The Raman Effect

A Unified Treatment of the Theory of Raman Scattering by Molecules

DerekA. Long Emeritus Professor ofStructural Chemistry

University of Bradford

Bradford, UK

JOHN WILEY & SONS, LTD

Vll

Contents

Preface xix

Acknowledgements xxiii

Part One Theory 1

1 Survey of Light-scattering Phenomena 3

1.1 Introduction 3 1.2 Some Basic Definitions 4 1.3 Rayleigh and Raman Scattenng 5

1.3.1 Description 5 1.3.2 Energy transfer model 7

1.4 Hyper-Rayleigh and Hyper-Raman Scattenng 10 1.4.1 Description 10 1.4.2 Energy transfer model 10

1.5 Second Hyper-Rayleigh and Second Hyper-Raman Scattering 11 1.5.1 Description 11 1.5.2 Energy transfer model 11

1.6 Coherent anti-Stokes and Coherent Stokes Raman Scattering 11 1.7 Stimulated Raman Gain and Loss Spectroscopy 13 1.8 Typical Spectra 14 1.9 Bases for the Theoretical Treatment of Rayleigh and

Raman Scattering 16 1.10 Historical Perspective 16

Contents

1.11 Caveat 17 References 17

Introduction to Theoretical Treatments of Incoherent Light Scattering 19 2.1 General Considerations 19 2.2 Induced Oscillating Electric Dipoles as Sources of

Scattered Radiation 21 2.3 Basis of the Classical Theory of Light Scattering 22 2.4 Basis of the Quantum Mechanical Treatment

of Incoherent Light-Scattering Phenomena: Electric Dipole Case 24 2.5 Extension of Quantum Mechanical Treatment of Incoherent

Light Scattering to Include Magnetic Dipole and Electric Quadrupole Cases 27

2.6 Comparison of the Classical and Quantum Mechanical Treatments of Light Scattering 28

2.7 The Way Ahead 29

Classical Theory of Rayleigh and Raman Scattering 31 3.1 Introduction 31 3.2 First-order Induced Electric Dipole 31 3.3 Frequency Dependence of the First-order Induced Electric

Dipole 34 3.4 Classical Scattering Tensors aRay and af™ 35 3.5 Selection Rules for Fundamental Vibrations 36

3.5.1 General considerations 36 3.5.2 Diatomic molecules 36 3.5.3 Polyatomic molecules 38

3.6 Selection Rules for Overtones and Combinations 43 3.7 Coherence Properties of Rayleigh and Raman Scattering 44 3.8 Limitations of the Classical Theory 45 3.9 Example of Rayleigh and Raman Scattering 45 3.10 Presentation of Raman Spectra 47

References 48

Quantum Mechanical Theory of Rayleigh and Raman Scattering 49 4.1 Introduction 49 4.2 Time-dependent Perturbation Theory and (a) / ; 50 4.3 Qualitative Discussion of {ctpa)fi 54

4.3.1 Frequency denominators - 55 4.3.2 Transition electric dipole numerators 56 4.3.3 Selection rules 58

4.4 Tensorial Nature of the Transition Polarizability and its Symmetry 58 4.5 Born-Oppenheimer Approximation and the Transition

Polarizability Tensor 61

Contents ix

4.6 Simplification of (oePa)efvf:eSvi: General Considerations 64 4.7 Simplification by Radical Approximation: the Placzek

Transition Polarizability 65 4.8 Simplification of (apo)efvf:ev ^y Stages 68

4.8.1 Introduction of Herzberg-Teller vibronic coupling 68 4.8.2 Identification of non-resonance and resonance

situations 75 4.9 Normal Electronic (and Vibronic) Raman Scattering 77 4.10 Normal Pure Vibrational Raman Scattering 78 4.11 Electronic (and Vibronic) Resonance Raman Scattering 81 4.12 Vibrational Resonance Raman Scattering 83 4.13 Units and Orders of Magnitude 83

References 84

5 Vibrational Raman Scattering 85 5.1 Introduction 85 5.2 The Placzek Vibrational Transition Polarizability:

Recapitulation 86 5.2.1 Cartesian basis 86 5.2.2 The spherical basis 88

5.3 Definition of Illumination-Observation Geometry 89 5.4 Intensity of Scattered Radiation: Some General Considerations 94

