Electronic structure of nanosystems and crystals

The electronic state of a system of spatially confined atoms is first studied within a single-particle Hamiltonian approach, starting from atomic orbitals with suited boundary conditions. Then, the case of a macroscopic crystal is obtained after determining the limiting behavior of the confined system while the number of atoms becomes infinitely large. As an alternative to the atomic orbitals, the same electronic states are as well determined starting from a different basis set, that is composed of free-particle wavefunctions. The Bloch theorem is introduced, as the fundamental tool establishing the form assumed by the wavefunctions representing the electronic states, it being a consequence of lattice translational invariance and periodicity. Specific methods suited to the computation of electronic states are thus illustrated, that are applied to selected semiconductors, graphene, and carbon nanotubes. Then, a few elementary concepts of many-particle physics are introduced, useful to describe the interactions of electrons in crystals and nanostructures and to introduce different types of correlated ground states. Indeed, interacting ground states are first investigated within the Hartree and Hartree-Fock methods and excitonic effects are considered. Eventually, ground states related to spin-related magnetic phenomena and to superfluidity and superconductivity are described at an elementary level.