Within the Born-Oppenheimer approximation[4], 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, 
.
