Table Of ContentJWDD011-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
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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
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TomywifeMali(Malvine)
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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
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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
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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
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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
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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
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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
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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
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