Wannier function

On the basis of this definition, the following properties can be proven to hold:[4]

• For any lattice vector R' ,

${\displaystyle \phi _{\mathbf {R} }(\mathbf {r} )=\phi _{\mathbf {R} +\mathbf {R} '}(\mathbf {r} +\mathbf {R} ')}$

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

${\displaystyle \phi (\mathbf {r} -\mathbf {R} ):=\phi _{\mathbf {R} }(\mathbf {r} )}$

• The Bloch functions can be written in terms of Wannier functions as follows:

${\displaystyle \psi _{\mathbf {k} }(\mathbf {r} )={\frac {1}{\sqrt {N}}}\sum _{\mathbf {R} }e^{i\mathbf {k} \cdot \mathbf {R} }\phi _{\mathbf {R} }(\mathbf {r} )}$,

where the sum is over each lattice vector R in the crystal.

• The set of wavefunctions ${\displaystyle \phi _{\mathbf {R} }}$ is an orthonormal basis for the band in question.

${\displaystyle {\begin{aligned}\int _{\text{crystal}}\phi _{\mathbf {R} }(\mathbf {r} )^{*}\phi _{\mathbf {R'} }(\mathbf {r} )d^{3}\mathbf {r} &={\frac {1}{N}}\sum _{\mathbf {k,k'} }\int _{\text{crystal}}e^{i\mathbf {k} \cdot \mathbf {R} }\psi _{\mathbf {k} }(\mathbf {r} )^{*}e^{-i\mathbf {k'} \cdot \mathbf {R'} }\psi _{\mathbf {k'} }(\mathbf {r} )d^{3}\mathbf {r} \\&={\frac {1}{N}}\sum _{\mathbf {k,k'} }e^{i\mathbf {k} \cdot \mathbf {R} }e^{-i\mathbf {k'} \cdot \mathbf {R'} }\delta _{\mathbf {k,k'} }\\&={\frac {1}{N}}\sum _{\mathbf {k} }e^{i\mathbf {k} \cdot \mathbf {(R-R')} }\\&=\delta _{\mathbf {R,R'} }\end{aligned}}}$

Wannier functions have been extended to nearly periodic potentials as well.[5]

The Bloch states ψ_k(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 e^iθ(k) to the functions ψ_k(r), for any (real) function θ(k), one arrives at an equally valid choice. While the change has no consequences for the properties of the Bloch states, the corresponding Wannier functions are significantly changed by this transformation.

One therefore uses the freedom to choose the phases of the Bloch states in order to give the most convenient set of Wannier functions. In practice, this is usually the maximally-localized set, in which the Wannier function ϕ_R is localized around the point R and rapidly goes to zero away from R. For the one-dimensional case, it has been proved by Kohn[6] that there is always a unique choice that gives these properties (subject to certain symmetries). This consequently applies to any separable potential in higher dimensions; the general conditions are not established, and are the subject of ongoing research.[7]

A Pipek-Mezey style localization scheme has also been recently proposed for obtaining Wannier functions.[8] Contrary to the maximally localized Wannier functions (which are an application of the Foster-Boys scheme to crystalline systems), the Pipek-Mezey Wannier functions do not mix σ and π orbitals.

The existence of exponentially localized Wannier functions in insulators was proved mathematically in 2006.[7]