Exact Solutions in the Density Functional Theory (DFT) and Time-Dependent DFT of Mesoscopic Systems
Density functional theory (DFT) and its time-dependent counterpart TDDFT are formally rigorous approaches to theoretically study the ground-states and the electronic excitations, respectively, of atomic, molecular, and condensed matter systems, including the mesoscopic ones. Practically, the usefulness of DFT and TDDFT is determined by the accuracy and the computational efficiency of the available approximations to the corresponding exchange-correlation (xc) functionals. In the three-dimensional, as well as in the purely low-dimensional, cases, the local-density approximation (LDA) to the xc functional of the corresponding dimensionality is traditionally and predominantly used. However, LDA, as well as its extensions of generalized gradient approximation (GGA) and meta-GGA, encounter severe difficulties when applied to the systems of intermediate dimensionality, i.e., mesoscopic systems. In this chapter, we review the recent progress in the use of the static and dynamic exact exchange (EXX) functional, which we show to be well fit to capture the characteristic features of mesoscopic systems. On this way, we find exact analytical solutions to the exchange-only DFT and TDDFT problems for specific systems of reduced-dimensionality: those of the quasi-2(1)D electron gas with one filled subband. These solutions provide us with important insights into the role of the interparticle interactions, allowing, in particular, to identify and separate the collective and the single-electron regimes in the quantum dynamics of the systems under study.