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 , 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  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.