Machine-learning the electronic density of states: electronic properties without quantum mechanics

The electronic density of states (DOS) quantifies the distribution of the energy levels that can be occupied by electrons in a quasiparticle picture and is central to modern electronic structure theory. It also underpins the computation and interpretation of experimentally observable material properties such as optical absorption, electrical conductivity, and heat capacity. It can be accurately computed through expensive first-principle calculations, limiting the size of the problems that can be simulated easily to a few thousand atoms. Machine-learning (ML) techniques are a promising alternative to these calculations, as they were successfully applied to study many atomic-scale problems by generalising information from small configurations to large and complex structures. However, most efforts focused on learning the ground-state Born-Oppenheimer energies and the atomic forces, which are scalar quantities, unlike the DOS, which is a multivariate function of the energy. In this thesis, we discuss the inherent challenges in constructing an ML framework that predicts the DOS as a combination of local contributions that depend, in turn, on the geometric configuration of neighbours around each atom. We compare different approaches to represent the DOS as a learning target and the accuracy of predicting quantities such as the Fermi level, the electron density at the Fermi level, or the band energy, either directly or as a side product of the evaluation of the DOS. As a first benchmark, we evaluate our model on a challenging case study that includes configurations of silicon spanning a broad set of thermodynamic conditions, ranging from bulk structures to clusters and from semiconducting to metallic behaviour. Then, we leverage the atom-centredness of the model to compute the DOS of large amorphous silicon samples, for which it would be prohibitively expensive to compute the DOS by direct electronic structure calculations. Besides the size transferability, we show that this decomposition of the DOS can extract physical insights into the connections between structural and electronic features to describe their transitions in disordered phases.Finally, we explore two approaches to using the DOS in integrated ML frameworks to model the properties of materials, where the DOS is used to incorporate the effect of thermal excitations of electrons. We propose to combine simulations from well-established ML interatomic potentials with band energy calculations extracted from DOS predictions on the already-produced trajectories. This procedure successfully describes the heat capacity of molten nickel and is in agreement with the experiments. However, we show that this method is only valid when the dynamics of the ions are, to a large extent, not affected by the electronic excitations, and it would fail in conditions with higher temperatures, such as those found in astrophysical settings. Therefore, we introduce an integrated ML framework that includes these thermal effects in constructing the interatomic potential. The novelty of this method is that the electronic temperature is an external parameter of the simulation because one only needs access to ground-state energies, forces and DOS. We successfully apply our model to study metallic hydrogen in the conditions of a young Jupiter core. We reconstruct its equation of state and its heat capacity and find that they are compatible with their first-principle-derived counterparts.