The two natural starting points to approach solids: i) the limit of almost free electrons and ii) the limit of almost free atoms.
The former limit connects seamlessly to problems encountered in basic quantum mechanics courses. Effects of the interactions between electrons are approximated by effective potentials and we remain with single particle Schroedinger equations. Indeed, until today the most successful method for the simulation of electronic structure in real materials – Density functional theory (DFT) – is based on such mapping to the “best possible effective single particle problem”.
The atomic limit is, in contrast, a focus on the building blocks, i.e. the atoms. Here the approximation is made for hybridization and interaction with the surrounding lattice. Exact diagonalization of finite size many-body Hamiltonians for atoms or molecular building blocks with partially filled shells – so called (ligand-) field theory – is the most basic approximation of this kind. While such approximations work well inside Mott-insulating phases, they are insufficient to capture coherent quasiparticle excitations around the Fermi level of a metal.
Dynamical Mean-Field Theory (DMFT) allows to bridge this gap and cross the insulator to metal phase transition from the localized state to the Fermi liquid metal. DMFT captures the metallic state as non-perturbative quantum superposition of different atomic configurations and yields the exact solution for the single particle self energy of the Hubbard model in the limit of infinite dimensions (or coordination number) at any correlation strength.