Below are examples of projects our group can offer as semester or Master thesis projects. This list is not exhaustive and other topics related to our research are equally possible. Several of the projects could also be adapted to the Bachelor level.

If you are interested in doing a project with us, **email the project proposer**, mentioning your grades in all relevant classes. Please be aware of the **general guidelines**. We will in general not accept or decline any project until six weeks before the semester begins.

Overview of projects:

- Accelerating structure optimisations using perturbative post-processing
- A mathematical look at modelling superconductor candidate materials with Hubbard corrections
- Performance optimisation of GPU routines in DFTK
- Tailored initial guesses for DFT using equivariant neural networks
- Mixed-precision approaches in density-functional theory
- Curation of a benchmark set for mathematical research

High-throughput studies, where thousands to tens of thousands of materials are simulated, are a powerful tool for broadening our knowledge of materials properties and discovering new and interesting functional materials. A key step in many of these studies is structure optimisation, in which an approximate arrangement of atoms in a crystal is optimized to the most stable configuration.

Since each iterative step of the associated with this optimisation is roughly as expensive as a single-point calculation of ground state energy, atomic structure optimisation accounts for a substantial amount of computational time in high-throughput workflows. Additionally, the obtained minimal-energy geometry can be highly dependent on the chosen numerical parameters for the calculation, such as the basis set cutoff. Therefore, a good compromise between the error of a too small cutoff and a too slow (but accurate) structure optimisation needs to be found.

Along this direction, mathematical research has provided a number of new tools in the past years to (a) estimate the numerical error due to basis set discretisations and (b) correct for this error using post-processing techniques. For the force as the key quantity of interest in structure optimisations, a promising perturbative approach has emerged recently^{[1]}. A preliminary implementation of this force-refinement strategy is already available in the density-functional toolkit (DFTK). DFTK is a software our group develops in collaboration with researchers all across the world and which enables joint research between mathematicians and scientists on first-principle materials simulations.

The goal of this project will be to integrate DFTK with AiiDA, a software developed at the THEOS group here at EPFL. AiiDA is a software framework written in Python which simplifies and automates workflows for high-throughput studies. By integrating DFTK with AiiDA, we want to both test the force refinement approach on a broader range of systems and unlock this cheaper route to structure optimisations for broader use.

**Requirements:** Strong programming skills in particular python; knowledge of Julia programming is a bonus, but can also be acquired as we go along; interest in learning about the numerical and mathematical underpinnings of first-principle based materials simulations.

**Contact:** Michael F. Herbst or Austin Zadoks

[1] | E. Cancès, G. Dusson, G. Kemlin and A. Levitt. SIAM J. Sci. Comp. 44 (2022). ArXiv 2111.01470 |

The study of transition-metal compounds using density-functional theory (DFT) is an established approach and has in the past been involved with the discovery of novel cathodes for Li batteries, thermoelectric devises or photovoltaic materials. Recently the literature discussion surrounding LK-99 as a promising candidate for high-temperature superconductivity is yet another example where DFT calculations are playing an important role to understand the effects of the copper (transition metal) doping^{[2]} in the lead phosphate apatite matrix. Unfortunately employing plain semi-local DFT is not sufficient to capture the physics of many transition-metal compounds due to the strongly localised and partially filled $d$-orbitals being inappropriately described. As a remedy the so-called Hubbard corrections^{[3]} have been proposed and excessively studied over the past years^{[4]}. Practically all theoretical studies of LK-99 employ Hubbard corrections to make their predictions (see e.g. the list on wikipedia).

Within DFTK Hubbard corrections are so far missing. Moreover many fundamental mathematical and numerical aspects surrounding Hubbard corrections are so far understudied. As part of this project you will (a) implement Hubbard corrections into DFTK, (b) use them to simulate the bands of LK-99, and (c) investigate numerical techniques to improve the SCF convergence of DFT calculations with Hubbard corrections (termed also DFT+U calculations). The project will be conducted in collaboration with the THEOS group here at EPFL.

**Requirements:** Strong programming skills; basic knowledge of quantum mechanics in solid-state physics; knowledge of Julia programming is a bonus, but can also be acquired as we go along; interest in learning about the numerical and mathematical underpinnings of first-principle based materials simulations.

