  In this work we investigate the Wigner localization of two interacting
electrons at very low density in two and three dimensions using the exact
diagonalization of the many-body Hamiltonian. We use our recently developed
method based on Clifford periodic boundary conditions with a renormalized
distance in the Coulomb potential. To accurately represent the electronic wave
function we use a regular distribution in space of gaussian-type orbitals and
we take advantage of the translational symmetry of the system to efficiently
calculate the electronic wave function. We are thus able to accurately describe
the wave function up to very low density. We validate our approach by comparing
our results to a semi-classical model that becomes exact in the low-density
limit. With our approach we are able to observe the Wigner localization without
ambiguity.
