The Wannier functions are a complete set of orthogonal functions used in solid-state physics. They were introduced by Gregory Wannier in 1937. Wannier functions are the localized molecular orbitals of 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 was proved in 2006. Specifically, these functions are also used in the analysis of excitons and condensed Rydberg matter. Definition 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 , where R is any lattice vector (i.e., there is one Wannier function for each Bravais lattice vector); N is the number of primitive cells in the crystal; The sum on k includes all the values of k in the Brillouin zone (or any other primitive cell of the reciprocal lattice) that are consistent with periodic boundary conditions on the crystal. This includes N different values of k, spread out uniformly through the Brillouin zone. Since N is usually very large, the sum can be written as an integral according to the replacement rule: where "BZ" denotes the Brillouin zone, which has volume Ω. Properties On the basis of this definition, the following properties can be proven to hold: For any lattice vector R' , In other words, a Wannier function only depends on the quantity (r − R). As a result, these functions are often written in the alternative notation The Bloch functions can be written in terms of Wannier functions as follows: , where the sum is over each lattice vector R in the crystal. The set of wavefunctions is an orthonormal basis for the band in question. Wannier functions have been extended to nearly periodic potentials as well. Localization The Bloch states ψk(r) are defined as the eigenfunctions of a particular Hamiltonian, and are therefore defined only up to an overall phase. By applying a phase transformation eiθ(k) to the functions ψk(r), for any (real) function θ(k),