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JWDD011-FM JWDD011-Baer February24,2006 7:20 CharCount=0 BEYOND BORN–OPPENHEIMER Conical Intersections and Electronic Nonadiabatic Coupling Terms MICHAEL BAER SoreqNuclearResearchCenter Yavne,Israel TheFritz–HaberResearchCenterforMolecularDynamics TheHebrewUniversityofJerusalem,Jerusalem,Israel AJOHNWILEY&SONS,INC.,PUBLICATION iii JWDD011-FM JWDD011-Baer February24,2006 7:20 CharCount=0 Copyright(cid:2)C 2006byJohnWiley&Sons,Inc.Allrightsreserved. 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Wileyalsopublishesitsbooksinavarietyofelectronicformats.Somecontentthatappearsinprintmay notbeavailableinelectronicformats.FormoreinformationaboutWileyproducts,visitourwebsiteat www.wiley.com. LibraryofCongressCataloging-in-PublicationData: Baer,M.(Michael) BeyondBorn–Oppenheimer:electronicnonadiabaticcouplingtermsandconical intersections/byMichaelBaer. p. cm. Includesindex. ISBN-13978-0-471-77891-2 ISBN-100-471-77891-5(cloth) 1.Moleculardynamics–Quantummechanics. 2.Born-Oppenheimertreatment. 3.Adiabaticanddiabaticstates. I.Title. QD96.M65B342006 541(cid:3).28–dc22 2005021350 PrintedintheUnitedStatesofAmerica 10 9 8 7 6 5 4 3 2 1 iv JWDD011-FM JWDD011-Baer February24,2006 7:20 CharCount=0 TomywifeMali(Malvine) v JWDD011-FM JWDD011-Baer February24,2006 7:20 CharCount=0 CONTENTS PREFACE xiii ABBREVIATIONS xvii 1 MATHEMATICALINTRODUCTION 1 1.1 HilbertSpace / 1 1.1.1 EigenfunctionandElectronicNonadiabatic CouplingTerm / 1 1.1.2 AbelianandNon-AbelianCurlEquations / 3 1.1.3 AbelianandNon-AbelianDivergenceEquations / 6 1.2 HilbertSubspace / 8 1.3 VectorialFirst-OrderDifferentialEquationandLineIntegral / 11 1.3.1 VectorialFirst-OrderDifferentialEquation / 12 1.3.1.1 StudyofAbelianCase / 12 1.3.1.2 StudyofNon-AbelianCase / 13 1.3.1.3 Orthogonality / 14 1.3.2 IntegralEquation / 14 1.3.2.1 IntegralEquationalonganOpenContour / 15 1.3.2.2 IntegralEquationalongaClosedContour / 16 1.3.3 SolutionofDifferentialVectorEquation / 20 1.4 SummaryandConclusions / 23 Problem / 23 References / 25 vii JWDD011-FM JWDD011-Baer February24,2006 7:20 CharCount=0 viii CONTENTS 2 BORN–OPPENHEIMERAPPROACH:DIABATIZATIONAND TOPOLOGICALMATRIX 26 2.1 Time-IndependentTreatment / 26 2.1.1 AdiabaticRepresentation / 26 2.1.2 DiabaticRepresentation / 28 2.1.3 Adiabatic-to-DiabaticTransformation / 30 2.1.3.1 TransformationforElectronicBasisSets / 30 2.1.3.2 TransformationforNuclearWavefunctions / 33 2.1.3.3 ImplicationsDuetoAdiabatic-to-Diabatic Transformation / 34 2.1.3.4 FinalComments / 38 2.2 ApplicationofComplexEigenfunctions / 39 2.2.1 IntroducingTime-IndependentPhaseFactors / 39 2.2.1.1 AdiabaticSchro¨dingerEquation / 39 2.2.1.2 Adiabatic-to-DiabaticTransformation / 40 2.2.2 IntroducingTime-DependentPhaseFactors / 41 2.3 Time-DependentTreatment / 43 2.3.1 Time-DependentPerturbativeApproach / 43 2.3.2 Time-DependentNonperturbativeApproach / 45 2.3.2.1 AdiabaticTime-DependentElectronic BasisSet / 45 2.3.2.2 AdiabaticTime-DependentNuclear Schro¨dingerEquation / 46 2.3.2.3 Time-DependentAdiabatic-to-Diabatic Transformation / 47 2.3.3 Summary / 49 Problem / 50 2AAppendixes / 51 2A.1 DressedNonadiabaticCouplingMatrixτ˜ / 51 2A.2 AnalyticityofAdiabatic-to-DiabaticTransformation MatrixA˜ inSpacetimeConfiguration / 52 References / 54 3 MODELSTUDIES 58 3.1 TreatmentofAnalyticalModels / 58 3.1.1 Two-StateSystems / 59 3.1.1.1 Adiabatic-to-DiabaticTransformation Matrix / 59 3.1.1.2 Topological(D)Matrix / 60 3.1.1.3 TheDiabaticPotentialMatrix / 61 JWDD011-FM JWDD011-Baer February24,2006 7:20 CharCount=0 CONTENTS ix 3.1.2 Three-StateSystems / 