectionruleistrulyuniversal,for specificsystemsitisstillnecessarytoassignirreplabelstoindividualquantumstatesand/orbasis functions–therebymakinganassociationwiththepertinent quantumnumbers forthosesystems. Onbalance,itisclearthat both thegrouptheoryandexplicitintegralapproachesarenecessaryfor interpretingandpredictingINSexperiments,astheyprovidecomplementaryunderstanding.

Since2015,Ba ˇ ci ´ c,Horsewill,Felker,andothershavecontinuedtoexploretheINSselection rule–togetherwithotherTReffectsofnanoconfinement–throughafruitfulsynergythat hasemergedbetweentheoryandexperiment.ThesehighlyinterestingTReffectshavenow beenobservedand/orpredictedacrossarangeofnanoconfinedsystems.RegardingtheINS selectionruleitself,theirrep-basedversionof[11]isinprinciplevalidforanymoleculeinany sphericalenvironment(section1.3.2).Intermsofspecificbasissetsandquantumstates,Ba

ci ´ c andcoworkershavegeneralizedtheruleforany diatomic moleculeinasphericalenvironment, usingtheirexplicitintegralapproach.TheyhavealsoappliedthismethodologytobothHD@C60 [4,7,12,15,17,18]andHF@C60 [14–16,19],anddiscoveredforbiddenINStransitionsinbothof thesenanoconfinedsystems.ExperimentsforH2 O@C60 [12,20]suggestaselectionruleforthis systemaswell.AlloftheserecentfindingsindicatethatforbiddenINStransitionsmaybemuch moreprevalentthanoriginallyrealized,withfuturestudiespotentiallyaddressinglargercage systemssuchasH2 @C70 [5],(H2 )2 @C70 [31],etc.Otherrefinementsincludethedevelopmentof improvedpotentialenergysurfaces(PESs)[32],and/ortheexplicitincorporationofvibrational effects.

Anotherimportantrecentdirectionhasbeentheconsiderationofthe largerenvironment surroundingagivenC60 cage,andtheroleof intercageinteractions [14,16,19,33,34].Forexample, FelkerandBa ˇ ci ´ chaveexaminedelectric-dipolecouplingbetweenthetwoguestH2 Omoleculesin (H2 O@C60 )2 [33],whereasRoyandcoworkershavestudiedthedipoleinteractionin(HF@C60 )n whenthe n cagesforma“peapod”arrangement[34].Onefascinatingandrelatedfindingpertains tothesmall(∼1cm 1 )butpuzzlinglevel-splittingobservedexperimentallyinthe j = 1statesof H2 @C60 ,HF@C60 ,andH2 O@C60 [6,13–16,19,20].Thesesplittingsbreaktheexpectedspherical oricosahedrallevelpattern,implyingareductioninsymmetrycausedbythelargerenvironment. Whereasvariousexplanationshavebeenprofferred(includingtheaforementionedguest–guest dipoleinteraction),theprecisecausewasrecentlydefinitivelyestablished[14–16]asbeingdueto

1EffectofConfinementontheTranslation-RotationMotionofMolecules

neighboringC60 cagesinthesolid–arrangedinanalternating“P”orientation–whichreducethe molecularpointgroupto S6 .

Fromagrouptheoryperspective,oneofthemostintriguingoftheaboverecentresultsarethe forbiddenINStransitionsthathavebeenpredictedforbothHD@C60 [12,15,18]andHF@C60 [19].RecallthatneutronsonlyinteracteffectivelywithHatomnuclei–ofwhichthesetwosystems eachcontainonlyone.Yet,fromgrouptheoryarguments[35,36],ithasbeenestablishedthat thereareno single-atomwavefunctionsthatbelongtotheirrepsnecessarytoeffectaforbiddenINS transition.Atleasttwoparticlesarenecessarytorealizetherequisiteirreps–inmuchthesameway thatindiatomicmolecules,atleasttwoelectronsarerequiredtorealizeanelectronicstatewith Σ character[37,38].Thisstateofaffairsunderscorestheneedforfurthertheoreticaldevelopmentin areconcilingvein–thereby,inpart,motivatingthepresentandfutureeffort.Theresolutionofthe dilemmawillbepresentedinsection1.2.1,butcanbesummarizedinasingleword: entanglement. Evidently,recentdevelopmentsinthestudyofquantumconfinementeffectsintheTRdynamics ofsmallmoleculesofferusplentyofmaterialtoponder.Inanyevent,thefieldisclearlyproceedingataveryrapidpace,beyondwhatcanbereviewedindetailinthissinglechapter.Accordingly, wechoosetofocushereprimarilyontheINSselectionruleasitappliestodiatomicmolecules confinedtospherical(orsphere-like)environments.Aspertheprecedingparagraph,animportant aimistoreconcilethegrouptheoryandexplicitintegralapproaches,inamannerthatwillstimulatemutualdevelopmentsgoingforward,andotherwisestreamlinefutureprogress.Amongother aspects,thiseffortnecessarilyrequiresthatanexplicitassociationbemadebetweenirreplabels andquantumstates/basisfunctions/quantumnumbers,forspecificmolecularsystems.Finally,we addressprospectsformoving beyond allpreviousINSselectionruleapplicationstodate–byconsideringguestmoleculesthatarelargerthandiatomic,and/orhostcagesotherthanspherical.Indeed, wealsoderivehereaquantum-number-basedINSselectionrulefor H2 O@C60 ,anduseittoprovide unambiguoustransitionassignmentsfortheexperimentalINSspectrum[12,20](section1.3.4.1).

