In this comparative study we benchmark a recently developed nonadiabatic exchange-correlation potential within time-dependent density-functional theory (TDDFT) [Phys. Rev. Lett. 120, 157701 (2018)] by (a) validating the transient dynamics using a numerically exact density-matrix renormalization-group approach as well as by (b) comparing the
RC time, a typical linear-response quantity, to up to second-order perturbation theory results. As a test bed we use the dynamics of the single-impurity Anderson model. These benchmarks show that the nonadiabatic potential yields quantitatively accurate results for the transient dynamics for temperatures of the order of the hybridization strength, while the TDDFT
RC times quantitatively agree with those from second-order perturbation theory for temperatures which are large compared to the hybridization strength. Both results are particularly intriguing given the relatively low numerical cost of a TDDFT calculation.
This thesis investigates different methods for treating strongly correlated systems, and discusses their respective strengths and weaknesses. Many of the results presented in this thesis come from comparing the different methods and approximations to exact results. Paper I: We studied the real-time dynamics of a trapped fermion gas, as it expands after removal of a trapping potential. Paper II: We constructed a new exchange-correlation potential, to be used in three dimensions. The potential is non-perturbative in the interaction, meaning that we could use it both for weakly interacting (metallic) systems, as well as for strongly interacting (Mott insulating) systems. Paper III: We studied transport of electrons through small disordered wires contacted to leads. Using TDDFT, we have studied how the inclusion of both disorder and large electron-electron interactions affect conduction. At finite bias, we saw that the effects where competitive, and that interactions could increase the current through disordered samples. Paper IV: We studied fermions in 3D, using our newly developed exchange-correlation potentials, presented in Paper II. We studied how a cloud of fermions expands when released from a trapping potential. The simulated systems were large enough to be relevant for actual experimental conditions. Paper V: We studied transport of electrons through small disordered wires contacted to leads. The setup was similar to the one in Paper III, but another method was used - NEGF. Many of the observed trends were similar to those seen in our previous investigations. The differences were attributed to non-local effects.