The physical and chemical properties of materials are intimately related to their underlying crystal structure: the detailed arrangement of atoms and chemical bonds within. This thesis uses computational methods to predict crystal structure, with a particular focus on structures and stable phases that emerge at high pressure. We explore three distinct systems.

We first apply the ab initio random structure searching (AIRSS) technique and density functional theory (DFT) calculations to investigate the high-pressure behaviour of beryllium, magnesium and calcium difluorides. We find that beryllium fluoride is extensively polymorphic at low pressures, and predict two new phases for this compound - the silica moganite and CaCl$2$ structures - to be stable over the wide pressure range 12-57 GPa. For magnesium fluoride, our results show that the orthorhombic `O-I' TiO$2$ structure ($Pbca$, $Z=8$) is stable for this compound between 40 and 44 GPa. Our searches find no new phases at the static-lattice level for calcium difluoride between 0 and 70 GPa; however, a phase with $P6―2m$ symmetry is energetically close to stability over this pressure range, and our calculations predict that this phase is stabilised at high temperature. The $P6―2m$ structure exhibits an unstable phonon mode at large volumes which may signal a transition to a superionic state at high temperatures. The Group-II difluorides are isoelectronic to a number of other AB$2$-type compounds such as SiO$2$ and TiO$2$, and we discuss our results in light of these similarities.

Compressed hydrogen sulfide (H$2$S) has recently attracted experimental and theoretical interest due to the observation of high-temperature superconductivity in this compound ($Tc$ = 203 K) at high pressure (155 GPa). We use the AIRSS technique and DFT calculations to determine the stable phases and chemical stoichiometries formed in the hydrogen-sulfur system as a function of pressure. We find that this system supports numerous stable compounds: H$3$S, H$7$S$3$, H$2$S, H$3$S$2$, H$4$S$3$, H$2$S$3$ and HS$2$, at various pressures. Working as part of a collaboration, our predicted H$3$S and H$4$S$3$ structures are shown to be consistent with XRD data for this system, with H$4$S$3$ identified as a major decomposition product of H$2$S in the lead-up to the superconducting state.

Calcium and oxygen are two elements of generally high terrestrial and cosmic abundance, and we explore structures of calcium peroxide (CaO$2$) in the pressure range 0-200 GPa. Stable structures for CaO$2$ with $C2/c$, $I4/mcm$ and $P21/c$ symmetries emerge at pressures below 40 GPa, which we find are thermodynamically stable against decomposition into CaO and O$2$. The stability of CaO$2$ with respect to decomposition increases with pressure, with peak stability occurring at the CaO B1-B2 phase transition at 65 GPa. Phonon calculations using the quasiharmonic approximation show that CaO$2$ is a stable oxide of calcium at mantle temperatures and pressures, highlighting a possible role for CaO$2$ in planetary geochemistry, as a mineral redox buffer. We sketch the phase diagram for CaO$2$, and find at least five new stable phases in the pressure/temperature ranges 0 $\le P\le$ 60 GPa, 0 $\le T\le$ 600 K, including two new candidates for the zero-pressure ground state structure.