This thesis presents an investigation of vibrational modes and phase transformations in functional materials, including graphite intercalation compounds and rutile TiO2 under negative pressure, using density functional theory. The vibrational modes play a crucial role in these studies. In the case of graphite intercalation compounds, the low-frequency Raman active modes are studied and compared with experimental operando Raman spectra. This includes analysing the stability of stacking sequences at various stages and the out-of-plane layer breathing modes. In the case of rutile TiO2, phase transitions induced by a zone-centre A2u mode and a zone- boundary transverse acoustic mode are investigated. These are described by Landau theory in terms of order parameters. The effect of dopant atoms on the phase transition pressure is also investigated. To study the effect of size of nanoparticles on the vibrational modes and frequencies of nanoparticles under negative pressure, a robust, and efficient first-principles-based method is developed and applied to rutile TiO2 nanoparticles. It is demonstrated that nanoparticles above a critical size exhibit unstable localised modes and their characteristic localisation length and decomposition with respect to bulk phonons are calculated. Additionally, the enhanced sampling method, metadynamics, and the isothermal-isobaric ensemble (NPT) method for finite systems are examined on a well-studied system of CdSe nanoparticles as a prototypical example. This provides a basis for future work on TiO2 nanoparticles with suitable force fields.