Usually the Coulomb potential (electron-electron interaction) can be Fourier transformed (aside from prefactors) like that:
$$
\frac{1}{|\vec r_1 -\vec r_2|} = \int \frac{\text d ^3 k}{(2\pi)^3} \frac{\text e^{\text i \vec k (\vec r_1 – \vec r_2)}}{\vec k^2}\;.
$$
But sometimes, when dealing with periodic crystals, I see things that:
$$
\frac{1}{|\vec r_1 -\vec r_2|} = \sum_{\vec G} \frac{\text e^{\text i \vec G (\vec r_1 – \vec r_2)}}{\vec G^2}\;,
$$
where the sum goes only over the reciprocal lattice vectors $\vec G$.
But I don't understand why? The electron-electron interaction is not periodic! Does it have something to do with the finite size of the system?
Best Answer
Much as Okarin wrote, in the crystal there is a periodic arrangement of Coulomb potentials. The Fourier transform of this periodic arrangement of Coulomb potentials is discrete; it is zero except for the reciprocal lattice vectors.
Like the Fourier transform of a single Dirac delta function is constant, but the Fourier transform of an infinite comb of Dirac delta functions is an infinite comb of Dirac delta functions. Using the convolution theorem you can see that the Fourier transform of the infinite comb of Coulomb potentials is the infinite comb of Dirac delta functions times the Fourier transform of the Coulomb potential.