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Applications of Group Theory to Atoms, Molecules, and Solids

Book Description

The majority of all knowledge concerning atoms, molecules, and solids has been derived from applications of group theory. Taking a unique, applications-oriented approach, this book gives readers the tools needed to analyze any atomic, molecular, or crystalline solid system. Using a clearly defined, eight-step program, this book helps readers to understand the power of group theory, what information can be obtained from it, and how to obtain it. The book takes in modern topics, such as graphene, carbon nanotubes and isotopic frequencies of molecules, as well as more traditional subjects: the vibrational and electronic states of molecules and solids, crystal field and ligand field theory, transition metal complexes, space groups, time reversal symmetry, and magnetic groups. With over a hundred end-of-chapter exercises, this book is invaluable for graduate students and researchers in physics, chemistry, electrical engineering and materials science.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright
  4. Contents
  5. Preface
  6. 1. Introductory example: Squarene
    1. 1.1 In-plane molecular vibrations of squarene
    2. 1.2 Reducible and irreducible representations of a group
    3. 1.3 Eigenvalues and eigenvectors
    4. 1.4 Construction of the force-constant matrix from the eigenvalues
    5. 1.5 Optical properties
    6. References
    7. Exercises
  7. 2. Molecular vibrations of isotopically substituted AB[sub(2)] molecules
    1. 2.1 Step 1: Identify the point group and its symmetry operations
    2. 2.2 Step 2: Specify the coordinate system and the basis functions
    3. 2.3 Step 3: Determine the effects of the symmetry operations on the basis functions
    4. 2.4 Step 4: Construct the matrix representations for each element of the group using the basis functions
    5. 2.5 Step 5: Determine the number and types of irreducible representations
    6. 2.6 Step 6: Analyze the information contained in the decompositions
    7. 2.7 Step 7: Generate the symmetry functions
    8. 2.8 Step 8: Diagonalize the matrix eigenvalue equation
    9. 2.9 Constructing the force-constant matrix
    10. 2.10 Green’s function theory of isotopic molecular vibrations
    11. 2.11 Results for isotopically substituted forms of H[sub(2)]O
    12. References
    13. Exercises
  8. 3. Spherical symmetry and the full rotation group
    1. 3.1 Hydrogen-like orbitals
    2. 3.2 Representations of the full rotation group
    3. 3.3 The character of a rotation
    4. 3.4 Decomposition of D[sup(l)] in a non-spherical environment
    5. 3.5 Direct-product groups and representations
    6. 3.6 General properties of direct-product groups and representations
    7. 3.7 Selection rules for matrix elements
    8. 3.8 General representations of the full rotation group
    9. References
    10. Exercises
  9. 4. Crystal-field theory
    1. 4.1 Splitting of d-orbital degeneracy by a crystal field
    2. 4.2 Multi-electron systems
    3. 4.3 Jahn–Teller effects
    4. References
    5. Exercises
  10. 5. Electron spin and angular momentum
    1. 5.1 Pauli spin matrices
    2. 5.2 Measurement of spin
    3. 5.3 Irreducible representations of half-integer angular momentum
    4. 5.4 Multi-electron spin–orbital states
    5. 5.5 The L–S-coupling scheme
    6. 5.6 Generating angular-momentum eigenstates
    7. 5.7 Spin–orbit interaction
    8. 5.8 Crystal double groups
    9. 5.9 The Zeeman effect (weak-magnetic-field case)
    10. References
    11. Exercises
  11. 6. Molecular electronic structure: The LCAO model
    1. 6.1 N-electron systems
    2. 6.2 Empirical LCAO models
    3. 6.3 Parameterized LCAO models
    4. 6.4 An example: The electronic structure of squarene
    5. 6.5 The electronic structure of H[sub(2)]O
    6. References
    7. Exercises
  12. 7. Electronic states of diatomic molecules
    1. 7.1 Bonding and antibonding states: Symmetry functions
    2. 7.2 The “building-up” of molecular orbitals for diatomic molecules
    3. 7.3 Heteronuclear diatomic molecules
    4. Exercises
  13. 8. Transition-metal complexes
    1. 8.1 An octahedral complex
    2. 8.2 A tetrahedral complex
    3. References
    4. Exercises
  14. 9. Space groups and crystalline solids
    1. 9.1 Definitions
    2. 9.2 Space groups
    3. 9.3 The reciprocal lattice
    4. 9.4 Brillouin zones
    5. 9.5 Bloch waves and symmorphic groups
    6. 9.6 Point-group symmetry of Bloch waves
    7. 9.7 The space group of the k-vector, g[sup(s)][sub(k)]
    8. 9.8 Irreducible representations of g[sup(s)][sub(k)]
    9. 9.9 Compatibility of the irreducible representations of g[sub(k)]
    10. 9.10 Energy bands in the plane-wave approximation
    11. References
    12. Exercises
  15. 10. Application of space-group theory: Energy bands for the perovskite structure
    1. 10.1 The structure of the ABO[sub(3)] perovskites
    2. 10.2 Tight-binding wavefunctions
    3. 10.3 The group of the wavevector, g[sub(k)]
    4. 10.4 Irreducible representations for the perovskite energy bands
    5. 10.5 LCAO energies for arbitrary k
    6. 10.6 Characteristics of the perovskite bands
    7. References
    8. Exercises
  16. 11. Applications of space-group theory: Lattice vibrations
    1. 11.1 Eigenvalue equations for lattice vibrations
    2. 11.2 Acoustic-phonon branches
    3. 11.3 Optical branches: Two atoms per unit cell
    4. 11.4 Lattice vibrations for the perovskite structure
    5. 11.5 Localized vibrations
    6. References
    7. Exercises
  17. 12. Time reversal and magnetic groups
    1. 12.1 Time reversal in quantum mechanics
    2. 12.2 The effect of T on an electron wavefunction
    3. 12.3 Time reversal with an external field
    4. 12.4 Time-reversal degeneracy and energy bands
    5. 12.5 Magnetic crystal groups
    6. 12.6 Co-representations for groups with time-reversal operators
    7. 12.7 Degeneracies due to time-reversal symmetry
    8. References
    9. Exercises
  18. 13. Graphene
    1. 13.1 Graphene structure and energy bands
    2. 13.2 The analogy with the Dirac relativistic theory for massless particles
    3. 13.3 Graphene lattice vibrations
    4. References
    5. Exercises
  19. 14. Carbon nanotubes
    1. 14.1 A description of carbon nanotubes
    2. 14.2 Group theory of nanotubes
    3. 14.3 One-dimensional nanotube energy bands
    4. 14.4 Metallic and semiconducting nanotubes
    5. 14.5 The nanotube density of states
    6. 14.6 Curvature and energy gaps
    7. References
    8. Exercises
  20. Appendix A: Vectors and matrices
  21. Appendix B: Basics of point-group theory
  22. Appendix C: Character tables for point groups
  23. Appendix D: Tensors, vectors, and equivalent electrons
  24. Appendix E: The octahedral group, O and O[sub(h)]
  25. Appendix F: The tetrahedral group, T[sub(d)]
  26. Appendix G: Identifying point groups
  27. Index