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.