The behavior of sp-valent solids and liquids under compression is a field of intense re- search. At high pressure, they often undergo phase transitions to new structures with novel properties such as superconductivity, high-energy density, and superhardness. Furthermore, knowledge of these materials is essential for understanding the structure and evolution of planets. Molecular systems such as nitrogen and carbon dioxide are particularly interesting as energetic materials: their strong molecular bonds break under compression spawning transformations to exotic polymeric phases. We have used first-principles theory and molecular dynamics to make predictions for the properties of dense nitrogen, carbon dioxide, magnesium silicate, and magnesium oxide. For nitrogen, we provide evidence for a rare first-order liquid-liquid phase transition; only the second such transition seen in an elemental fluid. New finite-temperature structure search techniques have been developed and applied to predict a thermodynamically stable polymeric metal phase of solid nitrogen. Regarding carbon dioxide, we have computed its high-pressure liquid phase diagram over a broad pressure-temperature range, revealing rich structural diversity. We have also designed new free energy methods to explore the stability of free CO2 under deep mantle conditions. Lastly, first-principles molecular dynamics and finite-temperature free energy methods were used to predict a high-pressure phase separation transition in liquid MgSiO3 and also characterize the high-pressure phase diagram of MgO, including its melting curve.

In this thesis I have studied some problems of condensed matter at high pressures and temperatures by means of numerical simulations based on Density Functional Theory (DFT). The stability of MgCO3 and CaCO3 carbonates at the Earth's mantle conditions may play an important role in the global carbon cycle through the subduction of the oceanic crust. By performing ab initio electronic structure calculations, we observed a new high-pressure phase transition within the Pmcn structure of CaCO3. This transformation is characterized by the change of the sp-hybridization state of carbon atom and indicates a change to a new crystal-chemical regime. By performing ab initio Molecular Dynamics simulations we show the new phase to be stable at 250 GPa and 1000K. Thus, the formation of sp3 hybridized bonds in carbonates can explain the stability of MaCO3 and CaCO3 at pressures corresponding to the Earth's lower mantle conditions. We have also calculated phase transition sequence in CaCO3, SrCO3 and BaCO3, and have found that, despite the fact that these carbonates are isostructural and undergo the same type of aragonite to post-aragonite transition, their phase transformation sequences are different at high pressures. The continuous improvement of the high-pressure technique led to the discovery of new composite structures at high pressures and complex phases of many elements in the periodic table have been determined as composite host-guest incommensurate structures. We propose a procedure to accurately describe the structural parameters of an incommensurate phase using ab initio methods by approximating it with a set of analogous commensurate supercells and exploiting the fact that the total energy of the system is a function of structural parameters. By applying this method to the Sc-II phase, we have determined the incommensurate ratio, lattice parameters and Wyckoff positions of Sc-II in excellent agreement with the available experimental data. Moreover, we predict the occurrence of an incommensurate high-pressure phase in Ca from first-principle calculations within this approach. The implementation of DFT in modern electronic structure calculation methods proved to be very successful in predicting the physical properties of a solid at low temperature. One can rigorously describe the thermodynamics of a crystal via the collective excitation of the ionic lattice, and the ab initio calculations give an accurate phonon spectra in the quasi-harmonic approximation. Recently an elegant method to calculate phonon spectra at finite temperature in a self-consistent way by using first principles methods has been developed. Within the framework of self-consistent ab initio lattice dynamics approach (SCAILD) it is possible to reproduce the observed stable phonon spectra of high-temperature bcc phase of Ti, Zr and Hf with a good accuracy. We show that this method gives also a good description of the thermodynamics of hcp and bcc phases of Ti, Zr and Hf at high temperatures, and we provide a procedure for the correct estimation of the hcp to bcc phase transition temperature.