Simple analytical parameterizations for the ground-state energy of the one-dimensional repulsive Hubbard model are developed. The charge dependence of energy is parameterized using exact results extracted from the Bethe-ansatz (BA). The resulting parameterization is shown to be in better agreement with highly precise data obtained from a fully numerical solution to the BA equations than previous expressions (Lima et al 2003 Phys. Rev. Lett. 90 146402). Unlike these earlier proposals, the present parameterization correctly predicts a positive Mott gap at half filling for any U > 0. The construction is extended to spin-dependent phenomena by parameterizing the magnetization dependence of the ground-state energy using further exact results and numerical benchmarking. Lastly, the parameterizations developed for the spatially uniform model are extended by means of a simple local-density-type approximation to spatially inhomogeneous models, e.g. in the presence of impurities, external 2 fields or trapping potentials. The results are shown to be in excellent agreement with independent many-body calculations, at a fraction of the computational cost.

Charge density and magnetization density profiles of one-dimensional metals are investigated by two complementary many-body methods: numerically exact (Lanczos) diagonalization, and the Bethe-Ansatz local-density approximation with and without a simple self-interaction correction. Depending on the magnetization of the system, local approximations reproduce different Fourier components of the exact Friedel oscillations.

The exact exchange-correlation (XC) potential in time-dependent density-functional theory (TDDFT) is known to develop steps and discontinuities upon change of the particle number in spatially confined regions or isolated subsystems. We demonstrate that the self-interaction corrected adiabatic local-density approximation for the XC potential has this property, using the example of electron loss of a model quantum well system. We then study the influence of the XC potential discontinuity in a real-time simulation of a dissociation process of an asymmetric double quantum well system, and show that it dramatically affects the population of the resulting isolated single quantum wells. This indicates the importance of a proper account of the discontinuities in TDDFT descriptions of ionization, dissociation or charge transfer processes. *

A systematic investigation of two approximate self-interaction corrections (SICs), Perdew-Zunger SIC and Lundin-Eriksson SIC, and the local-density approximation (LDA) is performed for a model Hamiltonian whose exact many-body solution and exact LDA are known. Both SICs as well as LDA are applied in the calculation of ground-state energies, ground-state densities, energy gaps, and impurity densities of one-dimensional Hubbard chains differing in size, particle number, and interaction strength. The orbital-dependent potentials arising from either SIC are treated within the optimized-effective potential method, which we reformulate for the Hubbard model. The delocalization tendency of LDA is confronted with the localization tendency of SIC. A statistical analysis of the resulting data set sheds light on the role of SIC for weakly and strongly interacting particles and allows one to assess the performance of each methodology.