1.2Diatomicsin C60 :Entanglement,TRCoupling,Symmetry,Basis Representation,andEnergyLevelStructure

1.2.1EntanglementInducedSelectionRules

Intheusualgrouptheoryprocedure,selectionrulesaredeterminedbysandwichingtheinteraction operatorbetweenbraandketsystemstates,anddecomposingtheresultanttriple-direct-productof irrepsintoitsownirreducibleform.(ThisprocedureisworkedoutindetailforthecaseofINSspectroscopyinsection1.3.1.)Quantummechanics,however,requirescarefulconsiderationofwhat, exactly,ismeantby“thesystem.”Inparticular,theapproachaboveisonlyvalidifthesystemis notappreciablyentangledwithitssurroundings.Putanotherway,itpresumesaseparableproduct formforthe“totalsystem”wavefunction:

Ψtotal = �� × ��surroundings , (1.1) where �� isthewavefunctionofthesystemitself.Therefore,whendefiningwhichpartofourexperimentalsampletoregardas“thesystem,”thisshouldbetakenasthesmallestpiecethatincludesall oftheconstituentparticlesofinterest, andalso leadstoanunentangledstateoftheEq.(1.1)form. Forthepresentpurpose,weareconsideringadiatomicguestmoleculetrappedwithinaC60 cage.ThestrongestINSsignalsareobservedwhenatleastoneofthetwoatomsishydrogen.Letus thereforeconsideradiatomicofthe“HX”variety,whereXisanyatom other thanhydrogen.Inthe

HX@C60 system,onemaypresumethatthereislittleentanglementbetweenthehostcageandthe guestdiatomic.Nevertheless,theHatomis strongly entangledwithX;HXisamolecule,afterall. Consequently,thesmallestunentangledunitthatcanprovideanINSsignalistheHXmolecule, whichmustbetakenas“thesystem”forgrouptheorypurposes.Wemustworkwithquantum wavefunctionsofthecombinedHXsystem–notofHitself–eventhoughthenucleusXismostly “dark,”insofarastheneutronbeamisconcerned.Thissituationisreminiscentofthemannerin whichtheexistenceofablackholemaybeinferredfromtheobservedbehaviorofitspartnerstar, eventhoughtheblackholeitselfcannotbedirectlyseen.

Includingboththe“bright”andentangled“dark”particlesinthedefinitionof“thesystem”can giverisetodifferentselectionrulesthanifjustthebrightparticlesalonewereused.Thisrather intriguingsituationgivesrisetoaneffectthatmightbecalled entanglementinducedselectionrules (EISR).Inthiswork,wedemonstratethattheforbiddenINStransitionspredictedforHD@C60 and HF@C60 areindeedduesolelytoentanglementwiththedarknucleus,X=DorF–andtherefore disappearentirelyforthesystemH@C60 .Inprinciple,theEISRideashouldnotberestricted toINSspectroscopy,butcouldalsobeapplied,e.g.,inopticalspectroscopy.SowhyisEISR evidentlynotdiscussedinthelattercontext?Infact,itturnsoutthattheEISReffectsometimes does manifestinopticalspectroscopy,butwhenitdoesso,itisknownbyother,specificnames inspecificinstances–e.g.,the“Herzberg-TellerEffect”[37].InterestinglyintheHerzberg-Teller context,EISRservesto remove anelectric-dipoleforbiddentransition,whereasforHX@C60 INS spectroscopy,EISRdoestheopposite!Moregenerally,thisintriguingnewphenomenon–or properly,newlensthroughwhichtointerpretphenomena–promisestoprovideinteresting insightandconnectionsacrossaverybroadrangeofapplications.EISRthereforedemandsfurther study,andwillserveasthefocusoffutureinvestigations.

1.2.2H@C60

ForH@C60 ,thesole“bright”nucleus(intheINSsense)isthatoftheguestHatom,whichwillnot beappreciablyentangledwiththeC60 host.Accordingly,aspersection1.1,therelevantquantum systemisjusttheHatomitself.ThecorrespondingquantumHamiltonian, ̂ H ,isconstructedby fixingtheC60 cage,andallowingthepositionoftheHatom, ⃗ r =(x , y, z),tovarywithinthecage. Theexplicitdimensionalityofthequantumdynamicalproblemisthusthree(3D),corresponding topuretranslationalmotiononly.

Thesymmetrygroupisdefinedasthesetofsymmetryoperationsthatleave ̂ H invariant[37]. NotethatthePEScontributionto ̂ H (i.e. ̂ V ),describingtheexternalfieldthatisfeltbytheHatom, arisesmainlyfrompair-wisevanderWaalsinteractionswithindividualCatomsofthehostcage, C60 .Consequently,keeping ⃗ r fixed,buttransformingtheC60 underanyofitsownpointgroup operations,clearlyleavestheH@C60 ̂ H invariant.Conversely,keepingtheC60 fixedbutapplying itspointgroupoperationsto ⃗ r alsoleaves ̂ H invariant.Therearenootheroperationsappliedto ⃗ r that alsodoso;consequently,theH@C60 symmetrygroupmustbenolargerthantheC60 pointgroup, Ih .Itmightinprinciplebeasubgroupof Ih ,ifmultiple Ih operationsappliedto ⃗ r weretoresult inthesametransformedpoint, ⃗ r t .Whereasthisistrueforcertain ⃗ r pointswithspecialsymmetric orientations,inthegeneralcase,each Ih operationyieldsadistinct ⃗ r t point–resultingin120such pointsinall.TheH@C60 symmetrygroupistherefore Ih

