The aim of an ab initio method is to find the solution to the many-body Schrödinger equation for the system being studied. The first simplification that we may make en route to this goal is the Born Oppenheimer approximation [6], whereby the electronic and nuclear degrees of freedom are separated. The justification for this is that the electrons are much less massive than the nuclei but experience similar forces and therefore the electrons will respond almost instantaneously to the movement of the nuclei. Thus, the energy for a given nuclear configuration will be that of the ground state of the electrons in that configuration. The equation we must solve is therefore,
where 
is the many body wavefunction for the N electronic
eigenstates, an anti-symmetric function of the electronic coordinates
, and E is the total energy. The
Hamiltonian 
is given by
where 
is the external potential imposed by the nuclear
configuration 
and 
is the electron -
electron interaction given by the Hartree term 
.
In principle, this equation may be solved to arbitrary accuracy by
representing as a direct product wavefunction and diagonalising
the Hamiltonian. However, the cost of this calculation scales
exponentially with the number of electrons in the system and is
intractable for all but the smallest of systems. Other approaches are
available for the solution of this equation by minimisation of
The variational principle states that the ground state is that given
by minimisation of E over all possible 
.
Quantum Monte Carlo techniques may be used to evaluate 
[7] for a given wavefunction and to perform the minimisation
over possible wavefunction configurations
[8, 9, 10]. However, the cost of such
calculations is still prohibitively high for complex systems such as
those studied in this thesis. Clearly, another approach is needed to
this problem.