Rotational Spectroscopy of Diatomic Molecules

1046 Pages·2003·9.637 MB·English

by  John M. Brown, Alan Carrington

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Rotational Spectroscopy of Diatomic Molecules
Writtentobethedefinitivetextontherotationalspectroscopyofdiatomicmolecules,
this book develops the theory behind the energy levels of diatomic molecules and
then summarises the many experimental methods used to study the spectra of these
moleculesinthegaseousstate.
Afterageneralintroduction,themethodsusedtoseparatenuclearandelectronic
motionsaredescribed.BrownandCarringtonthenshowhowthefundamentalDiracand
Breitequationsmaybedevelopedtoprovidecomprehensivedescriptionsofthekinetic
andpotentialenergytermswhichgovernthebehaviouroftheelectrons.Onechapteris
devotedsolelytoangularmomentumtheoryandanotherdescribesthedevelopmentof
the so-called effective Hamiltonian used to analyse and understand the experimental
spectraofdiatomicmolecules.Theremainderofthebookconcentratesonexperimental
methods.
Thisbookwillbeofinteresttograduatestudentsandresearchersinterestedinthe
rotationalspectroscopyofdiatomicmolecules.
   
SeriesEditors:
RichardJ.Saykally,DepartmentofChemistry,UniversityofCalifornia,Berkeley
Ahmed H. Zewail, Arthur Amos Noyes Laboratory of Chemical Physics, California
InstituteofTechnology
DavidA.King,DepartmentofChemistry,UniversityofCambridge
Rotational Spectroscopy of
Diatomic Molecules
 . 
ProfessorofChemistry,UniversityofOxford
FellowofExeterCollege,Oxford
 
FormerRoyalSocietyResearchProfessor,DepartmentofChemistry,
UniversityofSouthampton
HonoraryFellowofDowningCollege,Cambridge
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
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Published in the United States of America by Cambridge University Press, New York
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© John Brown and Alan Carrington 2003
This book is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2003
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Contents
Preface page xv
Summaryofnotation xix
Figureacknowledgements xxiii
1 Generalintroduction 1
1.1 Electromagneticspectrum 1
1.2 Electromagneticradiation 3
1.3 Intramolecularnuclearandelectronicdynamics 5
1.4 Rotationallevels 9
1.5 Historicalperspectives 12
1.6 Finestructureandhyperfinestructureofrotationallevels 14
1.6.1 Introduction 14
1.6.2 1(cid:2)+states 15
1.6.3 Openshell(cid:2)states 21
1.6.4 Openshellstateswithbothspinandorbitalangularmomentum 26
1.7 TheeffectiveHamiltonian 29
1.8 Bibliography 32
Appendix1.1 Maxwell’sequations 33
Appendix1.2 Electromagneticradiation 35
References 36
2 Theseparationofnuclearandelectronicmotion 38
2.1 Introduction 38
2.2 Electronicandnuclearkineticenergy 40

2.2.1 Introduction 40
2.2.2 Originatcentreofmassofmolecule 41
2.2.3 Originatcentreofmassofnuclei 43
2.2.4 Originatgeometricalcentreofthenuclei 44
2.3 ThetotalHamiltonianinfield-freespace 44
2.4 Thenuclearkineticenergyoperator 45
2.5 Transformationoftheelectroniccoordinatestomolecule-fixedaxes 51
2.5.1 Introduction 51
2.5.2 Spacetransformations 52
2.5.3 Spintransformations 54
2.6 Schro¨dingerequationforthetotalwavefunction 59
2.7 TheBorn–OppenheimerandBornadiabaticapproximations 60
vi Contents
2.8 Separationofthevibrationalandrotationalwaveequations 61
2.9 Thevibrationalwaveequation 63
2.10 RotationalHamiltonianforspace-quantisedelectronspin 67
2.11 Non-adiabaticterms 67
2.12 Effectsofexternalelectricandmagneticfields 68
Appendix2.1 Derivationofthemomentumoperator 71
References 72
3 TheelectronicHamiltonian 73
3.1 TheDiracequation 73
3.2 SolutionsoftheDiracequationinfield-freespace 76
3.3 Electronspinmagneticmomentandangularmomentum 77
3.4 TheFoldy–Wouthuysentransformation 80
3.5 TheFoldy–WouthuysenandDiracrepresentationsforafreeparticle 85
3.6 Derivationofthemany-electronHamiltonian 89
3.7 Effectsofappliedstaticmagneticandelectricfields 94
3.8 Retardedelectromagneticinteractionbetweenelectrons 97
3.8.1 Introduction 97
3.8.2 Lorentztransformation 98
3.8.3 Electromagneticpotentialsduetoamovingelectron 99
3.8.4 Gaugeinvariance 101
3.8.5 ClassicalLagrangianandHamiltonian 103
3.9 TheBreitHamiltonian 104
3.9.1 Introduction 104
3.9.2 ReductionoftheBreitHamiltoniantonon-relativisticform 105
3.10 ElectronicinteractionsinthenuclearHamiltonian 109
3.11 Transformationofcoordinatesinthefield-freetotalHamiltonian 110
3.12 TransformationofcoordinatesfortheZeemanandStarktermsinthe
totalHamiltonian 114
3.13 Conclusions 118
Appendix3.1 PowerseriesexpansionofthetransformedHamiltonian 121
References 122