Although Ih isformallythecorrectsymmetrygroupforH@C60 , Ih hasthegreatestsymmetryofallfinitepointgroups,andisotherwisealargesubgroupofthe sphericalpointgroup, Kh [35–37,39].Morespecifically,thestructureoftheC60 cageisitselfnearlyspherical;insolidangle space,oneisneververyfarawayfromaCatom.Moreover,thevanderWaalsnatureoftheexternal

1EffectofConfinementontheTranslation-RotationMotionofMolecules

fieldensuresthat ̂ V iseven more nearlyperfectlysphericalthanthefullereneitself.Thisimplies thatthetrueenergyeigenstatewavefunctionsareverynearlysphericallysymmetric,andtherefore well-describedbyasphericalbasisoftheform

where ⃗ r =(r ,��,��) isnowexpressedinsphericalcoordinates.Thisleadstoanaturalstatelabeling schemeintermsofthequantumnumbers (n,��, m�� ),whichcanbedirectlyrelatedtoirrepsofthe Kh pointgroup[35,37](Table1.1).

Notethatthesphericaleigenstateshavea (2�� + 1)-folddegeneracy,whereasnoirrepof Ih has agreaterthanfive-folddegeneracy.Thus,totheextentthatthetrueH@C60 eigenstatesdeviate fromsphericalsymmetry,theenergylevelstructureshouldexhibitaslightdegeneracy-lifting,as willbediscussedexplicitlyforH2 @C60 insection1.2.3.Wecanusegrouptheory–specifically,the correlationtablebetween Kh and Ih ,presentedinTable1.1–totelluspreciselyhowthisdecompositionoccurs,intermsof Ih irreps.Notethat,whereasingeneral,the Kh irrepsarelabeledbyboth �� and (g∕u) inversionparity,forthe Y��,m�� (��,��) sphericalharmonicbasisfunctions,only half ofthese irrepsareactuallyrealized,because (g∕u) parityisdeterminedbytheeven/oddnessof �� [35,36]. ThisfactwillbecomequiteimportantinourdiscussionofINSselectionrules(section1.3).

NotealsothatthespecificradialfunctioninEq.(1.2)isimmaterial,withrespecttothesymmetry characterofthecorrespondingeigenstate;inprinciple, Rn�� (r ) cantakeonanyform.Inallearlier studiesoftheH2 @C60 INStransitionintensities,anexplicit,isotropicharmonicoscillatorbasiswas presumed[3–5,9,12,14–16].Suchachoicemightbeexpectedtobeappropriatefortheverytight confinementinC60 –andindeed,turnsouttobereasonableforalldiatomicsystemsconsidered thusfar,exceptforHF@C60 .Forlargersphere-likecavities,however,thegreatestdensitywilllie

Table1.1 Correlationtable,indicatingthemultipoledecompositionof Kh irreps,withrespecttothe C ∞�� and Ih subgroups.Irrepsthatare not realizedbythesingle-particlesphericalharmonicbasisfunctions, Y��,m�� (��,��),are listedinbold;notethealternatingpatternwithincreasing ��.

Correlatesto:Correlatesto:

7 J g ⋮ T1g ⊕ T2g ⊕ G g ⊕ H g J u ⋮ T1u ⊕ T2u ⊕ G u ⊕ H u

8 K g ⋮ T2g ⊕ G g ⊕ 2H g K u ⋮ T2u ⊕ G u ⊕ 2H

9

1.2Diatomicsin C60 :Entanglement,TRCoupling,Symmetry,BasisRepresentation,andEnergyLevelStructure

notinthecenterofthecavityat r = 0,butatsomenonzeroradialvalue.Forsuchsystems,the harmonicbasiswillnolongerbeappropriate.Ontheotherhand,neitherisaharmonicbasisnecessary,inorderfortheINSselectionruletoremainvalid.Thisisanimportantgroup-theory-based generalizationthatgoesbeyondtheearlyexplicit-integralresults,althoughthelatterhavesince beengeneralizedaswell–e.g.,fortheHF@C60 system[19](section1.3.3.3).

1.2.3 H2 @C60

1.2.3.1Symmetry

ForH2 @C60 ,thetwoHatomswillinteractstronglywithanincidentneutronbeam.Theseatoms arealsostronglyentangledtoeachother–butagain,arenotexpectedtobeappreciablyentangled withtheC60 host.TherelevantquantumsystemisthustheH2 guestmoleculeitself.Thecorresponding ̂ H isconstructedbyfixingtheC60 cage,andallowingthepositionsofthetwoHatoms, ⃗ r 1 =(x1 , y1 , z1 ) and ⃗ r 2 =(x2 , y2 , z2 ),tovary.Theexplicitdimensionalityofthequantumdynamical problemisthussix(6D),correspondingtothreeoveralltranslationalcoordinates,plustworotationalcoordinates,plusasinglevibrationalcoordinate(whoseinclusionwouldformthe“TRV ̂ H ”). Inpractice,theH2 vibrationalmotionisoftenfrozen,sothat r = |⃗ r 1 ⃗ r 2 | = const–thereby reducingthedimensionalityoftheresultant“TR ̂ H ”tofive(5D)[3–5,9].Therationaleforthis simplificationisthatthefundamentalvibrationalfrequencyof(gasphase)H2 is4142cm 1 –which isordersofmagnitudelargerthanthetypicalnanoconfinedTRlevelspacing,andtheenergyrange oftheINSexperiments.Nevertheless,eventhegroundvibrationalstateofH2 @C60 isnotvery localizedin r –whichcouldleadtosmallcorrectionsinthelow-lyingTRlevels,thusproviding somemotivationforperformingTRVcalculationsinfull6D.Here,however,wefocusonlyon5D TRcalculations.

