MOLECULAR SPECTROSCOPY
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1 MOLECULAR SPECTROSCOPY First Edition Jeanne L. McHale University of Idaho PRENTICE HALL, Upper Saddle River, New Jersey 07458
2 CONTENTS PREFACE xiii 1 INTRODUCTION AND REVIEW Historical Perspective Definitions, Derivations, and Discovery Review of Quantum Mechanics The Particle in a Box: A Model for Translational Energies The Rigid Rotor: A Model for Rotational Motion of Diatomics The Harmonic Oscillator: Vibrational Motion Approximate Solutions to the Schrodinger Equation Variation Method Perturbation Theory Statistical Mechanics Summary \ Problems 30 BIBLIOGRAPHY 31 2 THE NATURE OF ELECTROMAGNETIC RADIATION Introduction The Classical Description of Electromagnetic Radiation Maxwell's Equations Polarization Properties of Light Electric Dipole Radiation Propagation of Light in Matter Refraction Absorption and Emission of Light 44
3 vi Contents Effect of an Electromagnetic Field on Charged Particles Quantum Mechanical Aspects of Light Quantization of the Radiation Field Blackbody Radiation and the Planck Distribution Law The Photoelectric Effect and the Discovery of Photons Summary Problems 53 BIBLIOGRAPHY 55 3 ELECTRIC AND MAGNETIC PROPERTIES OF MOLECULES AND BULK MATTER Introduction Electric Properties of Molecules Review of Electrostatics Electric Moments Quantum Mechanical Calculation of Multipole Moments Interaction of Electric Moments with the Electric Field Polarizability and Induced Moments Frequency Dependence of Polarizability Quantum Mechanical Expression for the Polarizability Electric Properties of Bulk Matter Dielectric Permittivity Frequency Dependence of Permittivity Relationships between Macroscopic and Microscopic Properties The Local Field Problem: The Onsager and Kirkwood Models Magnetic Properties of Matter Basic Principles of Magnetism Magnetic Properties of Bulk Matter Magnetic Moments and Intrinsic Angular Momenta Summary Problems 89 BIBLIOGRAPHY 91 4 TIME-DEPENDENT PERTURBATION THEORY OF SPECTROSCOPY Introduction: Time Dependence in Quantum Mechanics Time-Dependent Perturbation Theory 94
4 Contents vii First-order Solution to the Time-Dependent Schrodinger Equation Perturbation due to Electromagnetic Radiation: Momentum versus Dipole Operator " Fermi's Golden Rule and the Time-Energy Uncertainty Principle Rate Expression for Emission Photon Density of States Fermi's Golden Rule for Stimulated and Spontaneous Emission Perturbation Theory Calculation of Polarizability Derivation of the Kramers-Heisenberg-Dirac Equation Finite State Lifetimes and Imaginary Component of Polarizability Oscillator Strength Quantum Mechanical Expression for Emission Rate Time Dependence of the Density Matrix Summary Problems 116 BIBLIOGRAPHY THE TIME-DEPENDENT APPROACH TO SPECTROSCOPY Introduction Time-Correlation Functions and Spectra as Fourier Transform Pairs Properties of Time-Correlation Functions and Spectral Lineshapes The Fluctuation-Dissipation Theorem Rotational Correlation Functions and Pure Rotational Spectra Correlation Functions for Absorption and Light Scattering Classical Free-Rotor Correlation Function and Spectrum Reorientational Spectroscopy of Liquids Dielectric Relaxation Far-Infrared Absorption Depolarized Rayleigh Scattering Vibration-Rotation Spectra Spectral Moments Summary Problems 149
5 Vlil Contents BIBLIOGRAPHY EXPERIMENTAL CONSIDERATIONS: ABSORPTION, EMIS- SION, AND SCATTERING Introduction Einstein A and B Coefficients for Absorption and Emission Absorption and Stimulated Emission Absorption and Emission Spectroscopy Atomic Spectra Molecular Electronic Spectra Measurement of Light Scattering: The Raman and Rayleigh Effects Spectral Lineshapes Summary Problems 171 BIBLIOGRAPHY ATOMIC SPECTROSCOPY Introduction Good Quantum Numbers and Not So Good Quantum Numbers The Hydrogen Atom: Energy Levels and Selection Rules Many-Electron Atoms The Clebsch-Gordan Series Spin-Orbit Coupling Selection Rules for Atomic Absorption and Emission El, Ml, and E2 Allowed Transitions Hyperfine Structure I The Effect of External Fields The Zeeman Effect The Stark Effect Atomic Lasers and the Principles of Laser Emission Summary Problems 206 BIBLIOGRAPHY ROTATIONAL SPECTROSCOPY Introduction Energy Levels of Free Rigid Rotors 209
