German Research School for Simulation Sciences and RWTH Aachen University —
RWTHonline

tutorial: Thu 16:30-17:15, lecture: Thu 17:30-19:00 via Zoom (for details see RWTH Moodle)

tutorial: Thu 16:30-17:15, lecture: Thu 17:30-19:00 via Zoom (for details see RWTH Moodle)

**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**many-electron atom**

hydrogen atom

numerical integration, Numerov method, logarithmic mesh

many-electron problem

anti-symmetric wave-wave functions

slides and exercises**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**exchange-correlation energy**

non-interacting reference system

coupling constant integration

exchange-correlation energy, moments, sum-rule

local-density approximation

slides and exercises**Density-functional theory**

Hohenberg-Kohn theorem

variational calculus

examples

slides and exercises**Kohn-Sham approach**

variational equation for density functional

potential that gives non-interacting system with same density

self-consistency

slides and exercises**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**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**angular momenta**

angular momentum, eigenstates

adding angular momenta, Clebsch-Gordan coefficients

slides and exercises**Gaunt coefficients**

selection rules

recursion relations

calculating Gaunt matrix

slides and exercises**multiplets**

Coulomb interaction

symmetries, LS coupling

enumerating configurations and listing possible multiplets

slides**Hund's rules**

parallel spin and Pauli principle

Hund's rule multiplets

product state that is in the Hund's rule multiplet

multiplet calculator

slides**hydrogen atom: relativistic corrections**

spin, kinetic energy, spin-orbit coupling, Darwin term

slides and exercises**numerical solution of the Schrödinger equation**

1-dimensional potential, Numerov method

radial function, logarithmic mesh

slides and exercises**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**atomic multiplets**

atomic multiplets, LS-coupling

spin-orbit coupling, jj-coupling

Hund's rules

slides and exercises**atomic multiplets**

finding what multiplets appear in an open shell

construction of multiplet states

Coulomb exchange

calculation of multiplet energies

slides and exercises**Exchange interaction**

Hund's exchange

slides and exercises**Exchange mechanisms**

kinetic exchange

second quantization

slides and exercises**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`.**second quantization**

slides and exercises**crystal-field splitting**

crystal lattices, perovskites

crystal-field and symmetry

slides

Forschungszentrum Jülich, 2012

Erik Koch