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In this work we investigate the Wigner localization of two interactingelectrons at very low density in two and three dimensions using the exactdiagonalization of the many-body Hamiltonian. We use our recently developedmethod based on Clifford periodic boundary conditions with a renormalizeddistance in the Coulomb potential. To accurately represent the electronic wavefunction we use a regular distribution in space of gaussian-type orbitals andwe take advantage of the translational symmetry of the system to efficientlycalculate the electronic wave function. We are thus able to accurately describethe wave function up to very low density. We validate our approach by comparingour results to a semi-classical model that becomes exact in the low-densitylimit. With our approach we are able to observe the Wigner localization withoutambiguity.

Wigner localization in two and three dimensions: an \emph{ab initio} approach