6 Contents ix Diatomics Polyatomic Rotations Angular Momentum Coupling in Non- 1!] Electronic States Nuclear Statistics and J States of Homonuclear Diatomics Rotational Absorption and Emission Spectroscopy Rotational Raman Spectroscopy Corrections to the Rigid-Rotor Approximation Internal Rotation Free Rotation Limit, k B T» V o Harmonic Oscillator Limit, k B T «V o Summary Problems 245 BIBLIOGRAPHY VIBRATIONAL SPECTROSCOPY OF DIATOMICS Introduction The Born-Oppenheimer Approximation and Its Consequences The Harmonic Oscillator Model Selection Rules for Vibrational Transitions Infrared Spectroscopy Raman Scattering Beyond the Rigid Rotor-Harmonic Oscillator Approximation Perturbation Theory of Vibration-Rotation Energy The Morse Oscillator and Other Anharmonic Potentials Summary Problems 267 BIBLIOGRAPHY VIBRATIONAL SPECTROSCOPY OF POLYATOMIC MOLECULES Introduction Normal Modes of Vibration Classical Equations of Motion for Normal Modes Example: Normal Modes of a Linear Triatomic The Wilson F and G Matrices Group Frequencies 279
7 X Contents 10.3 Quantum Mechanics of Polyatomic Vibrations Group Theoretical Treatment of Vibrations Finding the Symmetries of Normal Modes, Symmetries of Vibrational Wavefunctions Selection Rules for Infrared and Raman Scattering Rotational Structure Anharmonicity Selection Rules at Work: Benzene Solvent Effects on Infrared Spectra Summary Problems 305 BIBLIOGRAPHY ELECTRONIC SPECTROSCOPY Introduction Diatomic Molecules: Electronic States and Selection Rules Molecular Orbitals and Electronic Configurations Term Symbols for Diatomics Selection Rules Examples of Selection Rules at Work: O 2 and I Vibrational Structure in Electronic Spectra of Diatomics Absorption Spectra Emission Spectra Dissociation and Predissociation Born-Oppenheimer Breakdown in Diatbmic Molecules Polyatomic Molecules: Electronic States and Selection Rules Molecular Orbitals and Electronic States of H 2 O Franck-Condon Progressions in Electronic Spectra of Polyatomics Benzene: Electronic Spectra and Vibronic Activity of Nontotally Symmetric Modes Transition Metal Complexes Emission Spectroscopy of Polyatomic Molecules Chromophores Solvent Effects in Electronic Spectroscopy Solvent-Induced Frequency Shifts Solvent Effects on Intensity 358
8 Contents xi Specific Solvent Effects in Electronic Spectra Summary Problems 360 BIBLIOGRAPHY RAMAN AND RESONANCE RAMAN SPECTROSCOPY Introduction Selection Rules in Raman Scattering Off-Resonance Raman Scattering Resonance Raman Scattering Polarization in Raman Scattering Polarization in Off-Resonance Raman Scattering Polarization in Resonance Raman Scattering Rotational and Vibrational Dynamics in Raman Scattering Analysis of Raman Excitation Profiles Transform Theory of Raman Intensity Time-Dependent Theory of Resonance Raman Spectra Raman Scattering as a Third-Order Nonlinear Process Summary Problems 406 BIBLIOGRAPHY 407 APPENDICES 408 A. MATH REVIEW 410 A.I Vectors and Tensors in Three Dimensions 410 A.2 Matrices 412 A.3 Operations with Cartesian and Spherical Tensors 415 A.4 Spherical Harmonics 417 A.5 Wigner Rotation Functions and Spherical Tensors 418 A.6 The Clebsch-Gordan Series and 3j Symbols 421 BIBLIOGRAPHY 423 B. PRINCIPLES OF ELECTROSTATICS 424 B.I Units 424 B.2 Some Applications of Gauss' Law 425
9 Xll Contents B.2.1 The Lorentz Model of the Atom 426 B.2.2 Electric Field within a Capacitor 426 B.3 Some Mathematical Details 427 C. GROUP THEORY 430 C.I Point Groups and Symmetry Operations 430 C.2 Information Conveyed by Character Tables 432 C.3 Direct Products and Reducible Representations 436 C.4 Character Tables 438 BIBLIOGRAPHY 448 SUBJECT INDEX 449
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