62 3.1.2.1 Adiabatic-to-DiabaticTransformationMatrix / 62 3.1.2.2 TopologicalMatrix / 63 3.1.3 Four-StateSystems / 64 3.1.3.1 Adiabatic-to-DiabaticTransformationMatrix / 64 3.1.3.2 TopologicalMatrix / 65 3.1.4 CommentsRelatedtoGeneralCase / 66 3.2 Studyof2×2DiabaticPotentialMatrixandRelatedTopics / 67 3.2.1 TreatmentofGeneralCase / 67 3.2.2 TheJahn–TellerModel / 70 3.2.3 EllipticJahn–TellerModel / 72 3.2.4 DistributionofConicalIntersectionsandDiabatic PotentialMatrix / 73 3.3 Adiabatic-to-DiabaticTransformationMatrixandWigner RotationMatrix / 75 3.3.1 WignerRotationMatrices / 76 3.3.2 Adiabatic-to-DiabaticTransformationMatrixand WignerdjMatrix / 77 Problem / 79 References / 82 4 STUDIESOFMOLECULARSYSTEMS 84 4.1 IntroductoryComments / 84 4.2 TheoreticalBackground / 85 4.3 QuantizationofNonadiabaticCouplingMatrix:Studyof AbInitioMolecularSystems / 87 4.3.1 Two-StateQuasiquantization / 87 4.3.1.1 {H ,H}System / 87 2 4.3.1.2 {H ,O}System / 91 2 4.3.1.3 {C ,H }System / 92 2 2 4.3.2 MultistateQuasiquantization / 96 4.3.2.1 {H ,H}System / 96 2 4.3.2.2 {C ,H}System / 99 2 References / 102 5 DEGENERACYPOINTSANDBORN–OPPENHEIMER COUPLINGTERMSASPOLES 105 5.1 RelationbetweenBorn–OppenheimerCouplingTermsand DegeneracyPoints / 105 JWDD011-FM JWDD011-Baer February24,2006 7:20 CharCount=0 x CONTENTS 5.2 ConstructionofHilbertSubspace / 108 5.3 SignFlipsofElectronicEigenfunctions / 109 5.3.1 Two-StateHilbertSubspace / 109 5.3.2 Three-StateHilbertSubspace / 110 5.3.3 GeneralHilbertSubspace / 114 5.3.4 MultidegeneracyPoint / 120 5.3.4.1 GeneralApproach / 120 5.3.4.2 ModelStudies / 121 5.4 TopologicalSpin / 122 5.5 ExtendedEulerMatrixasaModelforAdiabatic-to-Diabatic TransformationMatrix / 125 5.5.1 IntroductoryComments / 125 5.5.2 Two-DimensionalCase / 126 5.5.3 Three-DimensionalCase / 127 5.5.4 MultidimensionalCase / 130 5.6 Quantizationofτ MatrixanditsRelationtoSizeofConfiguration Space:MathieuEquationasaCaseofStudy / 131 5.6.1 MathieuEquationandItsEigenfunctions / 131 5.6.2 NonadiabaticCouplingMatrix(τ)andTopological Matrix(D) / 133 Problems / 134 References / 136 6 MOLECULARFIELD 139 6.1 SolenoidasaModelfortheSeam / 139 6.2 Two-State(Abelian)System / 141 6.2.1 NonadiabaticCouplingTermasaVectorPotential / 141 6.2.2 Two-StateCurlEquation / 143 6.2.3 (Extended)StokesTheorem / 144 6.2.4 ApplicationofStokesTheoremforSeveralTwo-State ConicalIntersections / 146 6.2.5 ApplicationofVectorAlgebratoCalculatetheFieldofa Two-StateHilbertSpace / 147 6.2.6 ANumericalExample:Studyof{H ,Na}System / 149 2 6.2.7 AShortSummary / 151 6.3 MultistateHilbertSubspace / 151 6.3.1 Non-AbelianStokesTheorem / 151 6.3.2 TheCurl–DivergenceEquations / 154 6.3.2.1 Three-StateHilbertSubspace / 155 JWDD011-FM JWDD011-Baer February24,2006 7:20 CharCount=0 CONTENTS xi 6.3.2.2 DerivationofPoissonEquations / 157 6.3.2.3 StrangeMatrixElementandGauge Transformation / 159 6.4 ANumericalStudyof{H ,H}System / 160 2 6.4.1 IntroductoryComments / 160 6.4.2 IntroducingFourierExpansion / 160 6.4.3 IntroducingBoundaryConditions / 161 6.4.4 NumericalResults / 162 6.5 MultistateHilbertSubspace:BreakupofNonadiabaticCoupling Matrix / 162 6.6 PseudomagneticField / 167 6.6.1 QuantizationofPseudomagneticFieldalongthe Seam / 168 6.6.2 Non-AbelianMagneticFields / 168 Problems / 168 References / 172 7 OPENPHASEANDBERRYPHASEFORMOLECULAR SYSTEMS 175 7.1 StudiesofAbInitioSystems / 175 7.1.1 IntroductoryComments / 175 7.1.2 OpenPhaseandBerryPhaseforSinglevalued Eigenfunctions:Berry’sApproach / 176 7.1.3 OpenPhaseandBerryPhaseforMultivalued Eigenfunctions:PresentApproach / 177 7.1.3.1 DerivationofTime-DependentEquation / 