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strongly correlated material is one where $\Sigma - V_{KS} $ is large at low frequencies. Strongly correlated materials, are those for which this is not the case, a famous example are materials such as LaCuO4 which are predicted to be metals in LDA but which are experimentally antiferromagnetic Mott insulators.   In quantum chemistry   Aron J Cohen, Paula Mori-Sánchez, Weitao Yang Publication date  2008/8/8  Journal (2008)  Science Volume  321  Issue  5890  Pages  792-794 321,792-794  there is a classification of the errors introduced by the use of approximate density functionals, as being of two types, static correlation and dynamic  correlation\cite{Yang_2012}. Similar ideas, also appear in the solid state context, but the nomenclature is exchanged. A {\it static } self energy   (i.e. a self energy which varies weakly with frequency at low energies) corresponds to the concept of dynamical correlations.  However, Strongly-correlated compounds have many unusual properties. For example,  there is another large class of materials called strongly-correlated compounds, whose distinguishing phenomenological feature is sensitivity are very sensitive  to external perturbation, which makes them technologically useful. Small changes in pressure, temperature or chemical doping often drives large changes in electronic or structural behavior. For example, changing the temperature by only several degrees Kelvin can drive a transition between a metallic and insulating state, behavior not observed in weakly-correlated compounds. In addition to metal-insulator transitions, these compounds display unusual magnetic properties, high-temperature superconductivity and strange-metal behavior. The basic feature of correlated materials is their electrons cannot be described as non-interacting particles. Often, this occurs because the material contains atoms with partially-filled $d$ or $f$ orbitals. The electrons occupying these orbitals retain a strong atomic-like character to their behavior, while the remaining electrons form bands; their interplay poses special challenges for theory. Consequently current implementations of DFT cannot describe their properties accurately. This led to the development of extensions to DFT such as LDA+U, and entirely more sophisticated approaches such as dynamical mean field theory (DMFT) and the GW approximation.