This thesis presents work done on the calculation of the effects of anharmonic nuclear motion on the properties of solid materials from first principles. Such anharmonic effects can be significant in many cases. A vibrational self-consistent field (VSCF) method is used as the basis for these calculations, which is then improved and applied to a variety of solid state systems.

Firstly, work done to improve the efficiency of the VSCF method is presented. The standard VSCF method involves using density functional theory (DFT) to map the Born-Oppenheimer (BO) energy surface that the nuclei move in, a computationally expensive process. It is shown that the accurate forces available in plane-wave basis DFT can be used to help map the BO surface more accurately and reduce the computational cost. This improved VSCF+f method is tested on molecular and solid hydrogen, as well as lithium and zirconium, and is found to give a speed-up of up to 40%.

The VSCF method is then applied to two different systems of physical interest. It is first applied to the case of the neutral vacancy in diamond, in order to resolve a known discrepancy between harmonic ab initio calculations and experiment -- the former predict a static Jahn-Teller distortion, whilst the latter leads to a dynamic Jahn-Teller effect. By including anharmonic corrections to the energy and nuclear wavefunction, we show that the inclusion of these effects results in agreement between first-principles calculations and experiment for the first time.

Lastly, the VSCF method is applied to barium titanate, a prototypical ferroelectric material which undergoes a series of phase transitions from around 400 K downwards. The nature of these phase transitions is still unclear, and understanding them is an active area of research. We describe the physics of the phase transitions of barium titanate, including both anharmonicity and the effect of polarisation caused by long wavelength vibrations, to help understand the important physics from first principles.