Electron correlation plays an important role in determining the properties of physical systems, from atoms and molecules to condensed phases. Recent theoretical progress in the field has proven possible using both analytical methods and numerical many-body treatments, for realistic systems as well as for simplified models. Within the models, one may mention the jellium and the Hubbard. The jellium model, while providing a simple, rough approximation to conduction electrons in metals, also constitutes a key ingredient in the treatment of electrons in condensed phases within density-functional formalism. The Hubbard- and the related Heisenberg-model Hamiltonians, on the other hand, are designed to treat situations in which very strong correlations tend to bring about site localization of electrons. The character of the interactions in these lattice models allows for a local treatment of correlations. This is achieved by the use of projection techniques that were first proposed by Gutzwiller for the multicenter problem, being the natural extension of the Coulson-Fischer treatment of the ${\mathrm{H}}_2$ molecule. Much work in this area is analytic or semianalytic and requires approximations. However, a full many-body treatment of both realistic and simplified models is possible by resorting to numerical simulations, i.e., to the so-called quantum Monte Carlo method. This method, which can be implemented in a number of ways, has been applied to atoms, molecules, and solids. In spite of continuing progress, technical problems still remain. Thus one may mention the fermion sign problem and the increase in computational time with the nuclear charge in atomic and related situations. Still, this method provides, to date, one of the most accurate ways to calculate correlation energies, both in atomic and in multicenter problems.

DOI:https://doi.org/10.1103/RevModPhys.66.445