Metal Clusters, Quantum Dots, and Trapped Atoms: From Single-Particle Models to Correlation

Abstract In this chapter, we discuss the electronic structure of finite quantal systems on the nanoscale. After a few general remarks on the many-particle physics of the harmonic oscillator, likely being the most studied example for the many-body systems of finite quantal systems, we turn to the electronic structure of metal clusters. We discuss Jahn–Teller deformations for the so-called “ultimate” jellium model which assumes a complete cancelation of the electronic charge with the ionic background. Within this model, we are also able to understand the stable electronic shell structure of tetrahedral (three-dimensional) or triangular (two-dimensional [2D]) cluster geometries, resembling closed shells of the harmonic confinement, but for Mg clusters being “doubly magic” as the electronic shells occur at precisely twice the atom numbers in the close-packed tetrahedra. Taking a turn to the physics of quantum dot artifical atoms, we discuss the electronic shell structure of the quasi 2D, harmonically confined electron gas. Between the clear shell closings, corresponding to the magic numbers in 2D, Hund's rule acts, maximizing the quantum dot spin at mid-shell. After a brief excursion to multicomponent quantum dots and the formation of Wigner molecules, we turn to finite quantal systems in strong magnetic fields or, equivalently, electron droplets that are set highly rotating. Working within the lowest Landau level, we draw the analogy between magnetic fields and rotation, commenting on the formation of the so-called maximum density droplet (MDD) and its edge reconstruction beyond the integer quantum Hall regime. Formation and localization of vortices beyond the MDD, as well as electron localization at extreme angular momenta, are discussed in detail. Analogies to the bosonic case, and the systematic build-up of the vortex lattice of a rotating Bose–Einstein condensate at high angular momenta, are drawn. With our contribution, we wish to emphasize the many analogies that exist between metallic clusters, semiconductor artificial atoms, and cold atoms in traps.