Bloch Functions
In crystalline solids, electronic states are most naturally described by Bloch functions, which are eigenstates of the periodic Hamiltonian. According to Bloch’s theorem, they have the form
- $$\psi_{n\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot\mathbf{r}}u_{n\mathbf{k}}(\mathbf{r})$$
where n is the band index, k is the crystal momentum, and $u_{n\mathbf{k}}(\mathbf{r})$ has the periodicity of the crystal lattice.
Bloch functions are delocalized: each state extends throughout the entire crystal. This makes them ideal for describing energy bands and periodic systems, but less convenient for understanding local chemical bonding or constructing compact tight-binding models.
Wannier functions
Wannier functions provide an alternative, equivalent representation of the same electronic states. They are obtained by performing a Fourier transform of the Bloch functions over the Brillouin zone,
- $$|w_{n\mathbf{R}}\rangle=\frac{1}{\sqrt{N}}\sum_{\mathbf{k}}e^{-i\mathbf{k}\cdot\mathbf{R}}|\psi_{n\mathbf{k}}\rangle$$
where $\mathbf{R}$ labels the lattice vector and N is the number of sampled $\mathbf{k}$-points.
Unlike Bloch functions, Wannier functions are localized in real space, typically around a single atom, bond, or group of atoms. They therefore resemble familiar atomic orbitals or chemical bonds and provide an intuitive picture of the electronic structure.
The sum over all $\mathbf{k}$-points in Eq. (2) transforms the delocalized Bloch states into a localized Wannier function. The phase factors are chosen so that the contributions from different Bloch states interfere constructively in the unit cell $\mathbf{R}$, while largely canceling each other in other cells.
This localization can be understood from the discrete Fourier orthogonality relation
- $$\langle w_{n\mathbf R}|w_{n\mathbf R'}\rangle = \frac{1}{N}\sum_{\mathbf{k}}e^{i\mathbf{k}\cdot(\mathbf{R}’-\mathbf{R})}=\delta_{\mathbf{R},\mathbf{R}’}$$
Thus, when the contributions from all $\mathbf{k}$-points are summed, they add coherently for the selected unit cell and cancel for the other lattice translations. In this sense, the Fourier sum over $\mathbf{k}$-points in Eq. (2) acts as a discrete delta function in real space, focusing the delocalized Bloch waves onto a particular unit cell.
The two representations contain exactly the same physical information—they are simply different choices of basis. Bloch functions are localized in momentum space and are best suited for describing band structures, whereas Wannier functions are localized in real space and are particularly useful for analyzing chemical bonding and constructing efficient tight-binding Hamiltonians.
For this reason, Wannier90 transforms the delocalized Bloch states obtained from first-principles calculations into Maximally Localized Wannier Functions (MLWFs), providing both a clear physical interpretation and an efficient representation of the electronic Hamiltonian.
Why “Maximally Localized”?
Wannier functions are not unique. The Bloch states at each $\mathbf{k}$-point can be multiplied by arbitrary phase factors, and, for a group of bands, they can also be mixed by unitary transformations without changing the band energies or the physical subspace that they span. However, different choices of these phases and unitary rotations produce different Wannier functions in real space: some are compact and strongly localized, while others may be spread over many unit cells.
Wannier90 resolves this freedom by searching for the set of Wannier functions with the smallest total spatial spread. For each Wannier function, the spread measures how far its probability density extends around its center,
$$\Omega_n = \langle w_n|r^2|w_n\rangle - \left| \langle w_n|\mathbf r|w_n\rangle \right|^2$$
and Wannier90 minimizes the total spread
$$\Omega=\sum_n\Omega_n$$
The resulting functions are called Maximally Localized Wannier Functions (MLWFs).
Why is strong localization important? When Wannier functions are localized in real space, the Hamiltonian matrix elements
$$H_{nm}(\mathbf R) = \langle w_{n\mathbf 0} | \mathbf{H} | w_{m\mathbf R} \rangle$$
usually decrease rapidly with the distance between Wannier functions. As a result, only a relatively small number of short-range hopping terms may be needed to describe the electronic structure accurately. The Hamiltonian therefore becomes compact in real space and can be efficiently transformed back to reciprocal space to interpolate the band structure and other electronic properties on very dense $\mathbf{k}$-point meshes.
In practice, Wannier90 starts from initial orbital projections supplied by the user and iteratively changes the unitary mixing of the Bloch states at each $\mathbf{k}$-point to reduce the total spread. If the target bands are entangled with other bands, an additional disentanglement step is performed before the final localization.
A converged Wannierization does not automatically guarantee a physically correct model. The resulting Wannier functions should always be validated by checking their spreads and centers, inspecting their real-space shapes, and, most importantly, comparing the Wannier-interpolated band structure with the original DFT bands in the energy range relevant to the problem.