Advanced Quantum Chemistry and Group Theory

55 lessons, pages 3601–3655.

  1. Symmetry as a Quantum-Chemistry Tool — Why molecular symmetry simplifies orbitals, spectra and calculations
  2. Symmetry Operations and Elements — Identity, rotations, reflections, inversion and improper rotations
  3. Combining Symmetry Operations — Operation products, closure and the group concept
  4. Assigning Molecular Point Groups — A systematic decision tree for common molecular geometries
  5. Linear and High-Symmetry Point Groups — C∞v, D∞h, Td, Oh and Ih classification
  6. The C2v Point Group in Water — Water's four operations and their geometric action
  7. The C3v Point Group in Ammonia — Threefold rotations and vertical mirror planes
  8. The D3h Point Group in BF3 — Principal axis, perpendicular axes and horizontal reflection
  9. Conjugacy Classes in Point Groups — Equivalent operations and class structure
  10. Representations and Matrices — How symmetry operations transform coordinates and basis functions
  11. Characters and Reducible Representations — Matrix traces as compact symmetry descriptions
  12. Irreducible Representations — Symmetry species and decomposition of a representation
  13. Reading Character Tables — Classes, symmetry species, coordinates and quadratic functions
  14. Reducing a Representation — Applying the character reduction formula to C2v
  15. The Great Orthogonality Principle — Why irreducible characters obey orthogonality relations
  16. Direct Products of Symmetry Species — Testing whether an integral or transition can be nonzero
  17. Symmetry-Adapted Linear Combinations — Constructing ligand orbital combinations with matching symmetry
  18. Projection Operators — Generating SALCs systematically from a trial function
  19. Water Molecular Orbitals by Symmetry — Matching H 1s SALCs with O atomic orbitals in C2v
  20. Ammonia Molecular Orbitals by Symmetry — A1 and E ligand combinations in C3v
  21. Sigma and Pi Orbitals in Diatomics — Orbital symmetry around the internuclear axis
  22. Inversion Parity in Homonuclear Diatomics — Gerade and ungerade labels and their consequences
  23. Term Symbols: Orbital and Spin Angular Momentum — L, S and J labels for atomic electronic states
  24. Diatomic Molecular Term Symbols — Lambda, spin multiplicity and reflection labels
  25. Born–Oppenheimer Approximation — Separating nuclear motion from electronic calculations
  26. Symmetry of Normal Vibrations — Building the 3N displacement representation
  27. Removing Translations and Rotations — Obtaining vibrational symmetry species from 3N motions
  28. Infrared Activity from Character Tables — Dipole derivatives and x, y, z symmetry
  29. Raman Activity from Character Tables — Polarizability derivatives and quadratic functions
  30. Mutual Exclusion in Centrosymmetric Molecules — IR and Raman parity selection rules
  31. Electronic Transition Selection Rules — Transition dipoles, direct products and spin restrictions
  32. Jahn–Teller Distortion — Why some electronically degenerate structures lower symmetry
  33. Hückel Theory: The Pi-Electron Model — Coulomb and resonance parameters for conjugated systems
  34. Hückel Secular Determinants — Setting up the coefficient equations and energy polynomial
  35. Ethene in Hückel Theory — Two pi orbitals, bonding and antibonding solutions
  36. Allyl Pi Molecular Orbitals — Three-centre eigenvalues, coefficients and a nonbonding level
  37. Butadiene Pi Molecular Orbitals — Four-centre energy ordering and occupied orbitals
  38. Benzene Pi Molecular Orbitals — Six-membered ring eigenvalues and degeneracy
  39. Hückel Aromaticity and the 4n + 2 Rule — Closed-shell cyclic pi systems in a simple orbital model
  40. Pi Bond Orders and Electron Densities — Extracting local chemical information from Hückel coefficients
  41. HOMO–LUMO Gaps in Conjugated Chains — How chain length changes electronic excitation energy
  42. Limits of the Hückel Approximation — Neglected overlap, electron repulsion and geometry effects
  43. Many-Electron Wavefunctions and Antisymmetry — Pauli principle and exchange of identical electrons
  44. Slater Determinants — Constructing antisymmetric wavefunctions from spin orbitals
  45. Hartree Product versus Hartree–Fock — Mean-field orbitals and the role of exchange
  46. The Fock Operator — Core Hamiltonian, Coulomb and exchange contributions
  47. Self-Consistent Field Iteration — Guessing orbitals, rebuilding the Fock operator and convergence
  48. Roothaan–Hall Equations — Matrix Hartree–Fock in a finite atomic-orbital basis
  49. Restricted and Unrestricted Hartree–Fock — Closed-shell pairing and open-shell spin treatment
  50. Orbital Energies and Koopmans' Theorem — Approximate ionisation energies and orbital relaxation limits
  51. Electron Correlation Beyond Hartree–Fock — What the mean-field approximation leaves out
  52. Configuration Interaction: The Basic Idea — Mixing determinants to represent correlated electronic states
  53. Symmetry in Computational Chemistry — Reducing matrix work and labelling calculated states
  54. Advanced Quantum Chemistry: Problem Workshop — Point groups, SALCs, Hückel levels and SCF reasoning
  55. Advanced Quantum Chemistry and Group Theory: Unit Review — Connecting symmetry, spectroscopy, pi models and Hartree–Fock