Electrons in model nanostructures

For calculating the properties of solids and molecules, density functional theory (DFT) has become extremely popular because of its inherent computational efficiency. However, despite being in principle exact, an approximation must be introduced into DFT in practice. The accuracy of DFT has been key to its popularity; however, even for some of the simplest systems, using common approximations to the exchange-correlation (xc) functional may give inaccurate results. Therefore, we aim to contribute to the development of improved approximate xc functionals. It is logical to begin by studying the most elementary of systems where the common approximate xc functionals require improvement, as one can model these systems exactly by solving the many-electron Schrödinger equation. By allowing us to study DFT and time-dependent DFT (TDDFT) in the absence of approximations for prototype systems, this approach provides insight into the fundamental principles of the theory, informing the development of new approximations. We show that steps arise in the level of the exact xc potential: steps are known to be important for giving accurate electron and current densities, yet little about their origin is understood. We show that steps form due to a change in the 'local effective ionisation energy' of the electrons: this concept is well defined for strongly localised electrons. We find that the tendency of an electron to exclude others from its vicinity (electron localisation) is surprisingly high in our finite systems; hence, we develop an approximate functional that uses a measure of localisation as an ingredient, with the analytical form of the Kohn-Sham potential in the limit of complete localisation. Our functional, termed the mixed localisation potential, gives accurate electron and current densities for our test systems where local approximations are less valid. The approximation’s success stems in part from its ability to reproduce steps in the xc potential