Correlated Electrons

German Research School for Simulation Sciences and RWTH Aachen University — RWTHonline
lecture: Thu 16:30-18:00, tutorials: Thu 13:30-14:30 Lecture Hall, GRS Jülich (how to reach, you will need a picture ID to enter the Forschungszentrum Jülich)


  1. Introduction
    Theory of Almost Everything
    periodic table (pse tool)
    Inorganic Crystal Structure Database (ICSD)
    atomic orbitals, shell structure: many-electron atoms
    atoms in crystals, atomic radius, formal charges
    example: magnetite (Nature 481, 173 (2012))
    atomic units
    nomenclature of inorganic compounds (IUPAC guide)
    reminder: spherical harmonics
    slides and exercises (Application Form)
  2. many-electron atom
    hydrogen atom
    numerical integration, Numerov method, logarithmic mesh
    many-electron problem
    anti-symmetric wave-wave functions
    slides and exercises
  3. reduced density matrices
    variational principle
    evaluating expectation values using reduced density matrices
    N-representability problem, Coulson's challenge
    pair correlation function and exchange-correlation hole
    Hartree energy
    slides and exercises
  4. exchange-correlation energy
    non-interacting reference system
    coupling constant integration
    exchange-correlation energy, moments, sum-rule
    local-density approximation (R.O. Jones, O. Gunnarsson and D.M. Ceperley and B. Alder)
    slides and exercises
  5. Density-functional theory
    Hohenberg-Kohn theorem
    variational calculus
    slides and exercises
  6. Kohn-Sham approach
    variational equation for density functional
    potential that gives non-interacting system with same density
    slides and exercises
  7. Many-electron wavefunctions
    many-particle wave-functions
    indistinguishability and spin-statistics connection
    Slater determinants, basis of many-electron Hilbert spaces
    second quantization
    slides, lecture notes and exercises
  8. Slater-Condon parameters
    mean-field potential in second quantization
    electron-electron repulsion in second quantization
    multipole expansion
    addition theorem for spherical harmonics
    Slater-Condon parameters
    slides and exercises
  9. angular momenta
    angular momentum, eigenstates
    adding angular momenta, Clebsch-Gordan coefficients
    slides and exercises
  10. Gaunt coefficients
    selection rules
    recursion relations
    calculating Gaunt matrix
    slides and exercises
  11. multiplets
    Coulomb interaction
    symmetries, LS coupling
    enumerating configurations and listing possible multiplets
    slides and exercises
  12. Hund's rules
    parallel spin and Pauli principle
    Hund's rule multiplets
    product state that is in the Hund's rule multiplet
    multiplet calculator
  13. hydrogen atom: relativistic corrections
    spin, kinetic energy, spin-orbit coupling, Darwin term
    slides and exercises
  14. numerical solution of the Schrödinger equation
    1-dimensional potential, Numerov method
    radial function, logarithmic mesh
    slides and exercises
  15. many-electron atom
    self-consistent field method, Hartree equation
    symmetry of the Hartee potential and spherical-field approximation
    self-consistent calculation for atoms
    slides and exercises
  16. atomic multiplets
    atomic multiplets, LS-coupling
    spin-orbit coupling, jj-coupling
    Hund's rules
    slides and exercises
  17. atomic multiplets
    finding what multiplets appear in an open shell
    construction of multiplet states
    Coulomb exchange
    calculation of multiplet energies
    slides and exercises
  18. Exchange interaction
    Hund's exchange
    slides and exercises
  19. Exchange mechanisms
    kinetic exchange
    second quantization
    slides and exercises
  20. correlated electrons
    introduction to second quantization and exchange mechanisms
    slides, lecture notes, and exercises
    An example of a realistic code for self-consistent electronic structure calculations. For the self-consistent calculation of an atom, look at ld1.f90, all-electron.f90, scr.f90, and ascheq.f90.
  21. second quantization
    slides and exercises
  22. crystal-field splitting
    crystal lattices, perovskites
    crystal-field and symmetry


Eva Pavarini, Erik Koch, Frithjof Anders, and Mark Jarrell (eds.)
Correlated Electrons: From Models to Materials
Forschungszentrum Jülich, 2012

book cover
Erik Koch