New Methodologies in Free Energy Calculations of Materials and for Solving the Kohn-Sham Equations

Resumen

This dissertation touches certain problems of modern computational condensed matter physics oriented to the accurate characterisation of properties of materials. There are two separate parts here, each of them deals with different aspects. The first part of the dissertation deals with the problem of evaluating the thermodynamical properties of materials, where in detail we mean those properties that are related with the first order phase transitions in the temperature and pressure domain. Mainly it is a quest for finding efficient and accurate methods allowing for reliable predictions of these phenomena by a numerical, computational calculation. Such an approach is a valuable alternative to the often expensive laboratory experiments. Here we try to verify the prospective usefulness of nonequilibrium free energy approaches being an alternative to the existing free energy methods. We provide a detailed analysis of this nonequilibrium approach and also introduce or verify new useful techniques allowing for example for the efficient tracing of the coexistence lines between two phases as a result of a single computational simulation. By applying these techniques in practice, we try to verify their predictive power in the context of the first order phase transitions in silicon modelled by several empirical, semi-quantum and quantum models. We also consider a possibility of analysis of the liquid-liquid phase transition in phosphorus described by the tight-binding model and density functional based approach as well. The second part of this dissertation deals with the problem of solving efficiently the Kohn-Sham equations. These equations constitute the famous density functional theory, which is a very successful approximation to the expensive many body description of the Fermi liquid. Density functional theory allows for very accurate determination of physical and chemical properties of materials by means of a numerical calculation. Still, however,