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, please make yourself familiar with **general guidelines** we follow during the project. 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
- Manifold optimisation methods in density-functional theory
- A mathematical look at modelling superconductor candidate materials with Hubbard corrections
- Reduced precision and GPU performance of DFTK
- 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.

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

The problem underlying density-functional theory (DFT) is a minimisation of an energy functional with respect to the density matrix. In turn the space of density matrices themselves has the structure of a smooth manifold. However, instead of solving this minimisation problem directly (termed *direct minimisation*, DM), most codes solve DFT by satisfying the first-order stationarity conditions, which leads to the self-consistent field (SCF) equations. For some settings the convergence of direct minimisation can be superior^{[2]}, which has been stimulating for the recent mathematical studies of DM approaches to DFT^{[3]}. In this project we want to build upon the readily available software stacks for optimisations on manifolds^{[4]} and explore the use of their readily available manifold optimisation routines in the context of DFT approaches.

[2] | E. Cancès, G. Kemlin, A. Levitt SIAM J. Mat. Anal. Appl. 42, 243 (2021). DOI 10.1137/20m1332864 |

[3] | X. Dai, S. de Gironcoli, B. Yang, A. Zhou. Multiscale Model. Simul. 21, 777 (2023). DOI 10.1137/22M1472103 |

[4] | https://www.manopt.org/ and https://manoptjl.org/ |

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 is yet another example where DFT calculations are playing an important role to understand the effects of the copper (transition metal) doping^{[5]} 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^{[6]} have been proposed and excessively studied over the past years^{[7]}.

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 implement Hubbard corrections into DFTK and use them to study open problems with respect to standard task in electronic structure theory, such as the SCF convergence or the geometry optimisation related to DFT+U methods.

**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.

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

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

[7] | 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 line we will further explore the opportunities of the Julia programming language to enable reduced precision (e.g. single precision) DFT computations, which are ideally suited to run on GPUs.

**Requirements:** Strong programming skills with experience in the implementation of algorithms for high-performance computing; knowledge of numerical linear algebra (e.g. Krylov methods); 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

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.

Last modified: February 29, 2024. Website built with Franklin.jl and the Julia programming language.