Theory of Molecular Collisions

294 Pages·2015·44.951 MB·English

by  Gabriel G Balint-Kurti, Alexander Palov, Jonathan Hirst

TheoryofMolecularCollisions
RSCTheoreticalandComputationalChemistrySeries
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Theory of Molecular
Collisions
GabrielG.Balint-Kurti
UniversityofBristol,UK
Email:[email protected]
and
AlexanderP.Palov
MoscowStateUniversity,Russia
Email:[email protected]
RSCTheoreticalandComputationalChemistrySeriesNo.7
PrintISBN:978-1-84973-830-9
PDFeISBN:978-1-78262-019-8
ISSN:2041-3181
AcataloguerecordforthisbookisavailablefromtheBritishLibrary
(cid:13)C GabrielG.Balint-KurtiandAlexanderP.Palov,2015
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Preface
Thebookprovidesanintroductiontothetheoryofmolecularcollisionsand
reactivescatteringandrelatesittocurrentexperiments.Theemphasisison
the quantum theory of scattering, although classical mechanics for atom–
atomscatteringisalsobrieflydescribed.Whilethereisinevitablyaconsider-
ableamountofmathematicsinthebook,itiskepttotheminimumrequired,
andthemoredifficulttopicsarerelegatedtoappendices.Themotivationfor
studying the topic is to provide the tools needed for investigating some of
themostbasicprocessesofnature:howchemicalreactionsoccurandhow
molecularbondsareformedandbroken.
Thefirsttwochaptersdiscusstheclassicalandquantumtheoryofatom–
atomscattering.Thesediscussionsintroduceimportantconceptsusedlater
inthebook.Chapters3and4dealwiththequantumtheoryofnon-reactive
atom–diatom collisions. The use of different coordinate reference frames,
the calculation of cross sections and the role of the scattering S matrix are
described.Thefollowingchaptersdealmainlywithreactivescattering.Both
historicandmodernexperimentalmethodsformeasuringdetailedreactive
crosssectionsandreactionprobabilitiesarereviewed.Thecurrenttheoreti-
calmethodsforthecalculationofreactioncrosssections,usingbothtime-
independent and time-dependent quantum treatments, are described. The
useoflasersinmodernexperimentsisreviewedandtheunderlyingtheory
of polarization and alignment experiments, which measure the correlation
of, say, the vector directions of the rotational angular momentum of the
productsofareactionwiththeirangleofscattering,isexplained.Thefinal
chapterofthebookdiscussesthedifficultiesencounteredinapplyingexact
quantummechanicstosystemsofmanyatoms.Thevariousmethodsavail-
able for the approximate quantum treatment of reactions involving many
RSCTheoreticalandComputationalChemistrySeriesNo.7
TheoryofMolecularCollisions
ByGabrielG.Balint-KurtiandAlexanderP.Palov
(cid:13)C GabrielG.Balint-KurtiandAlexanderP.Palov,2015
PublishedbytheRoyalSocietyofChemistry,www.rsc.org
v
vi Preface
atomsarediscussed.Alsodiscussedaremethodsforcomputingthermalrate
constantswithouttheneedtocomputethescatteringSmatrix.
Inadditiontothemainbodyofthebook,aglossaryoftermsisincluded
andextensivereferencestoliteraturesourcesareprovided.
The authors would like to thank and acknowledge their debt to past
supervisors, colleagues and students who have helped them in the field of
molecular scattering theory. On behalf of GGBK they include: M. Karplus,
R.D.Levine,M.Shapiro,R.N.Dixon,C.C.Marston,F.G¨ogtas,A.R.Offer,
S.K.Gray,M.Hankel,E.M.Goldfield,G.C.Schatz,O.S.Vasyutinskii,L.Eno,
B.M.D.D.JansenopdeHaar,M.M.Novak,J.vanLenthe,S.Bosanacand
R.Saktreger.OnbehalfofAPPtheycomprise:V.A.Mitrofanov,V.V.Pletnev,
B. I. Klain, T. V. Rakhimova, K. S. Klopovsky, A. A. Chukalovsky and
A.A.Sycheva.
G.Balint-KurtiandA.Palov
Contents
Chapter1 ScatteringExperimentsandClassicalTheoryof 1
Atom–AtomScattering
1.1 CrossedAtomicandMolecularBeams 1
1.2 ClassicalTheoryofAtom–AtomElasticScattering 2
1.2.1 HardSphereCollisions 2
1.2.2 ScatteringUndertheInfluenceofaPotential 5
1.2.3 Singularities of a Differential Scattering
CrossSection 11
1.3 The Laboratory and the Center-of-Mass Reference
Frames:NewtonDiagrams 14
References 17
Chapter2 QuantumTheoryofAtom–AtomElasticScattering 19
2.1 TheWavePictureoftheScatteringProcess 19
2.2 ThePhaseShift 21
2.3 PartialWaveExpansion 22
2.4 ScatteringofIdenticalParticles 25
2.5 TheSandTMatrices 28
2.6 Quasi-BoundStatesandtheLifetimeMatrix 29
2.7 IllustrativeExamples:Ne(cid:67)NeandAr(cid:67)(cid:67)Ar
Scattering 30
