Simultation methods in atomic-scale materials physics

The material presented in this book draws on many different disciplines and compromises have to be made to be able to present it in the linear form required by a book. The material has been organized to minimize the number of forward references: This has obvious advantages but the use of a particular device may not be apparent until several pages after it is presented. Let us first take a broad overview over the material presented in the rest of the book. In the first part of the book we survey the statistical mechanical basis for simulations and for the analysis of simulations and present a broad spectrum of algorithms for Monte Carlo and Molecular Dynamics simulations. The interaction potential is one starting point for the simulation. There is a hierarchy of interaction potentials from the simplistic generic potential to first principles quantum mechanical methods. For molecular dynamics some examples are: hard spheres, Lennard-Jones and other pairwise interactions, Effective Medium Theory, Embedded Atom Method and other semi-empirical manybody potentials, tight-binding and other approximate quantum mechanical methods, the Car-Parrinello method and other first principles methods. The principles and implementation of the simulation methods are almost independent of the interaction potential used. However, there is one aspect of the interaction potential, which is important for the implementation. At the simplest level, the interaction potential consists of a sum of pairwise interaction energies. Some examples of pair potentials are Morse, Lennard-Jones and Fumi-Tosi. For the quantum case, the energy of the system cannot be decomposed into a sum of pairwise interaction energies or even as a sum of energies for the individual atoms. At the intermediate level, the energy for the system consists of a sum of energies for the atoms, but the energy cannot be decomposed into a sum of pair-wise interaction energies. Modern approximate potentials for metallic systems are of this form. In many implementations a pairwise nature of the interaction potential 5 can be exploited to greatly increase the speed of the simulation. However, an implementation where a pairwise nature of the interactions has been used at an early state, is almost impossible to adapt the program to use a non-pair-potential. Careful analysis is needed to discover the implicit assumption of pair-wise interactions in e.g. the layout of loops. In the appendix we will discuss Effective Medium Theory as a concrete example of an approximative total energy method. EMT is not a pairpotential and this will give us the opportunity to display techniques for implementation of more general interactions than a simple pair-potential.