Density-functional theory (DFT), in its various forms, has become a near ubiquitous form of theoretical research used to benchmark and prototype solutions to many finite and extended state system. This is largely because DFT can both capture the rich physics that is present in these electronic systems, while remaining computationally cost-effective and interpretable. However, DFT also has the requirement that the density functional being used to iteratively converge towards a solution must be accurate and correct. While on the surface such a stipulation seems benign, in practice the density functionals can be overwhelmingly complex and error can be introduced that comes from either the density functional that is chosen or the approximations used to make a system more calculationally tenable. In this work, our focus is on the use of model systems to calculate and determine the usefulness and shortcomings of DFT. By simplifying the underlying system, while also retaining enough physical quantities from real systems, we can focus on how the approximations affect the outcomes that are produced. To begin, we show that charge-transfer dynamics can be described in unique and enlightening ways through the use of the particle-hole map (PHM). Using a one-dimensional, multi-well system, we effectively demonstrate how interesting electron dynamics can be uncovered by applying unitary transformations to the wavefunctions. By spatially localizing the electronic wavefunctions through the Foster-Boys method, which is analogous to Wannier localization in extended systems, the intermediating components of charge transfer systems can be examined to determine their effect on the system-at-large. From the simple one-dimensional system, we could quickly infer real molecular systems that could potentially be examined using the same method to surmise the role charge-transfer intermediaries play in such systems as organic photovoltaics. Beyond electron dynamics, the role of exchange-correlation (xc) scalar potentials and magnetic fields that are features of noncollinear spin Kohn-Sham (KS) and DFT was explored by comparing the exact Schroedinger solution to the KS and DFT approximations. By extending the Hubbard model to four sites, we can both solve the system exactly, while allowing for on-site and nearest-neighbor interactions. We were able to obtain benchmark solutions across a wide range of interaction strengths, determining that there are regimes where the xc magnetic fields play an increasingly larger role as the system becomes more correlated. In fact, there is a regime where the xc magnetic fields become larger than the external magnetic fields that are applied on the system. Through the model system, we could additionally compare the exact solutions against the approximated xc functionals and demonstrate that the weakly correlated regime can be adequately described by the xc functional approximations common to many real-systems. Moving beyond steady state observations, we can also describe time-dependent electron dynamics through real-time TDDFT and use a model system to compare the time-evolution of the exact and KS solutions. By allowing the xc potentials to propagate in real time, we could explore the role the xc torques played during the evolution of a triangular lattice under an applied, time-varying magnetic field. Additionally, by controlling the spin-orbit coupling present in the small model system, we determined that the spin orbit coupling plays a substantial role in keeping the spins more closely aligned with the exact system. In part, this was due to the spin-orbit coupling serving as a time-varying magnetic field, which tended to be larger than the xc potentials that were also present. The trimer can also be quickly and easily expanded with the added spin-orbit coupling and compared to real model systems through computational physics software, such as Octopus.
Topological viewpoint of two-dimensional group III-V and IV-IV compounds in the presence of electric field and spin-orbit coupling by density functional theory and tight-binding model
