N an7 I exp[ik l l

where we have introduced

that is, the Fourier transform of the energy in the k space that shows the overlap integral of H between two Wannier functions in l and l' for the same band. That is,

/a*n (r- l')Han(r~ I1) dr = J_Ie^ rJa*n' (r- l')an(r-Is) dr i'

Substituting (19) in (15), we obtain EEK»< ZnJ-V + Unn, (i 0}fn(i 0 = ^

where

is the matrix element of the perturbation potential.

Considering that the Fourier series for energy, in the k space, is en(k) = £ enlexp(ik^l'

where the enj coefficients are given by Eq. (18), and considering that k^-iV (23)

we can get the following product:

Zn(k)/(l) = Y.enf exp[i(-V) •l/(r) = ^ exp(! V)/(r)

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