5.4.1 Development of a symbol for scattered intensity 94 5.4.2 Scattering cross-section 95

5.5 Intensity Formulae and Polarization Characteristics for a General Vibrational Transition in Various Illumination-Observation Geometries 97 5.5.1 General considerations 97 5.5.2 Linearly polarized incident radiation 98 5.5.3 Natural incident radiation 102 5.5.4 Angular dependence of scattered intensity 103 5.5.5 Circularly polarized incident radiation 106 5.5.6 Symmetry and depolarization ratios, reversal

coefficients and degrees of circularity 109 5.6 Stokes Parameters for Scattered Radiation 113 5.7 Specific Vibrational Transitions 116 5.8 Vibrational Selection Rules 120 5.9 Patterns of Vibrational Spectra 123 5.10 Orders of Magnitude 126 5.11 Epilogue 127

References 131 Reference Tables for Chapter 5 132 Reference Table 5.1: Definitions for 1(6; ps,p{) 132

Contents

Reference Table 5.2(a) to 5.2(g): Intensities, Polarization Properties and Stokes Parameters for Vibrational Raman (and Rayleigh) Scattering 132 Reference Table 5.3: Symmetry classes for x, y, z, the rotations Rx, Ry and Rz, and the components of the cartesian basis tensor ca. 145

Rotational and Vibration-Rotation Raman Scattering 153 6.1 Introduction 153 6.2 Irreducible Transition Polarizability Components 154 6.3 Symmetrie Top 156

6.3.1 Selection rules 156 6.3.2 Placzek invariants ( # 0 ) ) / i 157 6.3.3 Intensities 167 6.3.4 Subsequent development 169

6.4 Rotational and Vibrational Terms 169 6.5 Statistical Distribution of Molecular Population 171 6.6 Diatomic Molecule 173

6.6.1 Introduction 173 6.6.2 Heteronuclear diatomic molecule: pure rotation 174 6.6.3 Heteronuclear diatomic molecule: vibration-rotation 175 6.6.4 Homonuclear diatomic molecule: nuclear spin degeneracy 179 6.6.5 Intensity distribution 180

6.7 Symmetrie Top Molecule 186 6.7.1 Introduction 186 6.7.2 Symmetrie top: pure rotation 187 6.7.3 Symmetrie top: vibration-rotation 191 6.7.4 Intensities 203

6.8 Linear Molecules 204 6.8.1 Rotation and vibration-rotation Raman spectra 204 6.8.2 Intensities 207

6.9 Contributions from Electronic Orbital and Spin Angular Momenta 208 6.10 Spherical Top Molecules 210 6.11 Asymmetrie Top Molecules 211 6.12 Epilogue 211

References 213 Reference Tables for Chapter 6 214 Introduction 214 Reference Tables 6.1 to 6.4 216

Vibrational Resonance Raman Scattering 221 7.1 Introduction 221 7.2 Vibrational Transition Polarizability Tensor Components in the

Resonance Case, Based on Perturbation Theory 222

Contents xi

7.3 Comparison of the AVI, ßV I , CVI and DW1 Terms 224 7.3.1 The AVI term 224 7.3.2 The ß V I term 227 7.3.3 The CVI term 229 7.3.4 The DVI term 229 7.3.5 Subsequent developments 230

7.4 AVI Term Raman Scattering from Molecules with Totally Symmetrie Modes 231 7.4.1 AVI term Raman scattering from molecules with one

totally Symmetrie mode 231 7.4.2 AVI term Raman scattering from molecules

with more than one totally Symmetrie mode: general considerations 237

7.4.3 ' AVI term Raman scattering from totally Symmetrie modes when Ak is very small 238

7.5 AVI Term Raman Scattering Involving Non-Totally Symmetrie Modes 239 7.5.1 General considerations 239 7.5.2 AVI term scattering involving a change of molecular

symmetry of the resonant excited State 239 7.5.3 AVI term scattering involving excited State Jahn-Teller

coupling 240 7.5.4 Summary of excited State Jahn-Teller effects in

resonance Raman scattering 240 7.6 ßV I Term Scattering Involving Vibronic Coupling of the Resonant

Excited State to a Second Excited State 241 7.6.1 Introduction 241 7.6.2 BV1 term scattering from molecules with non-totally

