Self-consistency involves redistribution of charge throughout the cluster until a minimum energy is reached, thereby hopefully producing an accurate simulation of the charge distribution in an equivalent real system. This equilibrium distribution of charge for a given set of atomic coordinates is achieved by iteratively solving the Kohn-Sham equations until the charge density distribution produced by the Kohn-Sham orbitals gives the same potential that was used to generate it.
In order to achieve this, a first guess set of charge density coefficients, c_{k} and d_{k,s} are required; initially we take these from the neutral atom cases but during structural optimisation they are taken from the result of the previous iteration. The Kohn-Sham equations are then solved to determine the density matrix, b_{ij,s}, which is then used to produce a set of output charge density coefficients, c_{k}^{0}, d_{k,s}^{0} (Equations 3.2.10 and 3.2.19).
The next choice of charge density coefficients are formed from a weighted combination of the previous two, i.e.
c'_{k} = c_{k} + w (c^{0}_{k} - c_{k}). | (38) |
The choice of weighting factor, w, is important. If we solve once for a particular value of w (`w_{1}') giving a specific charge density c^{0}_{1k}, then the deviation from self-consistency of c_{k} is given by
(39) |
This e_{k} can be used to form a pseudo-charge density, which has a corresponding electrostatic energy,
(40) |
w is then chosen by minimising this energy. In practise this can be generalised to include the coefficients from all of the previous iterations, and the cycle converges exponentially, normally in only a few iterations (six or so).