177 7.1.3.2 TreatmentofAdiabaticCase / 179 7.1.3.3 TreatmentofNonadiabatic(General)Case / 181 7.1.4 {H ,H}SystemasaCaseStudy / 183 2 7.2 Phase–ModulusRelationsforanExternalCyclic Time-DependentField / 187 7.2.1 DerivationofReciprocalRelations / 187 7.2.2 MathieuEquation / 189 7.2.2.1 Time-DependentSchro¨dingerEquations / 189 7.2.2.2 NumericalStudyofTopologicalPhase / 191 7.2.3 ShortSummary / 194 Problem / 194 References / 195 JWDD011-FM JWDD011-Baer February24,2006 7:20 CharCount=0 xii CONTENTS 8 EXTENDEDBORN–OPPENHEIMERAPPROXIMATIONS 197 8.1 IntroductoryComments / 197 8.2 Born–OppenheimerApproximationasAppliedtoaMultistate ModelSystem / 199 8.2.1 ExtendedApproximateBorn–OppenheimerEquation / 199 8.2.2 GaugeInvarianceConditionforApproximate Born–OppenheimerEquations / 201 8.3 MultistateBorn–OppenheimerApproximation:Energy ConsiderationstoReduceDimensionsofDiabaticPotential Matrix / 201 8.3.1 IntroductoryComments / 201 8.3.2 DerivationofReducedDiabaticPotentialMatrix / 202 8.3.3 ApplicationofExtendedEulerMatrix:Introducingthe N-StateAdiabatic-to-DiabaticTransformationAngle / 206 8.3.3.1 IntroductoryComments / 206 8.3.3.2 DerivationofAdiabatic-to-Diabatic TransformationAngle / 206 8.3.4 Two-StateDiabaticPotentialEnergyMatrix / 210 8.3.4.1 DerivationofDiabaticPotentialMatrix / 210 8.3.4.2 ANumericalStudyof(cid:3)WMatrixElements / 211 8.3.4.3 ADifferentApproach:Helmholtz Decomposition / 214 8.4 ANumericalStudyofaReactive(Exchange)Scattering Two-CoordinateModel / 214 8.4.1 BasicEquations / 214 8.4.2 ATwo-CoordinateReactive(Exchange)Model / 216 8.4.3 NumericalResultsandDiscussion / 217 Problem / 220 References / 221 9 ASUMMARY 224 INDEX 227 JWDD011-FM JWDD011-Baer February24,2006 7:20 CharCount=0 PREFACE ThestudyofmolecularsystemsisbasedontheBorn–Oppenheimertheory,whichdis- tinguishesbetweenrapidlymovingelectronsandslowlymovingnucleiandtherefore leadstotheformationoftheelectronic(adiabatic)eigenstatesandthecorresponding nonadiabatic coupling terms (NACTs). The existence of the adiabatic states seems to be compatible with a wide range of experimental studies, from photochemical processes through photodissociation and unimolecular processes and finally to bi- molecular interactions accompanied by exchange and/or charge transfer processes. Having available the adiabatic states, the dynamical treatment is frequently carried outapplyingtheBorn–Oppenheimerapproximation,whichassumestheexistenceof asingledecoupledstate,thusignoringtheNACTsthatcouplethevariousstates.The justificationforthisapproximationisinthefactthatNACTsareproportionaltothe massratio,namely,(m /m )(1/2)(wherem andm arethemassesofanelectronanda e p e p proton,respectively),andthereforeareexpectedtobeabouttwoordersofmagnitude smallerthanothercharacteristicmagnitudesthatappearinthenuclearSchro¨dinger equation.However,asmoreandmoremolecularsystemsarestudiedandthenumber ofabinitiotreatmentsincreases,itisrealizedthatthisapproximationholds,atmost, forsmallregionsinconfigurationspace,butotherwisecannotbesatisfied. Themainreasonforthismisfortuneistheexistenceofmoleculardegeneratestates, whichinturncausetheNACTsnotonlytobecomeinfinitelylargebutalsotodressas poles.Beingpoles,theNACTsbecomethesourcefornumerousphenomenathatare consideredastopologicaleffectsandleadtoseveralinterestingissues,includingthe well-knownLonguet–Higgins/Berryphaseandtheopen-pathphase,thelessknown quantizationfeatureoftheNACTs,theexistenceofmolecularfields,andfinallybut notlessinteresting,topologicalspin.Sincethepotentialenergysurfacesareexpected tobehavelinearlyinthevicinityofthesepoints,thedegeneracypointsarefrequently xiii