Symmetrie modes 241 7.6.3 BY1 term scattering from molecules with totally

Symmetrie modes 244 7.7 Symmetry, Raman Activity and Depolarization Ratios 246

7.7.1 General symmetry considerations 246 7.7.2 The AVI term 247 7.7.3 The ßV I term 250

7.8 Time-Dependent Formulation of Resonance Raman Scattering 262 7.8.1 Introduction 262 7.8.2 Transformation of the AVI term to a time-dependent

expression 263 7.8.3 The time-dependent Interpretation of resonance Raman

scattering 264 7.9 Continuum Resonance Raman Scattering 266

References 270

XU Contents

8 Rotational and Vibration-Rotation Resonance Raman Scattering 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10

Introduction General Expression for (a\-ß)fi for a Symmetrie Top Molecule

General Expression for (afi )fi

Contraction of General Expression for (a« ) / , The Quadratic Term Selection Rules Evaluation of \(cim ) / / | 2

Intensities and Depolarization Ratios An Illustrative Example Concluding Remarks Reference

271 271 272 274 274 275 276 277 279 283 287 287

9 Normal and Resonance Electronic and Vibronic Raman Scattering 9.1 9.2

9.3

Introduction Normal Electronic and Vibronic Raman Scattering 9.2.1 9.2.2 9.2.3 9.2.4 9.2.5

General considerations Affl-term scattering (Bm + Cin)-term scattering Dra-term scattering Transition tensor symmetry

Resonant Electronic and Vibronic Raman Scattering 9.3.1 9.3.2 9.3.3 9.3.4 9.3.5

General considerations Av-term scattering ßv-term scattering Cv-term scattering Dv-term scattering

9.4 Selection Rules in Electronic Raman Spectra 9.4.1 General symmetry considerations

9.5 Intensities and Polarization Properties of Electronic Raman Scattering 9.5.1 Intensities: general considerations 9.5.2 Excitation profiles 9.5.3 Depolarization ratios

289 289 289 289 290 291 292 292 292 292 293 296 297 297 297 297

301 301 301 302

10 Rayleigh and Raman Scattering by Chiral Systems 10.1 Introduction 10.2 Outline of the Theoretical Treatment 10.3 Intensities of Optically Active Rayleigh Scattering

10.3.1 General considerations 10.3.2 Intensity formulae 10.3.3 Stokes parameters

303 303 305 310 310 314 317

Contents xiii

10.4 Intensities of Optically Active Raman Scattering 321 10.4.1 General considerations 321 10.4.2 Discussion of intensities and isotropic invariants 323 10.4.3 Placzek polarizability theory and optically

active scattering 324 10.5 Symmetry Considerations 326 10.6 Concluding Remarks 327

Reference 327 Reference Tables for Chapter 10 328

Part Two Appendices 337

Introduction 339

AI The Right-handed Cartesian Axis System and Related Coordinate Systems 341 ALI Introduction 341 AI.2 The Right-handed Cartesian Axis System 342 AI.3 Cartesian Coordinate System 344 AI.4 Cylindrical Coordinate System 344 AI.5 Polar Coordinate System 345 AI.6 Complex Coordinate Systems 347

A2 The Summation Convention 349 A2.1 General Definitions 349

A3 Direction Cosines 351 A3.1 Introduction 351 A3.2 Definitions and Properties of Direction Cosines 351 A3.3 Definitions of Direction Cosines in Other Coordinate Systems 354

A4 Isotropic Averages of Products of Direction Cosines 355 A4.1 Introduction 355 A4.2 Specific Isotropic Averages of Products of Direction Cosines 356 A4.3 General Formulae 358

References 358

A5 The Euler Angles and the Rotation Operator 359 A5.1 Introduction 359 A5.2 Definitions of the Euler Angles and the Rotation Operator 359 A5.3 The Relationship of the Euler Angles to the Polar Coordinates 362 A5.4 Direction Cosines and Euler Angles 362

References 363

xiv Contents

A6 Complex Numbers and Quantities 365 A6.1 Introduction 365 A6.2 Definitions and Operations 365 A6.3 Graphical Representation of Complex Numbers 367 A6.4 Complex Numbers and Polar Coordinates 369 A6.5 Complex Quantities and Physical Phenomena 370 A6.6 Spherical Coordinates 371

