Integration in the FBZ

There is a problem with P, because it requires a summation over k. We do not like this, because the number of the permitted k is huge for large N (and N has to be large, because we are dealing with a crystal). We have to do something with it. 

Let us try a small exercise. Imagine, we have to perform a summation J2kf(k), where / represents a smooth function in the FBZ. Let us denote the sum to be found by X. Let us multiply Xhy a small number A = where VpQz stands 

Electronic Motion in the Mean Field Periodic Systems 

In other words we just cut the FBZ into tiny segments of volume A, their number equal to the number of the permitted fe s. It is clear that if N is large (as it is in our case), then a very good approximation of XA would be 

After applying this result to the bond order matrix we obtain 

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Secular Equation Integration in the FBZ Fock Matrix Elements Iterative Procedure Total Energy... 

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