A mathematical look at electronic structure theory
2021-07-22, 14:00–17:00 (UTC), Green

Electronic structure theory is a fascinating interdisciplinary field. Physics, chemistry, materials science, mathematics, high-performance computing ... they're all in it. Rooted at the quantum-mechanical description of electrons it is the backbone for quite a few simulation methods in the chemical and physical sciences. Here we'll focus on the numerical tools required to solve standard problems in the field like density-functional theory, which --- as we will see --- is challenging in itself.


Content

The material for the workshop is available at https://github.com/mfherbst/juliacon_dft_workshop.

I'll briefly introduce the setting of density-functional theory (DFT), in particular show the equation system and its mathematical structure. With that we are in good shape to tackle the main part of the workshop, which will be devoted to discussing the numerical techniques used for solving it.

Our main tool in this workshop will be the density-functional toolkit (DFTK),
a state-of-the-art DFT code written in Julia (of course ;)). This code will allow us to consider a number of reduced problems, where things are more tractable if you wish to interactively explore, visualise and understand. In particular we will use DFTK to inspect what's going on while the DFT problem is being solved. With that knowledge at hand we'll try to code up some simple DFT solvers on our own. Due to the scaffolding DFTK provides this is a fairly manageable task and (as a small bonus) the resulting algorithms could be directly applied to cutting edge problems (if we're careful with performance issues).

Depending on how our progress is I plan to cover the following topics:
- Problem setup: Mathematical structure of DFT
- Typical discretisation approaches: Gaussians versus plane waves
- Typical solution algorithms: Direct minimisation versus self-consistent field (SCF) iterations
- Numerical analysis of SCF problems
- Writing our own SCF, understanding why it badly fails and what we can do about it.
- Connections between the physical properties of matter and convergence properties of an SCF

Assumed background

I'll try to do my best to make this workshop accessible for a broad range of people: Those with the chemistry or physics background that always wanted to understand the maths behind DFT as well as those with the understanding on PDEs / linear algebra that are interested in getting an idea about this challenging application domain.

I will assume you safely know your way around Julia.

By training a chemist, who got more and more twisted towards the "dark side" of numerical analysis. Now working as a PostDoc at the Applied and Computational Mathematics lab of RWTH Aachen University, Germany. Lead developer of DFTK, a Julia code for mathematical research in on density-functional theory (a quantum chemistry method).

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