Reference 372

A7 Some Properties of Matrices 373 A7.1 Introduction 373 A7.2 Nomenclature 373 A7.3 Some Special Matrices 374 A7.4 Matrix Representation of Simultaneous Linear Equations 377 A7.5 Eigenvalues and Eigenvectors 377 A7.6 Example of Diagonalization of a Matrix 379

A8 Vectors, I 381 A8.1 Introduction: Sealars, Vectors and Tensors 381 A8.2 Basic Definition of a Vector 382 A8.3 Unit Vectors 383 A8.4 Vector Addition, Subtraction and Multiplication by a Scalar 385 A8.5 Multiplication of Two Vectors 385 A8.6 Triple Products of Vectors 390

A8.6.1 A(B-C) 391 A8.6.2 A (B x C) 391 A8.6.3 Ax(B x C) 392 A8.6.4 ABC 392

A8.7 Formal Definition of a Vector in Terms of its Transformation upon Rotation of Axes 393

A8.8 Polar and Axial Vectors: Time Even and Time Odd Vectors 394 A8.9 Vector Differentiation 395

A8.9.1 The Operator V 395 A8.9.2 The gradient 396 A8.9.3 The divergence 397 A8.9.4 The curl, V x V 397 A8.9.5 The divergence and the curl illustrated and compared 398 A8.9.6 Composite funetions involving V 398 A8.9.7 Successive applications of V 400 A8.9.8 Time derivative of a vector 401 A8.9.9 Caveat 401

A8.10 Change of Basis Vectors and Effect Upon Coordinates of a Fixed Vector 401

A8.ll The Effect of a Symmetry Operation on Vectors and Basis Vectors 404

Contents xv

A9 Vectors, II 407 A9.1 Introduction 407 A9.2 Cylindrical Coordinates and Basis Vectors 407 A9.3 Polar Coordinates and Polar Basis Vectors 408 A9.4 Spherical Components and Spherical Basis Vectors and Direction

Cosines 409 A9.5 Rotation of Vectors using Spherical Coordinates 413 A9.6 Vectors in «-Dimensional Space 415

References 416

A10 Tensors 417 A10.1 General Definitions 417 AI 0.2 Representation or Specification of a Tensor 419 AI0.3 Transformation of Tensors upon Rotation of Axes 422 A10.4 Some Properties of Tensors 425

AI 0.4.1 General 425 A10.4.2 Tensors of rank two 425 A10.4.3 Tensors of rank three: the alternating or

Levi-Civitä tensor 426 A10.4.4 Isotropie tensors of various ranks 427 A10.4.5 Tensor contraction 427

A10.5 Irreducible Tensorial Sets 428

A l l Electrostatics 433 AI 1.1 Introduction 433 AI 1.2 Force Between Charges 433 AI 1.3 Electric Field Strength 435 AI 1.4 Electrostatic Potential 436 AI 1.5 Gauss's Law 437 AI 1.6 The Equations of Poisson and Laplace 438

A12 Magnetostatics 439 AI 2.1 Introduction 439 A12.2 Magnetic Forces 441 A12.3 The Magnetic Induction 442 A12.4 The Lorentz Force on a Point Charge Moving in a

Magnetic Field 445 A12.5 The Divergence of the Magnetic Induction B 446 A12.6 The Vector Potential A 446

A13 The Interaction of a System of Electric Charges with Electric and Magnetic Fields 449 A13.1 Introduction 449 A13.2 Point Charges in Molecular Systems 450

XVI Contents

AI3.3 Electric Dipole in a Molecular System 450 A13.4 Basic Treatment of the Energy of a Distribution of Point Charges

at Rest in a Uniform Electric Field 451 AI3.5 Basic Treatment of Permanent and Induced Molecular Dipoles

in a Uniform Static Electric Field 454 AI3.6 Basic Treatment of Macroscopic Polarization and Electric

Susceptibilities 457 A13.7 Basic Treatment of the Electric Displacement for a Uniform

Static Electric Field 460 AI3.8 The Implications of Using Dynamic Electric Fields 461 AI3.9 More General Treatment of Energy of Interaction of Point Charges

at Rest in a Static Electric Field 462 AI3.10 Interaction of Charges in Motion with a Static Magnetic Field 466

