Within the Born-Oppenheimer approximation, the time independent Schrödinger equation for a fully interacting many-electron system is
where is the N-electron wavefunction, are the electron positions, are the positions of the ions and are the ionic charges. This equation is impossible to solve exactly so approximate solutions must be sought. One of the main challenges of condensed matter physics is to try to find good, workable approximations that contain the essence of the physics involved in a particular problem and to obtain the most accurate solutions possible. For the rest of this thesis all equations will be written in atomic units, .