The Schrödinger equation, which accurately models nuclei-electron system without unknown parameters, is the basis for condensed matter physics and computational chemistry. An important issue is its high computational complexity, e.g. already for a water molecule it means to solve a partial differential equation in 39 dimensions. Computational approximations are therefore needed and the goal of the course is to present, use and understand numerical methods for the important coarse-grained approximations.
The complexity is reduced by classical approximation of the nuclei, using Born-Oppenheimer dynamics. To computationally solve the quantum problem for the electrons the Hartree-Fock and Kohn-Sham density functional theory is important and leads to an ab initio molecular dynamics model. The ab initio molecular dynamics can be simplified by empirical potentials. Thermal fluctuations in an ensemble at constant temperature introduces stochastics into the dynamics which leads to the Langevin molecular dynamics, or variants thereof. On long time scales and in the high friction limit this dynamics can be described without the velocities by the Smoluchowski equation. The next step in the coarse-graining process is to derive partial differential equations for the mass, momentum and energy of a continuum fluid, which determines the otherwise unspecified stress tensor and heat flux.