Traditionally the Kohn Sham equations given by Equation
2.5 were solved by a self consistent iteration. This
requires expensive matrix diagonalisation for each iteration at a cost
of 
operations where 
is the number of plane waves.
A more efficient approach involves the direct minimisation of the
total energy functional (Equation 2.2). This was first suggested
by Car and Parrinello in 1985, who proposed minimisation by a
molecular dynamics treatment of the electronic degrees of freedom
[7]. This approach scales as 
where 
is
the number of bands. The direct minimisation approach used in the
CASTEP/CETEP codes uses a conjugate gradient technique to minimise the
Kohn-Sham functional [8]. The use of a plane wave basis set
and Fast Fourier Transform algorithms lead to a scaling of
for the majority of the calculation, although
for large systems the cost of maintaining orthogonality between the
orbitals, 
, dominates. Furthermore, the conjugate
gradients approach leads to a much faster convergence in practice than
the molecular dynamics approach.