Wannier function

The Wannier functions are a complete set of orthogonal functions used in solid-state physics. They were introduced by Gregory Wannier. Wannier functions are the counterpart of localized molecular orbitals for crystalline systems. The Wannier functions are a complete set of orthogonal functions used in solid-state physics. They were introduced by Gregory Wannier. Wannier functions are the counterpart of localized molecular orbitals for crystalline systems. The Wannier functions for different lattice sites in a crystal are orthogonal, allowing a convenient basis for the expansion of electron states in certain regimes. Wannier functions have found widespread use, for example, in the analysis of binding forces acting on electrons; the existence of exponentially localized Wannier functions in insulators has been proved in 2006. Specifically, these functions are also used in the analysis of excitons and condensed Rydberg matter. Although, like localized molecular orbitals, Wannier functions can be chosen in many different ways, the original, simplest, and most common definition in solid-state physics is as follows. Choose a single band in a perfect crystal, and denote its Bloch states by where uk(r) has the same periodicity as the crystal. Then the Wannier functions are defined by