Remember that N is the number of possible values of k for the nth band in the first Brillouin zone.

One can appreciate that the Wannier functions do not depend on the k wavevector, possessing also the following properties:

1. There are as many Wannier functions as points in the direct lattice and therefore as values of k in the first Brillouin zone.

Indeed, in Eq. (5) there are as many terms [and consequently many values of an(r, I)] as different values of J (actually N).

2. The Wannier functions are located at the proximity of each I node of the direct lattice.

3. The Wannier functions obey also that an(r, l) = an(r l)

which is almost evident if we take into account the periodicity of uj. (r) in Eq. (7). 4. The Wannier functions satisfy the relation

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