2.7.1 DifferentialandIntegralCrossSections 32
2.7.2 ShapeResonances 39
2.7.3 ExchangeSymmetry 41
RSCTheoreticalandComputationalChemistrySeriesNo.7
TheoryofMolecularCollisions
ByGabrielG.Balint-KurtiandAlexanderP.Palov
(cid:13)C GabrielG.Balint-KurtiandAlexanderP.Palov,2015
PublishedbytheRoyalSocietyofChemistry,www.rsc.org
vii
viii Contents
2.8 Extraction of Interaction Potentials from Crossed
BeamExperiments 42
2.9 LowEnergyScatteringandtheScatteringLength 42
References 43
Chapter3 InelasticScattering:BasicTheory 46
3.1 Atom–DiatomCollisions 46
3.2 Space-FixedFormulation 48
3.3 TheTotalAngularMomentum 50
3.4 TheCoupledEquations 51
3.5 TheSandTMatricesandDifferentialCrossSections 53
3.6 ConservationofParticleFlux,Symmetry
andUnitarityofSMatrix 55
3.7 TheIntegralCrossSections 56
3.8 Body-FixedFormulationandHelicities 59
References 62
Chapter4 InelasticScattering:ExactandApproximateSolutions 64
4.1 SolvingtheClose-CouplingEquations 64
4.1.1 PiecewiseAnalyticSolutionsMethod 64
4.1.2 Log-DerivativeMethod 70
4.1.3 ComparisonofTheoryandExperiment 71
4.2 ApproximateMethods 73
4.2.1 NeglectofCoupling 73
4.2.2 BornandDistortedWaveApproximations 75
References 83
Chapter5 RateConstants,CrossSectionsandReactiveScattering 86
5.1 CrossSectionsandRatesofChemicalReactions 86
5.1.1 DetailedBalance 89
5.2 DeterminationofDetailedRateConstants 90
5.3 MeasurementofDifferentialReactiveCrossSections 92
5.3.1 TheAlkaliAge 92
5.3.2 TheModernAge—UniversalDetectors 94
References 96
Chapter6 Time-IndependentQuantumTheoryofReactive 98
Scattering
6.1 PotentialEnergySurfaces 99
6.2 HypersphericalCoordinates 100
6.3 TheWavefunction,SMatrixandCrossSections 103
Contents ix
6.4 ExactReactiveScatteringCalculations
forAtom–DiatomReactions 105
6.4.1 H(cid:67)H ReactiveScattering 105
2
6.4.2 F(cid:67)HD→HF(cid:67)DReactiveScattering 108
References 111
Chapter7 WavepacketsandTime-DependentQuantumTheoryof 115
ReactiveScattering
7.1 SolvingtheTime-DependentSchr¨odingerEquation 117
7.1.1 ChebyshevExpansionoftheTimeEvolution
Operator 117
7.2 SettingUptheInitialWavepacket 119
7.3 AbsorbingtheWavepacketattheEdgeoftheGrid 121
7.4 The Scattering Equations and Calculation of the
CrossSections 122
7.5 AnExample:HD(cid:67)OH→H O(cid:67)DReactive
2
DifferentialCrossSection 125
References 126
Chapter8 TheRealWavepacketMethodandTime-Independent 129
Wavepackets
8.1 IterativeTimePropagation 129
8.2 ScalingandMappingtheHamiltonianOperator 130
8.3 AbsorbingtheWavepacket 131
8.4 CalculatingtheSMatrix 132
8.5 Applications 134
8.5.1 O(1D)(cid:67)H →OH(cid:67)H:Conical
2
IntersectionsandElectronically
Non-AdiabaticTransitions 134
8.5.2 N(4S)(cid:67)CH(X2(cid:80))→CN(cid:67)H:Reaction
ProceedingviaaDeepWell 136
References 139
Chapter9 LasersandthePhotolocMethod 141
9.1 PHOTOLOC—PhotoInitiatedReactionAnalyzedvia
theLawofCosines 141
9.2 DetectionMethods 145
9.2.1 Resonantly-EnhancedMultiphoton
Ionization(REMPI) 146
9.2.2 REMPI-Time-Of-Flight(REMPI-TOF) 148
9.2.3 Laser-InducedFluorescence(LIF) 149
9.2.4 DopplerSpectroscopy 149
References 151
x Contents
Chapter10 Polarization,AlignmentandVectorCorrelation 153
10.1 OrientationandAlignment 153
10.2 PolarizationDependentDifferentialCrossSections 156
10.2.1 SphericalMultipoles 159
10.2.2 Applications 160
References 163
Chapter11 CollisionofLargerMolecules 165
11.1 ReducedDimensionalityApproximations 166
11.2 Multi-ConfigurationTime-DependentHartree
Method(MCTDH) 167
11.3 ComputationofRateConstants:
TheFlux–FluxAuto-CorrelationMethod 170
11.3.1 StateAveragedMCTDHMethod 173
References 176
AppendixA EnergyNormalizationofPlaneWave 179
A.1 PlaneWaveinOneDimension 179
A.2 PlaneWaveinThreeDimensions 180
A.2.1 PropagationinZDirection 182
References 182
AppendixB EvaluationofthePhaseShiftandtheVariablePhase 183
Approach
B.1 PhaseShiftforl(cid:90)0 187
B.2 PhaseShiftforl(cid:79)0 187
B.3 EvaluationofPhaseShift 188
B.4 Levinson’sTheorem 190
References 190
AppendixC JacobiCoordinates 191
C.1 TriatomicSystem 191
C.2 GeneralizationtoannAtomSystem 194
References 196
AppendixD Body-FixedFormulationofInelasticScatteringTheory 197
D.1 TotalAngularMomentumEigenfunctions 197
D.2 UnitaryTransformationMatrixBetweenSpace-Fixed
andBody-FixedAngularBasisFunctions 199
D.3 TheCoupledEquations 199