A Spectroscopy Primer An Introduction to Atomic, Rotational, Vibrational, Raman, Electronic, Photoelectron and NMR Spectroscopy by Robert J. Le Roy Department of Chemistry, University of Waterloo Waterloo, Ontario N2L 3G1, Canada c Robert J. Le Roy, 2003-2011 i ii Preface Spectroscopy is the scientist’s window on the molecular world. As molecules are too small to be seen directly by the human eye, we rely on their interaction with light (or electromagnetic radiation), to determine their properties, how they are formed from their constituent atoms, and how they react. Interaction with light probes the molecules’ characteristic rotational and vibrational motions, and we then attempt to explain that behaviour in terms of theoretical models. This allows us to determine what atoms a particular molecule is composed of, the length and strength of its bonds, and more generally, the patterns in which atoms assemble to form the diverse and myriad molecules on which we rely for industrial applications, for modern drugs, and for life itself. These notes begin by examining the interaction between light and matter as predicted by models derived from quantum mechanics, and by outlining the principles of spectroscopy. We then study the types of spectra associated with several different regions of the electromagnetic spectrum, and see that absorption or emission of light in those distinct regions tends to be associated with different types of molecular motion. We will see how the structures of molecules found both in interstellar space and closer to home can be determined with rotational (or microwave) spectroscopy. We will then see how vibrational (or infrared) and Raman spectroscopy may be used to determine the strengths of bonds and to identify characteristic groups of atoms within molecules. We will also see that the electronic energy binding atoms together in molecules and molecular ions can be studied using electronic and photoelectron spectroscopy. Finally, we will see how nuclear magnetic resonance spectroscopy uses the magnetic properties of atomic nuclei within molecules to learn about the structures of complex molecules, such as proteins, and to image tissues within human beings. The credit for these notes rests not only with the author, but also with several colleagues who provided important input. In particular, Professor John Hepburn developed the first offerings of this material at the University of Waterloo, while Professor William Power’s renovated version of his course notes were a key template for the current document. In addition, Professors Fred McCourt and Peter Bernath have freely offered valuable and abundant suggestions and criticism throughout the development process, while Drs. Iain McNab and John Ogilvie have provided helpful comments and suggestions on early drafts of the manuscript. I am also particularly grateful to Professor Fred McCourt for a thorough, critical reading of the current version of this document. Finally, the curiosity and bafflement of several classes of Chemistry 129 and 209 students stimulated many revisions and improvements that now grace these pages. All surviving errors are, of course, mine. I invite all readers to help to improve these notes further by suggesting changes that they feel might be helpful. Robert Le Roy Waterloo, August 2011 iii iv PREFACE Contents Preface iii Contents vii List of Figures x List of Tables xi List of Symbols xiii 1 Light, Quantization, Atoms and Spectroscopy 1 1.1LightandtheElectromagneticSpectrum.............................. 1 1.1.1 Physics in 1900 . 1 1.1.2 WavePropertiesofLight.................................. 2 1.1.3 TheQuantumTheoryofLight............................... 3 1.1.4 ABriefNoteonUnits.................................... 7 1.2QuantumTheoryofMatter..................................... 8 1.2.1 TheSpectrumoftheHydrogenAtom........................... 8 1.2.2 TheBohrTheoryoftheAtom............................... 9 1.2.3 deBroglieWavelengths................................... 11 1.3 Wave Mechanics and the Schr¨odingerEquation.......................... 12 1.3.1 AParticleinaOne-DimensionalBox............................ 12 1.3.2 Orbital or Rotational Motion: A Particle on a Ring . 17 1.4ElectronicStructureofAtomsandMolecules........................... 18 1.4.1 HydrogenicAtomicOrbitals................................. 18 1.4.2 Multi-ElectronAtomsandAtomicSpectroscopy..................... 21 1.4.3 Molecular Energies and the Born-Oppenheimer Approximation . 22 1.5Spectroscopyatlast......................................... 24 1.6Problems............................................... 25 2 Rotational Spectroscopy 27 2.1ClassicalDescriptionofMolecularRotation............................ 27 2.1.1 Why Does Light Cause Rotational Transitions? . 