Reference 470

A14 The Polarizability Tensor 471 A14.1 Introduction 471 A14.2 The Polarizability Tensor in the Cartesian Basis 472

A14.2.1 General considerations 472 A14.2.2 Reduction of the tensor 473 A14.2.3 The polarizability ellipsoid 475 A14.2.4 Transformation of ca under rotation of axes 477

A14.3 The Polarizability Tensor in the Spherical Basis 477 AI4.3.1 General definitions 477 A14.3.2 Reduction of the tensor sa 479 A14.3.3 Transformation of s a under rotation of axes 481

A14.4 The Relation Between the ax-ß and the a ^ 481 A14.5 Irreducible Polarizability Tensors and their Components 482

A14.6 Transformation Properties of the am under Rotations 486 AI4.7 Isotropie Averages and Tensor Invariants 486

A14.7.1 General considerations 486 A14.7.2 Isotropie averages and rotational invariants # ^ in terms

of the atf 487 A14.7.3 Isotropie averages and rotational invariants # ( j ) in terms

of the dpa 488 A14.7.4 Isotropie averages and rotational invariants # ( v ) in terms

of the ax-v 490 A14.7.5 Isotropie averages and the rotational invariants #U\

a, 8 and y for the cartesian basis 490 A14.7.6 Isotropie averages and rotational invariants, &U\ a,

8 and y for the spherical basis 494 References 495

Contents xvu

A15 The Optical Activity Tensors, G, # , A and st 497 AI 5.1 Introduction 497 A15.2 Isotropie Averages of the Type (aC) 497 AI5.3 Isotropie Averages of the Type (aA) 499 AI5.4 Other Tensor Invariants 500

Reference 501

A16 Maxwell Equations in Vacuum and in Media 503 AI6.1 The Fundamental Equations 503 AI6.2 Case I: the Maxwell Equations in Vacuum 507 A16.3 Case II: the Maxwell Equations in a Linear Medium

with p = 0 and o = 0 508 AI6.4 Case III: the Maxwell Equations in a Non-Linear Medium

with p = 0 and o = 0 509 AI6.5 Case IV: the Maxwell Equations in a Linear Medium

with p # 0 and CT # 0 511

A17 Monochromatic Plane Harmonie Waves in Vacuum and in a Non-absorbing Linear Medium 513 AI7.1 General Wave Equation in Vacuum 513 AI7.2 Monochromatic Plane Harmonie Electromagnetic Wave in

Vacuum 514 A17.2.1 Solution of the wave equation for the vector E 514 A17.2.2 Solution of the wave equation for the vector B 518 A17.2.3 Energy considerations for a plane harmonic

electromagnetic wave in vacuum 522 A17.3 The Exponential Representation of Harmonic Waves 526 AI7.4 Monochromatic Plane Harmonic Wave in a Homogeneous,

Isotropie and Linear Medium 530

A18 The Transition Polarizability Tensor (a)/,- 533 AI 8.1 Introduction 533 AI8.2 The Restrictions r ^ i, f 533 AI8.3 The Relative Signs of \rr 535

References 536

A19 Clebsch-Gordan Coefficients and Wigner 3-y and 6-j Symbols 537 A19.1 Introduction 537 A19.2 Clebsch-Gordan Coefficients 538 A19.3 Wigner 3- ; Symbols 544 A19.4 Wigner 6-j Symbols 546

Reference 550 Reference Table A19.1 551

XV111 Contents

A20 Sources of Electromagnetic Radiation 555 A20.1 Introduction 555 A20.2 The Oscillating Electric Dipole as a Source 555 A20.3 The Oscillating Magnetic Dipole as a Source 561 A20.4 The Oscillating Electric Quadrupole as a Source 562 A20.5 Scattering from Chiral Molecules 564

A21 Polarization of Electromagnetic Radiation 565 A21.1 Introduction 565 A21.2 States of Polarization: Monochromatic Radiation 565

A21.2.1 Linear polarization 565 A21.2.2 Elliptical and circular polarization 566 A21.2.3 Stokes parameters 570 A21.2.4 Stokes parameters for scattered radiation 572

A21.3 States of Polarization: Quasi-Monochromatic Radiation 573 A21.4 Change of Polarization: Depolarization Ratios, Reversal

Coefficients and Degrees of Circularity 575

Further Reading 579

Index 585