Followinganalogousargumentsasinsection1.2.2,wefindthattheTR(orevenTRV) ̂ H isagain invariantwithrespecttoall Ih symmetryoperations,appliedtobothHatomssimultaneously.In addition,exchangeofthetwoHatompositions–i.e., ⃗ r 1 ↔ ⃗ r 2 –alsoleaves ̂ H invariant.Sincethis (12)exchangepermutationoperationcommuteswithallofthe Ih pointoperations,theH2 @C60 symmetrygroupisthus

(12) h = Ih ⊗ S2 = Ih ⊗ {E, (12)} (1.3)

If Γ isan Ih irrep,wedenotethecorrespondingpermutationsymmetric/antisymmetricirrepof theH2 @C60 symmetrygroupas Γ(s∕a) .

Alsoaspersection1.2.2,thelow-lying5DTReigenstatesofH2 @C60 havebeenfoundtobenearly sphericallysymmetric[9]–leadingagaintoanaturalangular-momentum-basedlabelingscheme. Considertheusualcenter-of-masscoordinatetransformation,

R =(⃗ r 1 + ⃗

Workinginsphericalcoordinates ⃗ R =(R, Θ, Φ) and ⃗ r =(r ,��,��),andrecallingthat r = const,anaturaluncoupledbasisisthus

Rnl (R)Ylml (Θ, Φ)Yjmj (��,��) (1.5)

1.2.3.2SphericalBasisandEigenstates

Bycombiningthetwoangularmomenta–i.e.,the“orbital” ⃗ l associatedwithtranslation,andthe “rotational” ⃗ j –oneobtainsaTRcoupledbasis, |nj�� lm�� ⟩ = |nl⟩|j�� lm�� ⟩,whichcanbegiven explicitlyasfollows:

⟨R, Θ, Φ,��,��|nj�� lm�� ⟩ = Rnl (R) ∑ ml ,mj C��m�� lml jmj Ylml (Θ, Φ)Yjmj (��,��) (1.6)

8 1EffectofConfinementontheTranslation-RotationMotionofMolecules

InEq.(1.6),the C��m�� lml jmj aretheClebsch-Gordancoefficients[40–42].AsinthecaseofH@C60 ,the actual5DTReigenstatewavefunctionsofH2 @C60 closelyresemblethesphericalbasisfunctions themselves[i.e.,Eq.(1.2)forH@C60 ,orEq.(1.6)forH2 @C60 ].Asacaveat,weagainnotethat any Rnl (R) functionsmightbeusedtoformabasis,ormightemergeasthe(nearly)sphericaleigenstate solutions.Asdiscussedinsection1.2.2,thechoiceof Rnl (R) hasnoimpactonsymmetryorINS selectionrules–althoughtodate,anisotropicharmonicoscillatorformhasalwaysbeenused. Finally,takingnuclearspinsintoaccount,wenotethatexchangeantisymmetryimplieseven/odd j valuesthatcorrespondto (s∕a) permutationsymmetry–i.e.,topara-(p-H2 )andortho-(o-H2 ), respectively.

InTable1.2,wepresenttheH2 @C60 eigenstateenergylevels,ascalculatednumericallyin[9],and usedasabasisforassessingexperimentalINStransitions[10].Variouslabelsareused,including thecoupledsphericalbasissetquantumnumbers, (n, j,��, l),togetherwiththecorresponding Kh and(throughTable1.1) Ih irreplabels.Notethatunlikethe Kh labels,the Ih labels exactly match theobservedleveldegeneracies.Incaseswheremultiple Ih irrepscorrelatetoasingle Kh irrep(i.e., for �� ≥ 3),thecorrespondinglevelsplittingprovidesaquantitativemeasureofthateigenstate’s deviationfromtruesphericalsymmetry–i.e.,oftheeffectoficosahedral“corrugation.”Thesecases areindicatedinTable1.2usingitalicisedpairs.Fromthetable,thesesplittingsareindeedseen tobeverysmallindeed–justafewtenthsorhundredthsofacm 1 ,onanenergyscaleranging overseveralhundredcm 1 .NotonlyarethetrueH2 @C60 eigenstatesveryclosetosphericalas predicted–implyingthat �� and m�� areexcellentquantumnumbers–butwealsofindthat n, j,and l aregoodquantumnumbers,aswell.

ThelevelstructureaspresentedinTable1.2hasmuchtotellusabouttheunderlyingTRquantum dynamics.Averydetailedaccountofthephysicalsignificance–including,even,adiscussionofthe preciseHamiltoniansourceofthetiniest Ih levelsplittingsdiscussedabove[13,15]–maybefound intheaforementionedpapersbyBa ˇ ci ´ c,Horsewill,andcoworkers.Inbroadterms,weseethatthe rotationallevelspacingsareontheorderof100cm 1 ,whereasthetranslationallevelspacingsare bothsmaller(in l)andlarger(in n).Finally,foragiven (n, j, l),weseethatthe �� levelspacingsare onlyaround10cm 1 orless,withthelevelsforthe extremal �� values[ortheeven (j + l ��) values] lying highest inenergy.

