Spectra and Dynamics of Small Molecules: Alexander von Humboldt Lectures

162 Pages·2015·3.43 MB·English

by  Robert W. Field (auth.)

Lecture Notes in Physics 900
Robert W. Field
Spectra and
Dynamics
of Small
Molecules
Alexander von Humboldt Lectures
Lecture Notes in Physics
Volume 900
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Robert W. Field
Spectra and Dynamics
of Small Molecules
Alexander von Humboldt Lectures
123
RobertW.Field
DepartmentofChemistry
MassachusettsInstituteofTechnology
Cambridge
Massachusetts
USA
ISSN0075-8450 ISSN1616-6361 (electronic)
LectureNotesinPhysics
ISBN978-3-319-15957-7 ISBN978-3-319-15958-4 (eBook)
DOI10.1007/978-3-319-15958-4
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Preface
I am a card-carrying,small-molecule spectroscopist. I like simple models and the
possibilityofcertaintythatananalysisiscorrect.
When I was a graduate student in William Klemperer’s laboratory, I received
a precious gift: I became a molecular spectroscopist. In my 40 years at MIT, my
students,postdocs,andcollaboratorskeptmefocusedonthecrucialquestion:“what
makesthisexperimentinteresting?”Often,theyprovidedbothquestionandanswer.
Ihavebeenenergizedandeducatedbytheirquestions,ideas,andah-hahmoments.
Always,theyhadmesayingbrilliantthingsIdidnotknowthatIknew.Butamirror
neverknowsthe beauty it reflects. I dedicate this bookto William Klempererand
myspecialsub-groupofhisscientificdescendants.
Frequency-domainexperimentalists observe transition frequencies and intensi-
ties.Wavefunctionsarenotdirectlyobservable.Thechallengesofspectroscopyare:
(a) assign the observed transitions; (b) fit the directly sampled energy levels to a
quantum mechanical model; (c) make predictions about other spectra; and most
importantly,(d)capturethedynamicsanddynamicalmechanismsthatareencoded
in the spectrum. The effective Hamiltonian, Heff, is an essential tool for meeting
all four of these challenges without license to take a peek at the wavefunction
(especially its nodal structure, which is tantamount to assignment, and its time
evolution, which reveals mechanism). This short book is an express user’s guide
forbeginnerswhoknowneitherthebasicsnortheelegantsimplicityandintuition-
guidingpowerofthemodelsthatliebeyondarchivalmolecularconstants.Itisnota
textbook.Itisneitherrigorousnorlogical.Mygoalinwritingthisbookistoprovide
asetofideas,tools,andchallengesthatwillignitetheabilityofbeginningstudents
toseewhatisintuitiveandmemorableinmolecularspectra.
The lecture format permits strong opinions, personal choices of topics, and
intentionally incomplete examples. My idea is to provoke my readers to explore
otherexamples,tocreatefreehandfiguresthatarequalitativelybutnotquantitatively
correct,andtopracticeusingtoolsandconceptsthatarederivedfrommyidiosyn-
cratic point of view rather than from a balanced and scholarly treatment. I have
attempted to make connections between disparate topics without being burdened
by lengthyjustification. I have presentedseveralof my favorite topics. My dream
v
vi Preface
is that mastery of the main topics in these eight chapters/lectures will serve the
studentaspreparationforproductiveyearsasaspectroscopist,andwillbeatleast
asenlighteningasasemester-longsojourninacoursebasedonacarefullyintegrated
formaltextbook.
ThevastmajorityofcitationsinthisbookaretorelevantsectionsofTheSpectra
andDynamicsofDiatomicMolecules[1],andtoalongpaperdevotedtomyfavorite
topic, effective Hamiltonians[2]. This excess of self-citation is one of the special
privilegesaccruingtoanauthoroflecturenotesratherthanatextbook.
