Abstract First-principles theoretical methods, based on the quantum mechanics of electrons in periodic atomic systems, have been an effective tool to predict the ground-state structural and energetic behaviour of crystals for at least two decades. However, more recently the computational implementation of such methods by very efficient computer software and the availability of cheap and powerful hardware have undergone impressive improvements. It is thus now possible to tackle quite complex problems of materials science, earth sciences, environmental and other applied disciplines involving crystalline compounds from a purely theoretical point of view ( Pisani, 1996 ). In this respect, while some quantum-mechanical techniques in principle are able to account also for thermal effects ( ab initio molecular dynamics), the most easily available ab initio methods neglect the role of temperature (athermal limit) but can include the physical parameter pressure very straightforwardly. A brief account of such methods will be given here, presenting some examples of applications in the fields of thermodynamics and kinetics of crystalline materials at high pressure. Comparisons of quantum-mechanical simulations with those based on atomistic potentials are available in the literature ( Catti et al. , 2000 ). The interest of ab initio predictions of the structural and elastic properties of minerals and technological materials, and of their stability ranges, at high pressure is clear. Measurement techniques in extreme non-ambient conditions are particularly challenging and often subject to large experimental errors ( cf. the Diamond-Anvil-Cell methods). The most severe problems are perhaps faced by calorimetric measurements at high pressure, and also the determination

Abstract Most of the Earth’s material exists at high pressures and temperatures inside the planet. Since experiments in this p-T regime often turn out to be difficult or plain impossible, it is often necessary to do simulations, which avoid some of the important problems encountered in experiments. For a more specific look on the topics of this chapter we refer the reader to Kohn (1999), Martin (2004) , Oganov et al. (2002) , Payne et al. (1992) and Stixrude et al. (1998) . In this chapter we describe how to calculate the energy of a crystal with ab initio methods. It will shortly touch the historical origins of today’s methods and will end with the state-of-the-art quantum-mechanical calculations. When we want to investigate a mineral system we start with the Gibbs free energy. Every system in equilibrium likes to be in the state with the lowest Gibbs free energy G at given pressure and temperature condition. The Gibbs free energy is then given by minimising the following equation: where E is the energy, p the pressure, V the volume, T the temperature and S the entropy of the system. From statistical mechanics, knowing the energy of different states of the system (i.e. the energies of different vibrational and electronic quantum levels or of different atomic configurations) one can calculate the entropy and the free energy 1 , the link being provided by the partition function Z : The total energy of a non-relativistic electron-nuclear system and all its energy levels can be calculated by solving the Schrodinger equation, where H is the Hamilton operator and ψ is the wave function for the N electrons and the M nuclei.