1.2.3.3EnergyLevelOrderingwithRespectto ��

Thelasttrendmentionedaboveisabitpuzzling,asitimpliesa non-monotonicenergylevelorderingwithrespectto ��.Thesituationbearsfurtherscrutiny,also,becauseitturnsouttoberelated totheINSselectionrule.OneexplanationcanbeprovidedthroughananalysisofEq.(1.6).Letus redefinejusttheangularpartofthisbasisasfollows:

Exchangingcoordinates,1 ↔ 2, andalsoindices, j ↔ l,

Wenextusetheidentity,

,easilyobtainedfromthewell-knownformula forexchangingtwocolumnsoftheWigner3-j symbol[40–42].SubstitutingintoEq.(1.8)above, andalsoflippingtheorderofthesphericalharmonicfactors,yields

1.2Diatomicsin C60 :Entanglement,TRCoupling,Symmetry,BasisRepresentation,andEnergyLevelStructure

Table1.2 Energylevels E anddegeneracies g for H2 @C60 ,from[9].Energiesareincm 1 ,relativetothe p-H2 groundstate.ColumnsIIIandIVlist (n, j ,��, l) andcorresponding Kh irreplabels;ColumnVlists I(12) h irreplabels.Italicsindicate Kh irrepsthatcorrelatetomultiple Ih levels.ColumnVIisthespectroscopic parity, �� =(−1)j+l+�� =(−1)j+l �� ,whereasColumnVIIis (j + l ��) itself.INStransitionsfrom (0, 0, 0, 0) tothe stateslistedinboldareforbidden,accordingtotherestrictedINSselectionruleof[9]andsection1.1. ReprintedwithpermissionfromPoirier,J.Chem.Phys. 143,101104(2015).Copyright2015American InstituteofPhysics.

, 1, 0, 1)

s) g +10 374.955 (2, 0, 2, 2)

(s) g +10 400.931 (2, 0, 0, 0) S

483.205 (��, ��, ��, ��) D u

492.654(2,1,3,2)Fu

493.163(2,1,3,2)Fu

494.663 (2, 1, 1, 2) P u

(a) u +10

(a) 2u +10

(a) 1u +12

522.483 (2, 1, 1, 0) P u T (a) 1u +10

522.565 (��, ��, ��, ��) D u

(s) u 11

528.494(1,2,3,1)Fu G(s) u +10

528.553(1,2,3,1)Fu T (s) 2u +10

533.453 (1, 2, 1, 1) P u T (s) 1u +12

583.784(3,0,3,3)Fu G(s) u +10

584.653(3,0,3,3)Fu T (s) 2u +10

625.423 (3, 0, 1, 1) P u

(s) 1u +10

TheleftandrightsidesofEq.(1.9)involveanexchangeofbothindicesandcoordinates.Wenow considerthespecialcasewherethecoordinatesare equal,i.e. (��1 ,��1 )=(��2 ,��2 )=(��,��).FromEq. (1.7),weseethat

SubstitutionintoEq.(1.9)thenyields

NotethatthetwofunctionsoneachsideofEq.(1.11)are identical,implyingthatthefunctionitself mustbe zero when (j + l ��) isodd.Consequently,theeigenstatewavefunction vanishes when ⃗ R and ⃗ r pointinthesamedirection[43].

Now,thegeometriesthatcorrespondtocollinear ⃗ R and ⃗ r vectorsarethoseforwhichtheH2 moleculepointsradiallyoutwardfromthecenteroftheC60 cage.Forfixed R = | ⃗ R| and r = |⃗ r |,

1EffectofConfinementontheTranslation-RotationMotionofMolecules

thesearethegeometriesforwhichoneHatomgetsclosesttotheC60 wall–andwhichtherefore havethehighestenergy.Sincetheyalsocorrespondtoevenvaluesof (j + l ��),itfollows(ina kindof“Hund’sRule”sense)thatthesestatesshouldliehigherinenergythantheodd-(j + l ��) states.Themathematicalderivationprovidedaboveistypicalof–andgenerallymuchsimpler than–thesortsofsphericalbasismanipulationsthatareappliedintheexplicitintegralcontext, forwhichlengthyappendicesareoftenrequired.Incontrast,a much simplerexplanationforthe �� trend–eventhanthatprovidedabove–canbeobtainedfromasimplesymmetryargument (section1.3.3.2).

1.2.4HX@C60

Finally,weconsidertheHX@C60 case,whichisperhapsthemostinterestingfromtheINSselectionruleperspective.Asdiscussedinsection1.2.4,duetoentanglement,wemusttreattheentire HXmoleculeastherelevantquantumsystem–eventhoughtheXnucleushasessentiallyzero interactionwiththeincidentneutronbeam.Evenso,itspresenceservestoinducenewselection rules,notobservedforH@C60 ,throughEISR.

Roughlyspeaking,wecanfollowthesameprocedureasinsection1.2.3.Thebiggestdifference isthatwenolongerhave(12)exchangepermutationsymmetry,asaconsequenceofwhichthe HX@C60 symmetrygroupisjust Ih .Thereisalsoanimportantdifferenceinthecenter-of-mass coordinatetransformation:

Workinginsphericalcoordinatesasbefore,wecangenerateasuitableuncoupledbasisofthesame formasEq.(1.5),andacoupledbasisoftheformofEq.(1.6).