I thank the Alexander von Humboldt Foundation for the Humboldt Research
AwardthatgavemetheopportunitytospendseveralmonthsasamemberofGerard
Meijer’s Molecular Physics Department at the Fritz Haber Institute of the Max
Planck Society in Berlin. I especially thank Gerard Meijer for inviting me into
hislaboratorywhile itwascracklingwith excitement,creativity,andexperimental
virtuosity.
IamthrilledthatthisbookwillbepublishedastheharbingerofanewSpringer
book series entitled “Alexandervon HumboldtLectures,” the purposeof which is
to make the lectures presented by Humboldt Awardees and Humboldt Professors
availabletostudentswhowereunabletoattendtheminperson.BretislavFriedrich
hadtheideaforthisseriesandIthankhimfortalkingmeintoputtingmylectures
intobookform.
ThisbookowesitsexistencetoPeterGiunta’singenuity,artistry,andefficiency.
ThetextforeachlecturewaswrittenlonghandinBerlinandFAXedtoPeteratMIT
in Cambridge on the day before the lecture. Peter created a LaTeX draft, which I
wouldproofread,markup,andsendbacktoPeter.BeforeeachlectureIwasableto
handoutasetofnotestotheclass!
Cambridge,MA,USA RobertW.Field
References
1.H.Lefebvre-Brion, R.W.Field, The Spectra and Dynamics of DiatomicMolecules (Elsevier,
Boston,2004)
2.R.W.Field,J.Baraban,S.H.Lipoff,A.R.Beck,EffectiveHamiltonians,inHandbookofHigh-
ResolutionSpectroscopies,ed.byM.Quack,F.Merkt(Wiley,Oxford,2010)
Contents
1 Introduction .................................................................. 1
1.1 Rotation:TheRigidRotor............................................. 1
1.2 Vibration:TheHarmonicOscillator.................................. 2
1.3 ElectronicStructure:TheParticlein a Boxandthe
HydrogenAtom........................................................ 3
1.4 Transition Selection and Propensity Rules: (cid:2)J,
Franck–Condon,and(cid:2)S.............................................. 4
1.5 RotationalBranches,VibrationalBands,andElectronic
Transitions ............................................................. 5
1.6 SomeSumRules....................................................... 8
1.7 EigenstatesareStationary............................................. 10
References..................................................................... 11
2 HierarchyofTermsintheEffectiveHamiltonian........................ 13
2.1 AdiabaticandDiabaticRepresentations.............................. 13
2.1.1 Introduction.................................................... 13
2.1.2 Adiabaticvs.DiabaticRepresentations ...................... 14
2.2 Hspin–orbit................................................................ 16
2.3 HROT,theRotationalOperator........................................ 17
2.4 Hund’sCases........................................................... 18
2.4.1 H.0/vs.H.1/ ................................................... 22
2.5 TwoBasisSetsforthe2(cid:2)2“Two-Level”Problem................. 22
2.6 SomeReasonsforPatterns............................................ 23
2.7 StraightLinePlots..................................................... 24
2.8 StackedPlots........................................................... 25
2.9 AngularMomenta:ABriefSummary................................ 29
2.10 WhereHaveWeBeenandWhereareWeGoing?................... 30
References..................................................................... 30
3 SpectroscopicPerturbations:HomogeneousandHeterogeneous ...... 33
3.1 WhatIsaPerturbation?................................................ 33
3.2 LevelShiftsandIntensityBorrowing................................. 41
vii
viii Contents
3.3 TwoQualitativelyDistinctClasses ofPerturbation:
HomogeneousandHeterogeneous.................................... 42
3.4 Franck–CondonFactors............................................... 42
3.5 WhichFranck–CondonFactorsShouldIUse?....................... 43
3.6 IntensityBorrowedfromaNearbyBrightState ..................... 44
3.7 IntensityBorrowedfromanEnergeticallyRemoteBrightState .... 44
3.8 IntensityInterferenceEffects.......................................... 45
References..................................................................... 46
4 TheEffectiveHamiltonianforDiatomicMolecules...................... 47
4.1 Introduction............................................................ 47
4.2 MainTopicsofThisLecture.......................................... 49
4.2.1 R-Dependence................................................. 49
4.2.2 How DoWe AccountforInteractionswith
