When an electron gas has low enough density, the electons' Coulomb repulsion can be strong enough relative to their kinetic energy that they spontaneously form a Wigner crystal. Since the electrons repel and there is no underlying lattice to give any structure, I would assume that the electrons just want to get as far away from each other as possible, so Wigner crystal lattice would be the closest sphere-packed lattice. In one and two dimensions this is true, but the Wikipedia article claims that in 3D, the Wigner crystal forms a body-centered cubic lattice rather than the closer-packed face-centered cubic lattice. Why is this?
I believe it could be because of the following. When you plot the quantum direct correlation function (QDCF) (obtained using the local correction field factor of the electron liquid convolved with the Coulomb potential) and obtain the structure factor, you can see that only the first mode of the QDCF is much more dominant than the others, these modes determine dominant lenghts. Because only one mode is dominant only one characteristic length is dominant, which in 3D produces BCC structuresm, while in 2D this produces a hexagonal lattice. This type of lattice formation due to these dominant lengths has been observed in classical density functional approaches. So, from a structural point of view the reason is that only characteristic length is dominant. I hope this answer helps.