Basics of first-principles simulation of matter under extreme conditions
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 energy1, 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.
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Extreme conditions and their effects on matter and materials are currently fashionable topics in modern science. Perhaps the fascination derives from the unimaginable dimensions that grab our attention and push the boundaries of our imagination. Imagine the pressures in extremely dense neutron stars where electrons and protons are fused together and atoms collapse to the density of an atomic nucleus; imagine temperatures of thousands of degrees Kelvin at the solar surface, or multimegabar and terapascal pressures deep within the interior of our planets. But even a simple droplet of water represents an extreme environment when it comes into contact with an otherwise stable crystal of rock salt, causing the crystal to dissolve as external conditions are drastically changed. We have an inherent desire to understand these diverse kinds of phenomena in nature, the mechanisms of the material changes involved, as well as the extreme conditions which are becoming increasingly demanded to achieve the extraordinary performance of new engineering materials. This rapidly evolving area of science is necessarily interdisciplinary, as it combines fundamental physics, chemistry and biology with geoplanetary and materials science, in addition to increasingly becoming one of the keys to engineering and technology aimed at process optimisation. Current experimental methods permit materials to be studied at pressures of several megabars, temperatures of tens of thousands of degrees Kelvin, and to achieve magnetic fields of several thousand teslas. Moreover, the rapid surge in computer technology has, in turn, permitted the solution of many previously intractable problems, and now even allows the behaviour of matter to be predicted far beyond the range of conditions currently accessible to experimentation. Previously unknown phenomena such as the formation of new phases, new forms of electronic and magnetic order, melting, atomic and electronic excitation, ionisation or the formation of a plasma state might result from exposing matter to extreme conditions well beyond those which were characteristic of the equilibria at the time of formation. With this volume of EMU Notes in Mineralogy we have endeavoured to provide up-to-date reviews of our understanding of the behaviour of minerals and geomaterials at exterior conditions that are sufficiently extreme to induce changes. In total 18 chapters reflect the diversity of this theme, but also demonstrate how strongly interdisciplinary this domain of modern mineralogy has become, bringing together physicists, chemists and geologists as well as experimentalists and computer scientists. The present volume contains the contributions of the lectures presented at the 7th EMU School, held at the University of Heidelberg from June 19 to June 25, 2005.