4 Interactionsarisingfromnuclearmagneticandelectricmoments 123
4.1 Nuclearspinsandmagneticmoments 123
4.2 Derivationofnuclearspinmagneticinteractionsthroughthemagnetic
vectorpotential 125
4.3 DerivationofnuclearspininteractionsfromtheBreitequation 130
4.4 Nuclearelectricquadrupoleinteractions 131
4.4.1 SphericaltensorformoftheHamiltonianoperator 131
4.4.2 CartesianformoftheHamiltonianoperator 133
4.4.3 MatrixelementsofthequadrupoleHamiltonian 134
4.5 Transformationofcoordinatesforthenuclearmagneticdipoleand
electricquadrupoleterms 136
References 138
Contents vii
5 Angularmomentumtheoryandsphericaltensoralgebra 139
5.1 Introduction 139
5.2 Rotationoperators 140
5.2.1 Introduction 140
5.2.2 Decompositionofrotationaloperators 142
5.2.3 Commutationrelations 142
5.2.4 Representationsoftherotationgroup 143
5.2.5 Orbitalangularmomentumandsphericalharmonics 144
5.3 Rotationsofarigidbody 146
5.3.1 Introduction 146
5.3.2 Rotationmatrices 148
5.3.3 Spin1/2systems 150
5.3.4 Symmetrictopwavefunctions 150
5.4 Additionofangularmomenta 152
5.4.1 Introduction 152
5.4.2 Wigner3-j symbols 154
5.4.3 Couplingofthreeormoreangularmomenta:Racahalgebra,
Wigner6-j and9-j symbols 155
5.4.4 Clebsch–Gordanseries 157
5.4.5 Integralsoverproductsofrotationmatrices 158
5.5 Irreduciblesphericaltensoroperators 159
5.5.1 Introduction 159
5.5.2 Examplesofsphericaltensoroperators 160
5.5.3 Matrixelementsofsphericaltensoroperators:theWigner–Eckart
theorem 163
5.5.4 Matrixelementsforcompositesystems 165
5.5.5 Relationshipbetweenoperatorsinspace-fixedandmolecule-fixed
coordinatesystems 167
5.5.6 Treatmentoftheanomalouscommutationrelationshipsofrota-
tionalangularmomentabysphericaltensormethods 168
Appendix5.1 Summaryofstandardresultsfromsphericaltensoralgebra 171
References 175

6 Electronicandvibrationalstates 177
6.1 Introduction 177
6.2 Atomicstructureandatomicorbitals 178
6.2.1 Thehydrogenatom 178
6.2.2 Many-electronatoms 181
6.2.3 Russell–Saunderscoupling 184
6.2.4 Wavefunctionsfortheheliumatom 187
6.2.5 Many-electronwavefunctions:theHartree–Fockequation 190
6.2.6 Atomicorbitalbasisset 194
6.2.7 Configurationinteraction 196
6.3 Molecularorbitaltheory 197
viii Contents
6.4 Correlationofmolecularandatomicelectronicstates 203
6.5 Calculationofmolecularelectronicwavefunctionsandenergies 206
6.5.1 Introduction 206
6.5.2 ElectronicwavefunctionfortheH+molecularion 207
2
6.5.3 ElectronicwavefunctionfortheH molecule 208
2
6.5.4 Many-electronmolecularwavefunctions 212
6.6 CorrectionstoBorn–OppenheimercalculationsforH+andH 219
2 2
6.7 Couplingofelectronicandrotationalmotion:Hund’scouplingcases 224
6.7.1 Introduction 224
6.7.2 Hund’scouplingcase(a) 225
6.7.3 Hund’scouplingcase(b) 226
6.7.4 Hund’scouplingcase(c) 228
6.7.5 Hund’scouplingcase(d) 228
6.7.6 Hund’scouplingcase(e) 229
6.7.7 Intermediatecoupling 230
6.7.8 Nuclearspincouplingcases 232
6.8 Rotationsandvibrationsofthediatomicmolecule 233
6.8.1 Therigidrotor 233
6.8.2 Theharmonicoscillator 235
6.8.3 Theanharmonicoscillator 238
6.8.4 Thenon-rigidrotor 242
6.8.5 Thevibratingrotor 243
6.9 Inversionsymmetryofrotationallevels 244
6.9.1 Thespace-fixedinversionoperator 244
6.9.2 Theeffectofspace-fixedinversionontheEuleranglesandon
molecule-fixedcoordinates 245
6.9.3 ThetransformationofgeneralHund’scase(a)andcase(b)func-
tionsunderspace-fixedinversion 246
6.9.4 Paritycombinationsofbasisfunctions 251
6.10 Permutationsymmetryofrotationallevels 251
6.10.1 Thenuclearpermutationoperatorforahomonucleardiatomic
molecule 251

6.10.2 ThetransformationofgeneralHund’scase(a)andcase(b)func-
tionsundernuclearpermutation P 252
12
6.10.3 Nuclearstatisticalweights 254
6.11 Theoryoftransitionprobabilities 256
6.11.1 Time-dependentperturbationtheory 256
6.11.2 TheEinsteintransitionprobabilities 258
6.11.3 Einsteintransitionprobabilitiesforelectricdipoletransitions 261
6.11.4 Rotationaltransitionprobabilities 263
6.11.5 Vibrationaltransitionprobabilities 266
6.11.6 Electronictransitionprobabilities 267
6.11.7 Magneticdipoletransitionprobabilities 269