Sum over all filled states

4) Sum Over All Filled States. The sum over filled states in condensed matter is in fact an integral over a continuous variable, which is the momentum k in a crystal. Judicious choices of how to approximate the 

For insulators, however, the integrals over filled bands can be carried out very efficiently using the special points of Baldereschi ° and Chadi and Cohenand cast in a more systematic form by Monkhorst and Pack °. This defines sets of points which are rational fractions of 

5) Self-Consistency. The methods to achieve self-consistency are to a large extent independent of the other choices in solving the LDF equations. They are, in fact, a part of general problems encountered in the numerical analysis, and arise in many problems in phyisics. The simplest procedure, which is often used, is to construct the potential at the n-f1 

There are many cases in which convergence is rapid, i.e., cc can be chosen small enough ( / ), and this simplest approaoh has been considered adequate. These cases are ones, such as Si in the diamond structure, in which the potential 9 is dominated by the ionio potentials. The simplest improvement, letting cc vary for different parts of the potential, can also be incorporated in this framework.  

Another difficulty that the change in 9 (r) or its Fourier transform 9n(G) is a function of 9° -9 at all r or G. This means that the oc in Eq. (21) should in fact be a matrix. It is clear that the needed matrix is in fact closely related to the dielectric matrix. Several authors have implemented approximate calculations of the dielectric matrix in order to improve convergence. 

---