Localization and Delocalization in Solids from Electron Distribution Functions

The extent of electron localization and delocalization in molecular and condensed phases has been the subject of intense scrutiny over the years. In Chemistry, where real, instead of momentum space viewpoints are many times closer to intuition, a plethora of localization descriptors exist, including a family of indices invariant under orbital transformations that rely only on an underlying partition of the physical space into meaningful regions. These localization and delocalization indices measure the fluctuation of the electron population contained in such domains, and have been rigorously related to the insulating or conductive character of extended systems. Knowledge of the full electron population probability distribution function is also available in molecules, where it has provided many meaningful results as well as uncovered exotic interaction regimes in excited states. Electron distribution functions (EDFs), which can be seen as real space analogs of Pauling resonance structures, are now reported in periodic systems. In agreement with what is known in finite systems, ionic compounds display narrow EDFs that get wider as covalency sets in. Contrarily to conventional wisdom, most electrons delocalize over their nearest neighbors, even in quasi electron-gas metals like sodium, and it is only in the decay rate of the probability distribution where conductors and insulators can be distinguished.