Shape of polygonal quantum dots and ground-state instability in the spin polarization

We theoretically investigate the ground-state electronic structure and the spin polarization of four electrons confined in two-dimensional (2D) polygonal quantum dots. We employ standard mean-field theory approaches using unrestricted Hartree-Fock (UHF) and density functional theory (DFT) calculations. Resonant UHF configuration interaction (res-UHF Cl) calculations were also performed to incorporate the electron correlation more intuitively. Odd polygons (trigons and pentagons) preferentially generate the ground-state triplet as predicted by Hund's rule, whereas even polygons (tetragons, hexagons, and octagons) promote ground-state instability in the spin multiplicity and tend to produce an anti-Hund state of the ground-state singlet with strengthening of the interelectron interaction. The circle, a limited polygon having an infinite number of apexes, divides these odd and even polygons, and the ground-state instability can be well classified by the area of the polygon apexes that protrudes from an equisized circle.