Shell structure and paramagnetism of 3-DN-eanisotropic (ellipsoidal) quantum dots: Exact multi-pole expansion of coulomb interaction under fermionic exchange symmetry

As the density of electrons (N=2, 3, 4, 5, 6, 7, 8,.) increases, complexity arises due to coulomb interactions, inclusion of fermionic exchange symmetry and anisotropy. Consequently, Schrödinger equations of anisotropic quantum dots become non-trivial. Recasting such non-relativistic quantum equations into Whittaker-M basis functions unifies coulomb (exchange) correlation and antisymmetric nature of electrons in two-centered integrals of exact, finite, single-summed, terminated and simplest Lauricella functions (F2) via multi-pole expansion and the subsequent Chu-Vandermonde identity. Coulomb correlations interplay with in-plane and out-of-plane electrical and magnetic confinements in their moderate fields. On the other hand, full-scale fermionic exchange symmetry is incorporated through multiple number of Slater determinants, unlike Hartree-Fock method. In this context, both open-shell and restricted closed-shell configurations are considered for trial Hartree-products which are composed of the basis set of oscillator spin-orbitals. Thus optimized bound states can easily capture large number of mixed term symbols. Consequently, level clustering/accidental degeneracies occur in energy level diagram due to competition among strength of anisotropy in electrical confinements, magnetic field, mass of the carrier and dielectric constant. It brings about orbital induced paramagnetism (T∼(0-1)K), signature of fractional quantum Hall effect (FQHE) in chemical potential cusps (μ) and formation of different ′shell structures′ in capacitive energy respectively, spanning over wide dielectric range of materials (atomic like quantum dot, ZnO, GaAs, CdSe (Cadmium Chalcogenide) and PbSe (Lead Chalcogenide) etc).