**Contact:** Michael F. Herbst, Iurii Timrov

[2] | S. M. Griffin. Origin of correlated isolated flat bands in copper-substituted lead phosphate apatite. ArXiv 2307.16892 |

[3] | V. I. Anisimov, J. Zaanen and O. K. Andersen. Phys. Rev. B 44, 943 (1991). DOI 10.1103/physrevb.44.943 |

[4] | I. Timrov, N. Marzari and M. Cococcion. Phys. Rev. B 98 (2018). DOI 10.1103/physrevb.98.085127 |

DFTK currently provides preliminary support for running calculations on CUDA-based graphics processing units (GPUs). Support for HIP-based GPU (e.g. the ones by AMD) is on the way. In its current stage the GPU code still requires substantial performance improvements to be competitive. The main task of this project is to employ Julia's profiling and benchmarking tools to assess and improve the GPU performance of DFTK. Along the lines you will learn how GPU programming in Julia works and how to improve the performance of a Julia code towards the HPC regime.

If time permits we might also target an extension of the overall GPU support in DFTK towards new platforms (e.g. Intel) or new routines in DFTK.

**Requirements:** Strong programming skills with experience in the implementation of algorithms for high-performance computing; knowledge of Julia programming is a bonus, but can also be acquired as we go along; interest in learning Julia's hardware abstractions for GPU computing

**Contact:** Michael F. Herbst

Most commonly the energy minimisation underlying DFT is solved using the self-consistent field procedure (SCF), which uses an iterative procedure to find the converged DFT density as a fixed-point. To start the SCF iterations an initial guess density is required, which most commonly is generated by simply adding the density of the atoms of a corresponding molecule or solid (superposition guess). This is not only very simple, but also works very well and for many practical applications SCF calculations converge in less than 20 iterations.

However, in particular whenever the electronic structure is non-trivial (e.g. magnetism, unusual spin polarisation), completely neglecting the electronic interaction between atomic densities in the initial guess represents a severe drawback. In practice often 40 or more SCF iteration can be required for such cases to obtain convergence. Especially for this setting a data-driven approach, which takes the *converged* densities from previous calculations into account to predict an initial guess for a similar chemical system is a promising approach. As has been noted in previous works^{[5]} this requires considerable care to ensure that the data-driven initial guess is truly an improvement over the simple, yet effective superposition guess. Moreover especially for the tricky cases it is crucial to ensure that natural symmetries in the problem are respected when the initial guess is predicted, to not misguide the numerical procedure.

To overcome these challenges in this project we will employ equivariant neural networks as implemented in the e3nn package^{[6]} in conjunction with DFTK. This package has in particular the capabilities to take appropriate symmetries into account both when learning the network and when predicting. Additionally we will employ strategies to monitor the uncertainties of the predicted density guesses such that we can safely target the challenging compounds, where most can be gained from a better initial guess. This project will be conducted in collaboration with the atomic architects group from MIT, which develops e3nn.

**Requirements:** Strong programming skills, ideally in Julia or Python; basic knowledge of quantum mechanics in solid-state physics; basic experience with data-driven methods or machine learning; interest in learning about the numerical and mathematical underpinnings of first-principle based materials simulations.

**Contact:** Michael F. Herbst

[5] | A. Fowler, C. Pickard, J. Elliott. J. Phys. Mater. 2, 034001 (2019) DOI 10.1088/2515-7639/ab0b4a |

[6] | M. Geiger and T. Smidt e3nn: Euclidean Neural Networks. ArXiv 2207.09453 |

In this project we want to explore the opportunities of the Julia programming language for performing mixed-precision computation is density functional theory. For this we will explore where single precision computations introduce inaccuracies and develop countermeasures using e.g. specific mixed-precision algorithms or iterative post-processing techniques.

**Requirements:** Strong background in numerical linear algebra in particular Krylov methods; considerable programming skills in python or Matlab; knowledge of Julia programming is a bonus, but can also be acquired as we go along

**Contact:** Michael F. Herbst

The aim of this project is to use AiiDA in order to curate a dataset of test systems, which can be used to automatically test and benchmark algorithms for density-functional theory. This will be used both to compare the performance of DFTK against other standard software in the field and their respective algorithms.

**Requirements:** Experience in running first-principle simulations in standard codes such as VASP, ABINIT, QuantumEspresso; experience with AiiDA or DFTK is a bonus, but can also be acquired as we go along.

**Contact:** Michael F. Herbst

Last modified: November 23, 2023. Website built with Franklin.jl and the Julia programming language.