Sections through have outlined a series of approximations for the evaluation of in the eigenvalue problem represented by Equation . We now discuss the solution of these equations to find the ground state energy and corresponding Kohn-Sham orbitals.
The original methods relied on a self consistent iteration to find the ground state. In this approach a trial charge density is used to calculate . The resulting effective Hamiltonian is the diagonalised to find its eigenstates and from these a new charge density is constructed. This is then used as the input to the procedure and the loop is repeated until the output charge density is consistent with the previous iteration. This procedure is very inefficient, as the cost of each iteration is dominated by the diagonalisation of the effective Hamiltonian, which is , where M is the number of basis functions. This diagonalisation yields M eigenstates, but the calculation of the total energy only requires the lowest N of these states. As N is usually significantly smaller than M, this represents a gross inefficiency.
The approach we use is that of direct minimisation of the energy functional. This was first proposed by Car and Parinello  who use a fictional electron dynamics to perform the minimisation. In contrast, we use a preconditioned conjugate gradient technique [5, 32]. This is an iterative technique which successively `improves' a set of trial wavefunctions by evaluating a step at each iteration that will reduce the total energy. This involves the evaluation of the functional derivative of the energy with respect to each of the wavefunctions . However, the required derivative is simply . As the kinetic energy is most efficiently calculated in Fourier space and the potential energy in real space, the cost of evaluating this derivative is dominated by transforming between these representations. The use of fast Fourier transforms  implies that the cost of this is . However, the minimisation must be carried out within the constraint that the must be orthogonal. Thus, the asymptotic cost of this procedure is dominated by the cost of orthogonalisation which is using the Gram-Schmidt technique . A smaller prefactor for the cost of this procedure means that this term only dominates for large system sizes. Nonetheless, it may be seen that this approach offers a significant efficiency saving over self-consistent iterative matrix diagonalisation.