Correlation and spin polarization in quantum dots: Local spin density functional theory revisited

Abstract

ABSTRACT:Using quantum dot artificial atoms as a simple toy model, we reflect on the question of whether spin density functional theory (SDFT) can accurately describe correlation effects in low-dimensional fermion systems. Different expressions for the local density approximation of the exchange-correlation energy for the twodimensional electron gas, such as the much-used functional of Tanatar and Ceperley, and the recent suggestion by Attaccalite et al., are compared with the results of a numerical diagonalization of the many-body Hamiltonian matrix in the limit of small electron numbers. For systems with degeneracies, as shown in the present work for the example of a spin triplet with S ϭ 1, the direct comparison with configuration interaction (CI) methods demonstrates that the spin representation of SDFT may, under certain circumstances, produce artificial energy splittings between states that belong to the same spin multiplet. For a singlet ground state with S ϭ S z ϭ 0, however, the correlation functions of the CI solutions confirm the spin-density wave states found earlier within the SDFT method.

Incipient Wigner localization in circular quantum dots

“…23,27,36,38,42 We do find many violations of Hund's second rule -the value of the orbital angular momentum is not necessarily a maximum -but a modified second rule holds for small r s . (iv) The addition energy curve first becomes smoother as interactions strengthen -the mesoscopic fluctuations are damped by correlation -and then starts to show features characteristic of the classical addition energy, indicating incipient Wigner crystallization.…”

“…21 Early calculations using density functional theory (DFT) within the local spin density approximation showed a spin density wave (SDW) signature for r s as small as 2. 22,23 Unrestricted Hartree-Fock (UHF) 24 yielded both SDW and charge density wave (CDW) features for r s ∼ 1. These were initially identified as signatures of strong correlations related to the analog of a WC in a finite system, often called a "Wigner molecule".…”

“…Indeed, later calculations confirmed that these effects are largely artifacts of the approximations used. 23,25,26,27,28,29,30 Projection methods were then used to restore symmetries as a second stage of the UHF calculations. 31,32,33,34,35 DFT has been pushed toward the large r s limit through the average spin density approximation.…”

Kohn-Sham density functional theory for quantum wires in arbitrary correlation regimes

“…(27), t c (ρ) and v d ee (ρ) are the kinetic correlation energy and the electron-electron decorrelation energy per particle of an electron gas with uniform density ρ, corresponding to…”

Electron correlation in metal clusters, quantum dots and quantum rings

“…Instead, one then has to investigate the pair correlation functions, i.e., conditional probabilities, to study the internal structure [35,36]. Indeed, the direct comparison to spin-dependent pair correlation functions in the configuration interaction method, clearly demonstrated that SDW-like states in question can appear in the internal structure of the exact many-body state [39]. However, the SDFT results remain questionable for the discussion of spin multiplets, as already shown by von Barth in 1979 [33].…”