In this thesis, I explore the possibility of constructing machine-learning models of the interacting
density-density response function (DDRF) and quantities derived from it. Accurate models of
the DDRF are a crucial ingredient to enabling GW quasiparticle calculations of more complex
systems. Model DDRFs bypass the expensive calculation and inversion of the dielectric matrix,
which is the origin of the poor scaling of the GW method with the number of atoms.
The thesis is organized as follows:
• Chapter 2 systematically reviews common descriptors used for machine-learning physical
quantities. The key ideas behind the construction of such descriptors are discussed. First,
I introduce several descriptors that systematically incorporate symmetry transformations
that leave the target quantity invariant. These descriptors can be used for learning
quantities such as the ground-state energy, atomization energies and scalar polarizabilities.
Next, I discuss several descriptors and models that are equivariant under transformations
of the molecular structure. These descriptors are ideal for learning quantities which
transform in a defined way under the action of a transformation, such as vectors, tensors
and functions, including the DDRF.
• In Chapter 3, I introduce the key electronic structure methods employed throughout the
thesis. I start by introducing density functional theory, followed by a detailed introduction
to the GW method and the DDRF.
• In Chapter 4, I develop a machine-learning model of an invariant quantity derived from
the random phase approximation (RPA) DDRF: the scalar polarizability. In this chapter, I
calculate the DDRF of 110 hydrogenated silicon clusters. The results of these calculations
are then used to train a model of the scalar polarizability based on the SOAP descriptor [16].
The resulting model is then used to predict the scalar polarizability of clusters with up to
3000 silicon atoms while converging to the correct silicon scalar polarizability bulk limit.
The findings of this chapter indicate that the scalar polarizability - even though derived
from the non-local DDRF - can be accurately predicted from structural descriptors that
only encode the local environment of each atom. These results indicate that the response
of a non-metallic system to an external potential described by the DDRF may also be
approximated as a sum of localized atomic contributions, which forms the motivation for
the following two chapters.
• In Chapter 5, I develop an approximation to the DDRF of the silicon clusters based on a
projection onto atom-centred auxiliary density-fitting basis sets. The results of this chapter
indicate that the plane-wave DDRF can be efficiently represented by a small localized basis,
thus significantly reducing the size of the DDRF. At the end of this section, I develop a
simple neural-network model of the DDRF in this localized basis, highlighting the necessity
for using an equivariant descriptor and motivating the next chapter’s developments.
• In Chapter 6, I develop a new approximation to the DDRF, which allows a decomposition
into atomic contributions. I further introduce the neighbourhood density matrix (NDM),
a non-local extension of the SOAP descriptor, which transforms under rotations in
the same way as the atomic contributions to the DDRF. The developed method is then
applied to the silicon clusters from the previous chapters. Using the NDM, I develop a
neural-network model capable of accurately predicting the atomic contributions to the
DDRF. These atomic contributions are transformed into a plane-wave basis and summed
to obtain the DDRF of a silicon cluster. The predicted DDRFs are then used in GW
calculations, which show that the model DDRFs accurately reproduce the quasiparticle
energy corrections from GW calculations, as obtained within the atomic decomposition
of the DDRF. This methodology can be used to construct arbitrarily complex model
DDRFs based on purely structural properties of clusters and nanoparticles, paving the
way towards GW calculations of complex systems, such as disordered materials, liquids,
interfaces and nanoparticles.