Here,however,weencounterabigdifferencewithH2 @C60 ,whichisthattheHX@C60 eigenstatesare not generallywelldescribedbyEq.(1.6)[4,12,15,16,19].Fromagrouptheory perspective,whereasthenearperfectsphericalsymmetryof ̂ H ensuresthat �� and m�� willbe goodquantumnumbers, therearenosuchassuranceswithregardton, jandl.Indeed,through explicitcalculation,Ba ˇ ci ´ candcoworkersfirstdemonstratedthat j is not agoodquantumnumber forHD@C60 [4,12,18],–andveryrecently,thatboth landj arenotgoodquantumnumbersfor HF@C60 [19].ForHF@C60 ,theisotropicharmonicoscillator n wasobservednottobegoodeither, althoughthisisnoguaranteethatsomeotherchoicemightnotworkbetter.

Despitethefactthat j and l arenotgoodquantumnumbers,Ba ˇ ci ´ candcoworkershaveshown thatthe“mostgeneralINSselectionrule”fromsection1.1isstillvalid,eveninthismoregeneral context.Thisapproachworks,becauseintheEq.(1.6)basissetexpansionsofthetrueeigenstates, onefindsthatonlyevenorodd (j + l) valuesareincluded–atleastforHD@C60 andHF@C60 .But whataboutmoregeneralHX@C60 systems?Whatguaranteeistherethatthepropertyholdsfor all suchsystems?

Thisexampleunderscorestheadvantagesofagrouptheoryapproach–andmoreproperly,of workingwiththe symmetrycharacters orirrepsofquantumstates,ratherthanwithaparticular choiceofbasissetorquantumnumbers.Theformerare always reliable,andcanbeuniversally applied;thelatter,aswehaveseenabove,arenot.Inthiscase,itturnsoutthatthegeneralprocedure proposedabove is,infactalwaysvalid,becausetheeven/oddrestrictionof (j + l) isalwaysenforced bythe inversionparity ofthecorrespondingirreps.Furtherdiscussionwillbedeferredtosection1.3.

1.3INSSelectionRuleforSpherical(K h )Symmetry

1.3.1InelasticNeutronScattering

Asdiscussedinsection1.1,INS(andalsoquasielasticneutronscattering)hasproventobean extremelypowerfulandprecisetoolforanalyzingnanoconfinedhydrogen[7,8,10,12,24,26,28, 44,45].InINSexperimentsthe“bright”hydrogenatomnucleiinteractstronglywiththeincident monochromaticneutronbeam,resultingininelasticscatteringofthelatter.Acomparisonofincidentandscatteredbeamenergiesthenprovidestransitionenergiesbetweentheinitialandfinal quantumstatesofthetarget.

AnimportantfeatureoftheINSinteractionoperator ̂ M isthatithasa preferredspatialdirection, definedbythe“neutronmomentumtransfervector,” ⃗ k =( ⃗ kscattered ⃗ kincident ).However,fortarget systemswith(nearly)perfectsphericalsymmetry,whichserveasthemainfocusofthischapter, alldirectionsare(essentially)equivalent.Theorientationofthebeamisthuslargelyimmaterialin practice,althoughweshallrevisitthisissueinsection1.4.

TheINSinteractionitselfinvolvesthenuclearforce,andoperatesonascaleof10 14 –10 15 m. Ontheotherhand,thewavelengthoftheneutronbeamisontheatomicormolecularscale–i.e., typicallyaround10 10 m.Tocomputescatteringcrosssections,itisthereforeappropriatetouseS wavescatteringandthefirstBornapproximation,intermsofwhichtheINSinteractionoperator takestheform[9,26,28–30]

InEq.(1.13)above,thesummationrunsonlyoverthe bright nuclei(i.e.,Hatoms)inthequantum system.

The ̂ bn operatorsinEq.(1.13)are“scatteringlengthoperators.”Theyconsistofbothspinand scalarcontributions,correspondingrespectivelytocoherentandincoherentINSscatteringcross sections.Inpracticalterms,insofarasdeterminingtheforbidden(i.e.zero-intensity)transitions isconcerned,itsufficestoignorethenuclearspinstatesaltogetherandconcentratesolelyonthe spatial wavefunctionsofthequantumsystem–asarepresumedthroughoutthiswork.Atransition frominitialstate |Ψi ⟩ tofinalstate ⟨Ψf | isthenforbiddenifandonlyifthetransitionintegralwith thespatialpartoftheINSinteractionoperator, ̂ I n ,iszeroforall n:

I f ←i n = ⟨Ψf | exp (i ⃗ k ⃗ r n ) |Ψi ⟩ = 0forall n impliesforbidden f ← i transition(1.14)

ForH2 andHX,theindividual ̂ I n interactionscanberewrittenintermsofthecenter-of-masscoordinatevectors ⃗ R and ⃗ r ,usingEqs(1.4)and(1.12):

̂ I n = exp (i ⃗ k ⋅ ⃗ R) exp[i(−1)n 1 ⃗ k ⋅ ⃗ r ∕2] forH2 guest(1.15)

̂ I 1 = ̂ I = exp (i ⃗ k ⃗ R) exp (i�� ⃗ k ⃗ r ) forHXguest(1.16)

InEq.(1.16)above, �� = m1 ∕(m1 + m2 ),where m1 = mH and m2 = mX .Thedifferencesbetween Eqs(1.15)and(1.16),andbetween ̂ I n and ̂ I 1 areminor;intruth,theyallbehaveidentically,insofar astheINSselectionruleofEq.(1.14)isconcerned.Itthereforesufficestoconsiderjust ̂ I 1 = ̂ I with �� = 1∕2,withoutlossofgenerality.

