This thesis presents a study of vibrational modelling of periodic solids, focussing in particular on vibrational self-consistent field theory (VSCF).

Firstly, the mathematical framework within which vibrational modelling takes place is detailed. The approximations which must be made are introduced, and the limitations imposed by these approximations are explored. Then the existing approaches to vibrational modelling are reviewed, and the modelling process is separated into three interrelated problems: representing the vibrational potential energy surface, fitting the free parameters of that representation using electronic structure theory, and using the fitted representation to calculate the vibrational wavefunctions and the vibrational free energy.

A number of developments to the VSCF method are described. These focus on exploiting the notions of symmetry and size consistency to reduce the computational cost of the method and to improve its accuracy and reliability. The discussions of symmetry and size consistency also reveal a number of fundamental problems with the VSCF method. Solutions are proposed to several of these problems, but other problems remain unsolved, and it is proposed that these problems may imply that VSCF is not well suited for modelling crystals.

A software package which implements the ideas contained within this thesis is introduced, and the methods used to encode VSCF as accurate and efficient software are detailed. The results of an example application of this software package are also presented.