27 2.1.2 RelativeMotionandtheReducedMass.......................... 28 2.1.3 MotionofaRotatingBody................................. 29 2.2QuantumMechanicsofMolecularRotation............................ 31 2.2.1 TheBasics.......................................... 31 2.2.2 EnergyLevels,SelectionRules,andTransitionEnergies................. 32 2.2.3 IllustrativeApplications................................... 33 2.3Complications!............................................ 35 2.4DegeneraciesandIntensities..................................... 39 2.5RotationalSpectraofPolyatomicMolecules............................ 42 v vi CONTENTS 2.5.1 LinearMoleculesare(Relatively)EasytoTreat!..................... 42 2.5.2 IllustrativeApplications................................... 44 2.5.3 Non-LinearPolyatomicMoleculesareMoreDifficult...... 45 2.6ConcludingRemarks......................................... 47 2.7Problems............................................... 49 3 Vibrational Spectroscopy 51 3.1ClassicalDescriptionofMolecularVibrations........................... 51 3.1.1 WhyDoesLightCauseVibrationalTransitions?..................... 51 3.1.2 TheCentreofMassandRelativeMotion......................... 52 3.1.3 The Classical Harmonic Oscillator . 53 3.2 Quantum Mechanics of the Harmonic Oscillator . 54 3.3 Anharmonic Vibrations and the Morse Oscillator . 57 3.3.1 EigenvaluesandPropertiesoftheMorsePotential.................... 57 3.3.2 OvertonesandHotBands.................................. 59 3.3.3 Higher-Order Anharmonicity and the Dunham Expansion . 60 3.4DissociationEnergiesandBirge-SponerPlots........................... 62 3.5VibrationsinPolyatomicMolecules................................. 64 3.6RotationalStructureinVibrationalSpectraofDiatomics.................... 67 3.7WhyAreVibrationalLevelSpacingssoLarge?.......................... 71 3.8Problems............................................... 72 4 Raman Spectroscopy 75 4.1“Light-As-A-Wave”DescriptionofRamanScattering....................... 75 4.2“Light-As-A-Particle”DescriptionofRamanScattering..................... 78 4.3RotationalRamanSpectra...................................... 79 4.4VibrationalRamanSpectra..................................... 80 4.5RamanSpectraofPolyatomicMolecules.............................. 81 4.6Problems............................................... 82 5 Electronic Spectroscopy 83 5.1WhyDoesLightCauseElectronicTransitions?.......................... 83 5.2Vibrational-RotationalStructureinElectronicSpectra...................... 84 5.3VibrationalPropensityRulesinElectronicTransitions...................... 88 5.4Problems............................................... 93 6 Photoelectron Spectroscopy 95 6.1 Photoelectron Spectroscopy: ThePhotoelectricEffectRevisited................................. 95 6.2Koopmans’Theorem......................................... 97 6.3VibrationalFineStructureinPhotoelectronSpectra....................... 97 6.4MolecularOrbitalsandPhotoelectronSpectra...........................100 6.5SomeComplicationsinPhotoelectronSpectra...........................103 6.6X-RayPhotoelectronSpectroscopy(XPS).............................105 6.7AugerElectronSpectroscopy(AES)................................107 6.8Problems...............................................108 7NMRSpectroscopy 111 7.1BasicsofNMRSpectroscopy....................................111 7.1.1 AngularMomentumandNuclearSpin..........................111 7.1.2 Magnetic Moments and Nuclei in a Magnetic Field . 112 7.1.3 NMRSpectra.........................................113 7.2ChemicalShifts............................................117 CONTENTS vii 7.2.1 ElectronicShieldingofNucleiand‘ChemicalShifts’...................117 7.2.2 What Determines Chemical Shifts, and The Chemical Shift Scale . 117 7.2.3 WorkingWithChemicalShifts...............................119 7.3Spin-SpinCoupling..........................................120 7.3.1 Basics:CouplingfromaSingleNeighbour ........................120 7.3.2 ‘Equivalence’, and Coupling from Multiple Equivalent Nuclei . 121 7.3.3 Spin-Spin Coupling to More Than One Type of Neighbour . 123 7.4MolecularStructuresfromNMRSpectra..............................124 7.5Problems...............................................125 viii CONTENTS List of Figures 1.1 The electric field of light oscillates in space and in time . 3 1.2 Black-body radiation: observed distributions and the Rayleigh-Jeans law prediction . 4 1.3 The photoelectric effect: A.Theexperiment; B.Theobservations............... 5 1.4Thehydrogenatomemissionspectrum............................... 9 1.5Hydrogenatomenergylevelsandtransitions........................... 10 1.6Propertiesofthreesquare-wellparticle-in-a-boxsystems..................... 14 1.7 Level energies and wave functions for four particle-in-a-box