EnergeticallyRemoteStates?................................. 49
4.2.3 VanVleckTransformation.................................... 50
4.3 R-DependenceIsEncodedinv;J Dependence..................... 51
4.3.1 TransitionMoments:(cid:2).R/!Mv0;v00........................ 51
4.3.2 CentrifugalDistortion,D .................................... 52
e
4.3.3 Vibration-RotationInteraction,˛e:ASmallSurprise....... 54
4.4 VanVleckTransformationforNon-1†C States...................... 55
4.4.1 CentrifugalDistortion......................................... 56
4.4.2 TheVanVleckTransformation............................... 57
4.4.3 ExampleofCentrifugalDistortionina3…State............ 60
4.4.4 ƒ-Doubling.................................................... 61
4.5 Summary............................................................... 67
References..................................................................... 68
5 RotationofPolyatomicMolecules.......................................... 69
5.1 Introduction............................................................ 69
5.2 RotationalEnergyLevelsofRigidPolyatomicRotors .............. 74
5.2.1 SymmetricTop ................................................ 74
5.2.2 AsymmetricTop............................................... 75
5.3 CorrelationDiagrams:WHY? ........................................ 79
5.3.1 Prolate–OblateTopCorrelationDiagram.................... 79
5.3.2 AssignmentsofRotationalTransitions....................... 81
5.4 VibrationalDependenceofRotationalConstants.................... 83
References..................................................................... 84
6 QuantumBeats............................................................... 85
6.1 Introduction............................................................ 85
6.2 Time-DependentSchrödingerEquation(TDSE)..................... 87
6.3 “Bright”and“Dark”States............................................ 87
6.4 Dynamics............................................................... 90
6.5 QuantumBeats......................................................... 98
6.5.1 SimpleTwo-LevelQuantumBeats........................... 98
6.5.2 Two-LevelTreatmentofQBwithComplexEnergies....... 100
Contents ix
6.5.3 WhatDoesaQuantumBeatSignalLookLike?............. 103
6.5.4 PopulationQuantumBeats.................................... 104
6.5.5 Level-Crossingvs.Anticrossing.............................. 110
References..................................................................... 110
7 TheEffectiveHamiltonianforPolyatomicMoleculeVibration ........ 113
7.1 TheEffectiveVibrationalHamiltonianforPolyatomic
Molecules.............................................................. 113
7.2 HarmonicOscillator................................................... 114
7.2.1 MatrixElementsofPandQ.................................. 114
7.2.2 DimensionlessForms:HO,QO,PO ............................... 114
7.3 PolyatomicMolecules................................................. 118
7.3.1 Basis Set as Productof 3N (cid:3)6 Harmonic
OscillatorEigenstates......................................... 118
7.3.2 Matrix Elements of V.QO ;QO ;:::QO /
1 2 3N-6
inthe .0/ BasisSet ................................ 119
v1;v2;:::;3N(cid:2)6
7.3.3 BreakdownofNon-DegeneratePerturbationTheory........ 120
7.3.4 Polyads......................................................... 121
7.3.5 Patterns for Spectral Assignment and
Mechanismsof IntramolecularVibrational
Redistribution(IVR)andUnimolecularIsomerization...... 123
7.4 PolyadsintheAcetyleneElectronicGroundState(S ) ............. 124
0
References..................................................................... 127
8 IntramolecularDynamics:Representations,Visualizations,
andMechanisms ............................................................. 129
8.1 Fromthe“Pluck”att D0totheTime-EvolvingState.............. 130
8.2 PerturbationTheory ................................................... 130
8.3 Toluene:AHinderedRotor.AFullyWorkedOutExample......... 132
8.4 ThePluck:‰.Q;t D0/.............................................. 139
8.5 ‰.Q;t/ContainstooMuchInformation............................. 140
8.5.1 MotioninRealSpace ......................................... 141
8.5.2 MotioninStateSpace......................................... 143
8.6 Mechanism............................................................. 145
References..................................................................... 153