Potential energy surfaces (PESs) or landscapes provide a conceptual framework for structure prediction, and a detailed understanding of their topological features is necessary to develop efficient methods for their exploration. The ability to visualise these surfaces is essential, but the high dimensionality of the corresponding configuration spaces makes this visualisation difficult. This thesis is concerned with the potential energy surfaces sampled during the structure prediction of materials and molecules, with a focus on random structure searching. The key contribution is a novel approach to energy landscape visualisation. The proposed method, SHEAP (stochastic hyperspace embedding and projection), is an algorithm for dimensionality reduction constructed by adapting existing, state-of-the-art manifold learning approaches to better deal with the structural datasets obtained from searching. Also discussed is a method for basin volume computation, previously applied to model periodic systems with fixed unit cells. Adapting this method to deal with realistic crystalline systems with variable unit cells was the initial intention for the research reported here, and we provide an overview of the early difficulties encountered. Ultimately, a different direction was pursued.

In this thesis, the common framework of state-of-the-art manifold learning-based approaches to dimensionality reduction for visualisation, such as t-SNE and UMAP, is described. Following this outline, we provide motivation for adapting these existing approaches in order to deal with structural data, before presenting our new method designed for application to PESs: SHEAP. SHEAP is compared to the methods on which it is based using test datasets standard in the machine learning community, showing comparable performance.

SHEAP is then applied to a variety of structural datasets. First considered are simple model systems: Lennard-Jones clusters, for which the energy surfaces are generally well understood. These systems are used to demonstrate how SHEAP can reproduce well-known features, such as funnels, and provide detail beyond, or complementary to, previous methods. For the 38-atom cluster, the impact of varying the structure descriptor on SHEAP’s representation of the dataset is explored. Also studied are periodic and finite systems of atoms described by first-principles density functional theory. Through the visualisations provided by SHEAP, key features of the energy landscapes considered are discussed and compared to one another. In addition, the number of dimensions required for SHEAP to faithfully depict the captured features of these energy landscapes is examined and revealed to often be no more than three, the maximum that we can visually comprehend.

Two structure searching case studies, which use SHEAP to aid in the analysis and depiction of the search results, are also presented. The first is a search for metallic structures with random structure searching, using orbital-free density functional theory for the electronic structure calculations. This study demonstrates how orbital-free density functional theory can be used to drive accelerated searching for metallic systems. SHEAP maps for solid-state Li, Na, Mg, and Al provide clear and concise summaries of the search results for each system, and facilitate comparison between key features of their energy landscapes. Of particular note, this analysis highlights that each PES possesses a band of low-energy, close-packed structures, with face-centred cubic and hexagonal close-packed at either end.

The second case study is a search for lithium-ion battery cathode materials, conducted using random structure searching with conventional Kohn-Sham density functional theory. Study of LiCoO2, a well-established cathode material, is used to demonstrate the impact of varying the parameters of a search, and the advantage that prior knowledge of a system can provide. SHEAP proves to be an effective tool through which to illustrate and interpret the difference between various searches on this system. Using the findings for this system as a basis, a systematic method for approaching entirely unexplored systems, for which one has no prior knowledge, is also discussed. In addition, an established polyanionic cathode material, LiFePO4, is studied, revealing a more complex PES than LiCoO2, and demonstrating the benefit of using pre-defined structural units with AIRSS, where appropriate. A relatively unexplored potential LIB cathode material is also considered: Li2Fe(C2O4)2. Using the exploratory searching framework laid out in the discussion of LiCoO2, a new layered phase of Li2Fe(C2O4)2 is discovered, with lower energy and higher theoretical rate capability than the previously discovered experimental phase.