In principle, all knowledge about a system can be obtained from the quantum mechanical wave function. This is obtained (non-relativistically) by solving the Schrödinger equation of the complete many electron system. However in practice solving such an N-body problem proves to be impossible. This chapter will give a brief description of earlier approximations made to solve this many-body problem and a description of the important physical features omitted from these theories. For these reasons it is necessary to use density functional theory developed by Kohn and Sham based on the theory of Hohenberg and Kohn which, in principle, is an exact ground state theory. As the name suggests, the fundamental variational parameter is the electron charge density rather than the electronic wavefunctions. In this formalism, the N-electron problem is expressed as N one-electron equations where each electron interacts with all other electrons via an effective exchange-correlation potential. These interactions are then calculated using the local density approximation to exchange and correlation. Plane wave basis sets and total energy pseudopotential techniques are then used to solve the Kohn-Sham one electron equations. The Hellmann-Feynman theorem can then be used to calculate the forces required to integrate the ionic equations of motion within a molecular dynamics simulation.
There are now many articles on this subject which review the topics involved in total energy calculations in depth such as those in references [31, 32, 33, 34, 35, 36]. Therefore in this chapter only a brief description of the methods used for the total energy calculations used in later chapters will be discussed.