Distributed Gaussian orbitals for the description of electrons in an external potential

The Emergence of the Hexagonal Lattice in Two-Dimensional Wigner Fragments

At very low density, the electrons in a uniform electron gas spontaneously break symmetry and form a crystalline lattice called a Wigner crystal. But which type of crystal will the electrons form? We report a numerical study of the density profiles of fragments of Wigner crystals from first principles. To simulate Wigner fragments, we use Clifford periodic boundary conditions and a renormalized distance in the Coulomb potential. Moreover, we show that high-spin restricted open-shell Hartree-Fock theory becomes exact in the low-density limit. We are thus able to accurately capture the localization in two-dimensional Wigner fragments with many electrons. No assumptions about the positions where the electrons will localize are made. The density profiles we obtain emerge naturally when we minimize the total energy of the system. We clearly observe the emergence of the hexagonal crystal structure, which has been predicted to be the ground-state structure of the two-dimensional Wigner crystal.

Wigner localization in two and three dimensions: An ab initio approach

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.

The localization spread and polarizability of rings and periodic chains

The localization spread gives a criterion to decide between metallic and insulating behavior of a material. It is defined as the second moment cumulant of the many-body position operator, divided by the number of electrons. Different operators are used for systems treated with open or periodic boundary conditions. In particular, in the case of periodic systems, we use the complex position definition, which was already used in similar contexts for the treatment of both classical and quantum situations. In this study, we show that the localization spread evaluated on a finite ring system of radius R with open boundary conditions leads, in the large R limit, to the same formula derived by Resta and co-workers [C. Sgiarovello, M. Peressi, and R. Resta, Phys. Rev. B 64, 115202 (2001)] for 1D systems with periodic Born-von Kármán boundary conditions. A second formula, alternative to Resta's, is also given based on the sum-over-state formalism, allowing for an interesting generalization to polarizability and other similar quantities.