1EffectofConfinementontheTranslation-RotationMotionofMolecules

1.3.2GroupTheoryDerivationoftheINSSelectionRule

1.3.2.1Group-Theory-BasedINSSelectionRuleforCylindrical(C∞�� )Environments Standardgrouptheorydictates[37]thatthevanishingoftheINStransitionmatrixelementofEq. (1.14)canbedeterminedfromthetriple-direct-productrepresentationformedfromtheirrepsof ⟨Ψf |, ̂ I ,and |Ψi ⟩,inthesymmetrygroupforthequantumsystem.Theproductrepresentationis generallynotirreducible,butcanbereducedtoadirectsumofanintegernumberofcopies, aj ,of everygroupirrep, Γj : Γ=Γ(Ψ

Thetransitionintensityisthenidenticallyzeroprovidedthat a0 = 0–wherebyconvention, Γ0 labelsthetotallysymmetricirrep.

Atthispointweencounteranimmediatetechnicaldifficultyforsphericalsystems[11],whichis that theINSinteraction ̂ IdoesnotbelongtoanyKh irrep! Thisstemsfromtheaforementionedfact that ̂ I hasapreferreddirection–i.e.,thatofthevector ⃗ k.(Incontrast,thefamiliarelectricdipole operatorfromopticalspectroscopyhasnopreferreddirection,andcanthereforeberepresentedin anymolecularpointgroup.)Torectifythesituation,onestrategyistoreplace Kh withthelargest subgroupinwhich ̂ I canberepresented.Taking ⃗ k todefinethepositive z axis,thelargestpoint groupwithapreferreddirectionisclearlythecylindricalgroup, C∞�� .Inthisgroup,allofthevarious ̂ I n formsofsection1.3.1belongtothetotallysymmetricirrep, Σ+ .

Thefactthat ̂ I belongsto Γ0 isa“luckybreak,”inthatweneedonlyconsiderthesimplerdirect productrepresentation, Γ(Ψf ) ⊗ Γ(Ψi ) inEq.(1.17).Theright-handsideofEq.(1.17)isthusprovidedbythe C∞�� directproducttable,Table1.3. Only productsthatcontainatleastonecopyof Σ+ correspondto allowed transitions.Fromthetable,thesearefoundtocorrespondtothediagonal entries.Thisleadstothefollowing,cylindricalversionoftheINSselectionrule:

mostgeneralINSselectionruleforanyguestmoleculeinanycylindrical(C∞�� )environment:Transitionsforwhich Γ(Ψf )=Γ(Ψi ) areallowed;allothersareforbidden.

Fromagrouptheorystandpoint,thekeysymmetryoperationassociatedwiththeaboveINSselectionruleisthe verticalreflection operation.Thecharacterofagivenirreporquantumstateunder verticalreflectioniscalledthe (e∕f ) or spectroscopicparity, �� [37,38].Thesinglydegenerateirreps Σ± havedefinite �� =±1characterunderverticalreflection.Theremainingirreps,whichareall doubly-degenerate,havezero �� character–meaningthatone +1stateandone 1statecanalways beconstructedthroughsuitablelinearcombinationsofanygivendegeneratepair.

1.3.2.2Group-Theory-BasedINSSelectionRuleforSpherical(Kh )Environments

Theaboveisallfineandgood,foracylindricalsystemthatisperfectlyalignedwith ⃗ k;however, wemuststilldealwiththefactthatinreality,our Ψi and Ψf statesbelongtothe Kh group,notto C∞�� .Thissituationisaddressedusingthecorrelationtablebetween Kh and C∞�� –whichisalso presentedinTable1.1.Fromthetable,weseethateverydegenerate(��> 0)irrepof Kh contains onecopyofthe C∞�� irrep, Π.Thus,thedirectproductof any two(i.e.notnecessarilythesame) degenerate Kh irrepswillincludeatleastonecopyof Σ+ ,through Π ⊗ Π=Σ+ ⊕ Σ ⊕ Δ (where thelatterdecompositioncomesfromTable1.3).Therefore,alltransitionsbetweendegenerate Kh irrepsareallowed.

Forthecasewhereatleastoneofthe Γ(Ψi ) and Γ(Ψf ) Kh irrepsissinglydegenerate(i.e., �� = 0), wefirstnotethatevery Kh irrepdecompositioninTable1.1containsexactlyonecopyofeither Σ+

Table1.3 Directproducttableforthe C ∞�� pointgroup.The Γ(Ψf ) irreps arelistedinthefirstcolumn;the Γ(Ψi ) irrepsarelistedinthefirstrow. Thetotallysymmetricirrep, Γ0 =Σ+ ,isrealizedinthe Γ(Ψf ) ⊗ Γ(Ψi ) directproductonlywhen Γ(Ψf )=Γ(Ψi ).

or Σ .AccordingtoTable1.3,forbiddentransitionswillthereforebethoseforwhichoneofthe two Kh irrepscontains Σ+ ,andtheothercontains Σ .Notethatverticalreflectionisagainthekey; thissymmetryoperationbelongstothe Kh groupasmuchastothe C∞�� group.Furthermore,the characterofagiven Kh irrepunderverticalreflectioniseither ±1,sothatwecancontinuetotake thisasourdefinitionof �� ,thespectroscopicparity.InTable1.1,the Kh irrepswith �� =−1(i.e., thosethatcontain Σ )arepresentedinbold,whereasthosewith �� =+1arenotinbold.

