Preface xvii
Acknowledgments xxiii
About the Companion Website xxv
1 Vectors and Coordinate Systems 1
1.1 Introduction 1
1.2 Vector Spaces and Linear Independence 5
1.3 Vector Calculus and Fields 6
1.4 Line and Surface Integrals (Conservative Fields) 14
1.5 Orthogonal Curvilinear Coordinates 21
1.6 Spherical Polar Coordinates 25
1.7 Cylindrical Coordinates 26
1.8 Summary 27
Further Reading 29
Exercises 29
2 Linear Vector Spaces 33
2.1 Introduction 33
2.2 What Is a Linear Vector Space? 33
2.3 Dirac Notation 47
2.4 Hilbert Spaces 48
2.5 Some Useful Inequalities 51
2.6 Summary 55
Further Reading 56
Exercises 56
3 Linear Transformations and Operators 59
3.1 Introduction 59
3.2 Linear Transformations 60
3.3 Applications to Quantum Chemistry 62
3.4 Matrices as Linear Operators 65
3.5 Outer Products 68
3.6 Geometrical Operations 69
3.7 Affine Transformations 76
3.8 Affine Transformations as the Building Blocks of Learning: Neural Networks 79
3.9 Summary 89
3.10 Key Connections 89
Further Reading 90
Exercises 90
4 Matrix Characteristics and Quantum-mechanical Operators 95
4.1 Introduction 95
4.2 Some Definitions 96
4.3 Special Types of Matrices and Their Physical Roles 99
4.4 The Core Idea: Finding the âAxesâ of a Transformation 106
4.5 Properties of Hermitian Matrices 107
4.6 Eigenvectors and Eigenvalues of Unitary Matrices 109
4.7 Commuting Operators and Simultaneous Measurement 110
4.8 Orthogonality of Eigenvectors of Normal Matrices 112
4.9 Transition to Eigenvalue Problems 113
4.10 Summary 114
Further Reading 114
Exercises 114
5 Eigenvalue Problems, Orthogonality, and Completeness 119
5.1 Introduction 119
5.2 Determination of Eigenvalues and Eigenvectors of a Matrix 120
5.3 Algebraic Versus Geometric Multiplicity: The Key to Diagonalization 125
5.4 Degenerate Eigenvalues 127
5.5 Gramâ"Schmidt Orthogonalization Process 129v 5.6 Completeness 136
5.7 Summary 139
Further Reading 140
Exercises 140
6 Diagonalization of Matrices 143
6.1 Introduction 143
6.2 Change of Basis and Similarity Transformation 144
6.3 Matrix Diagonalization 148
6.4 Quadratic and Hermitian Forms 152
6.5 Quadratic Surfaces 156
6.6 Simultaneous Diagonalization 161
6.7 Summary 170
Further Reading 170
Exercises 170
7 Group Theory and Symmetry 175
7.1 Motivation and Overview 175
7.2 Introduction 177
7.3 Symmetry Operations and Point Groups 177
7.4 Mathematical Foundations: What Is a Group? 188
7.5 Group Multiplication Table 190
7.6 Classification of Molecules into Point Groups 193
7.7 Matrix Representation of Symmetry Operations 200
7.8 Matrix Representation of a Group 203
7.9 Order, Similarity Transformations, and Matrix Characters 205
7.10 Reducible and Irreducible Representations 209
7.11 The Great Orthogonality Theorem (GOT) 210
7.12 Character Tables 211
7.13 Applications of Group Theory in Chemistry 213
7.14 Symmetry in Periodic Structures: A Minimal Mathematical Note 214
7.15 Summary 219
Further Reading 219
Exercises 219
8 Applications of Group Theory 223
8.1 Introduction and Motivation 223
8.2 Vibrational Analysis: General Principles 223
8.3 Vibrational Analysis: The Water Molecule (;;2;;) 225
8.4 Infrared and Raman Activity 232
8.5 MOs and Symmetry 239
8.6 The Ï-systems 249
8.7 Additional Conceptual Applications 255
8.8 Summary 258
8.9 Practice Problems 258
Further Reading 258
Exercises 259
9 Limits, Series, and Power-series Expansions 263
9.1 Limit of a Function 263
9.2 Taylor and Maclaurin Series 266
9.3 Binomial Expansion 268
9.4 Tests of Series Convergence 269
9.5 Fourier Series 274
9.6 Summary 290
Further Reading 290
Exercises 291
10 Fourier Transforms 295
10.1 Introduction 295
10.1.1 From Discrete to Continuous Spectrum 296
10.2 The Dirac Delta Function 303
10.3 Time and Frequency Domains 305
10.4 Properties of the FT 310
10.5 Fourier Sine and Cosine Transforms 311
10.6 Applications of Fourier Transforms 313
10.7 Summary 330
Further Reading 331
Exercises 331
11 Laplace Transforms 335
11.1 Introduction 335
11.2 Laplace Transform 336
11.3 Inverse Laplace Transforms 343
11.4 Some Properties of the Laplace Transform 345
11.5 Solution of Initial-value Problems 349
11.6 Applications in Electrochemistry and Chemical Kinetics 357
11.7 Summary 362
Further Reading 362
Exercises 363
12 Linear Differential Equations 367
12.1 Introduction 367
12.2 Linear Independence and Wronskian 368
12.3 Power-Series Method 371
12.4 Classification of Points 378
12.5 Case Study: Legendreâs Equation 381
12.6 Connections to Chemistry and Physics 394
12.7 Summary 395
Further Reading 396
Exercises 396
13 Special Functions 401
13.1 Introduction 401
13.2 Properties of Legendre Polynomials 402
13.3 Associated Legendre Polynomials 411
13.4 Other Orthogonal Polynomials 416
13.5 Summary 431
Further Reading 434
Exercises 434
14 Partial Differential Equations 437
14.1 Introduction 437
14.2 The âBig Threeâ Partial Differential Equations 437
14.3 The Stretched String: Wave Equation 438
14.4 The Unified View: Separation of Variables for the âBig Threeâ 446
14.5 Molecular Vibrations: From Strings to Spectra 447
14.6 Fourier Superposition and Physical Interpretation 448
14.7 Fundamentals and Overtones 448
14.8 Three-dimensional Wave Equation and Helmholtz Form 448
14.9 From Classical Waves to Quantum Operators 449
14.10 In-depth Applications and Examples: The Heat and Laplace Equations 451
14.11 Summary and Conclusions 459
Further Reading 460
Exercises 460
Appendix A: Character Tables of Common Molecular Point Groups 465
Appendix B: Fourier Transform Pairs and Properties (Unitary Convention) 471
Appendix C: Laplace Transform Table (Compatible With Book Convention) 475
Appendix D: Special Functions: Definitions, Orthogonality, and Recurrence Relations 479
Appendix E: Differential Operators in Cartesian, Cylindrical, and Spherical Coordinates 483
Appendix F: Vector Calculus Identities 487
Appendix G: Linear Algebra Identities and Matrix Decompositions 491
Appendix H: Numerical Methods Reference Sheet 497
Appendix I: Physical Constants and Conversions 501
Appendix J: Extended Proofs and Derivations 505
Appendix K: Notations and Conventions 525
Index 527
