Background for Computational chemistry and materials modeling

(This list is not for students, but for curriculum development)

Mathematical background see here

Quantum mechanics

• Fundamentals:
  • Observables and wave-function in different representations, density operator (including open systems)
  • Schrodinger equation and other evolution equations, Green's function and propagator
  • Spin
  • Many-body wave-function, density matrices (e.g. transition)
  • One-particle orbitals, Slater determinant
  • Second quantization – post-HF methods and empirical Hamiltonians
• Methods:
  • Variational method (including linear basis set expansion)
  • Perturbation theory (including degenerate, periodic, step-like, Fermi's golden rule)
  • Hartree-Fock method (including spin)
  • Density functional theory
  • Many-body perturbation theory
• Models:
  • Free particle (multidimensional) – conceptual model
  • Particle in a box (multidimensional) – conceptual model
  • Particle in a uniform field – atomic motions
  • Harmonic oscillator – vibrations
  • Rigid rotor (at least spherically symmetric) – molecular rotations
  • Hydrogen atom (nonrelativistic) – atoms
  • Periodic potential (Dirac comb or Kronig-Penney model or Huckel model of trans-polyacetylene) – solids
  • Two-state quantum system – conceptual model
  • Landau-Zener model – nonadiabatic transitions
  • Double-well potential – various transitions

Statistical physics

• Fundamentals:
  • Basics of equilibrium and nonequilibrium thermodynamics
  • Hamiltonian, phase space, partition function, averaging, detailed balance
  • Thermodynamic potentials
  • Phase diagram, phase transitions
  • Kinetic equations (at least Master equation, also Boltzmann equation)
• Models:
  • Two-state quantum system – conceptual model
  • Harmonic oscillator – vibrations
  • Rigid rotor – molecular rotations
  • Ideal gas (classical and quantum) – conceptual model
  • Real gas (at least van der Waals) – gases
  • Equation of state of solids (e.g. Murnaghan) – solids
  • Emission and absorption of boson waves (e.g. electromagnetic waves, Einstein coefficients)
  • Lattice models (e.g. lattice gas or spin model) – conceptual model

Materials science (broadly defined including atoms and molecules)

• Most common structural types of molecules and solids, crystallographic notations, bonding
• Electronic structure of atoms, state notations, periodic table
• Electronic structure of molecules (e.g. Molecular orbital theory), state notations, molecular orbital notations
• Electronic structure of solids, k-point notations
• Vibration of molecules and solids, symmetry notations
• Most common observables: ionization potential, electron affinity, bandgap, electronic and vibrational spectroscopy
• See also list of concepts for Survey of Materials