Notethealternatingpattern,whichresultsinthefollowingusefulrelationbetween �� ,andthe Kh irreplabels, �� and p:

= p(−1)�� = p(−1) �� (1.18)

InEq.(1.18)above, �� isthe“totalangularmomentum,”and p =±1isthetotalor (g∕u) inversionparity.ItmustbestressedthatEq.(1.18)isa universalrelation,sinceallofthequantitiesare group-theorybased.NotethatbothformsgiveninEq.(1.18)areequivalent,since �� isaninteger. However,bothformsareuseful;thesecondconnectswiththeearlierselectionrulesaspresentedin section1.1,whereasthefirstgeneralizesinthecasewheretherearehalf-integerangularmomenta quantitiesduetospin.

Puttingtogetherthevariouspiecesabove,intermsofspectroscopicparity,weobtainthefollowingcompletelygeneralsphericalINSselectionrule:

mostgeneralINSselectionruleforanyguestmoleculeinanyspherical(Kh )environment:Transitionsforwhich Γ(Ψf ) and/or Γ(Ψi ) aresinglydegenerate,andforwhich ��f ��i =−1,areforbidden;allothersareallowed.

FromTable1.1,aversionintermsofexplicit Kh irrepscanalsobeprovided:

mostgeneralINSselectionruleforanyguestmoleculeinanyspherical(Kh ) environment:Transitionsbetween Sg and {Su , Pg , Du , Fg , Gu ,…} areforbidden;transitions between Su and {Sg , Pu , Dg , Fu , Gg ,…} areforbidden;allothertransitionsareallowed.

1.3.3SpecificSystems,QuantumNumbers,andBasisSets

TheINSselectionrulesgivenattheendofsection1.3.2.2are alwayscorrect forsphericalenvironments,regardlessofthespecificdetailsofthehostcageorguestmolecule.Nevertheless,itcan

1EffectofConfinementontheTranslation-RotationMotionofMolecules

alsobeusefultohavespecificformsthataretailoredtothequantumnumbersandbasissetsused inspecificapplications.Notethat p and �� are always well-definedquantumnumbersforany Kh system;moreover,thesequantitiesareusuallystraightforwardtoobtainforagivenexpliciteigenstateorbasisrepresentation.UsingEq.(1.18),itisthenstraightforwardtoobtainasystem-and basis-set-specificformof �� .Theindividualcasesarepresentedbelow,andinsection1.3.4.

1.3.3.1H@sphere

ForH@spheresystems,alleigenstatewavefunctionsarenecessarilyoftheformofEq.(1.2),for which �� isspecified.Furthermore, p =(−1)�� [35,36],ascanbeshownbytransformingthesolid anglecoordinatesandsphericalharmonicsundertheinversionsymmetryoperation:

Inanyevent,fromEq.(1.18),wefindthat all H@spherestateshave �� =+1.Thus,onlythe Kh irreps thatare not inboldinTable1.1arephysicallyrealized.Moreimportantly,therearenotransitions thatcanchangethesignof �� ;consequently, alltransitionsareallowed

1.3.3.2 H2 @sphere

ForH2 @C60 (andlikelyothersphere-likecavities),thetrueeigenstatesarewell-approximatedby acoupledbasisoftheformofEq.(1.6)–forwhich �� againdirectlyappearsasaquantumnumber. Asfor p,letusfirstconsidertheuncoupledbasisofEq.(1.5).Here,thetotalinversionsymmetry operationappliesseparatelytoboth (��,��) and (Θ, Φ).Accordingly,thetotalinversionparityisjust theproductoftheparitiesforeachindividualsphericalharmonicfactor–i.e., p =(−1)j (−1)l = (−1)j+l .Importantly,Eq.(1.6)doesnotcombineuncoupledbasisfunctionsofdifferent j or l values. Thus, j and l aregoodquantumnumbersforthecoupledbasis(andfortheeigenstatesthemselves) andsotheaboveexpressionfor p isstillvalid.

FromEq.(1.18),thisleadsatonceto

=(−1)j+l+�� =(−1)j+l �� . (1.20)

Thesignof �� isthusdeterminedbytheeven/oddnessof (j + l ��).Fromtheendofsection1.3.2.2, theforbiddentransitionsarethusthosethatchangethischaracter,andforwhichatleastoneof thetwostatesissinglydegenerate.Thisgivesrisetothe mostgeneralINSselectionrulefor H2 @C60 asgiveninsection1.1.

Asspecialcases,wecanconsiderforbiddentransitionsstartingfromthep-H2 ando-H2 ground states.Thep-H2 groundstatehas Sg character;thus,transitionstoallstateswith �� =−1areforbidden.ThesearethestatesthatcorrespondtothefirstlistinthesecondINSselectionrulein section1.3.2.2–oralternatively,the Kh irrepslistedinboldinTable1.1.InTable1.2,the �� =−1 energylevelsofH2 @C60 arealsolistedinbold.INStransitionsfromthe �� =+1p-H2 groundstate (0, 0, 0, 0),tothethreeboldstatesindicatedinthetable,arethus forbidden Theo-H2 caseismoreinteresting.Here,thegroundstatehas Pu character,sothatweobtain:

INSselectionruleforH2 inanyspherical(Kh )environment(fromo-H2 ground state):Transitionsareforbiddentoall Su states;allothertransitionsareallowed.

NotethatforH2 , therearenoSu states,andthereforenoforbiddentransitionsfromtheo-H2 ground state.Nevertheless,theaboveformmaybeusefulinmoregeneralsituations.

Finally,werevisitthetrenddiscussedattheendofsection1.2.3,wherebytheeven (j + l ��) statesliehigherinenergythantheodd (j + l ��) states.Asimplesymmetry-basedexplanation

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