# The difference between the Wannier function and atomic orbit in a tight binding model

In a tight binding model, we usually start from the atomic orbits and linearly combine them to get the wave function of the crystal energy band.

My questions are:

1. Since this kind of tight binding is an approximate method due to using atomic orbits, is it exact to use the Wannier function formalism? If so, how do I get the Wannier function systematically?

2. What is the use of maximally localized Wannier functions?

3. Why can't we get the maximum localized Wannier function when the Berry phase is not zero?

4. Also, in tight binding, formalism, taking atomic orbits or Wannier functions as the basis function, respectively, what does position operator (diagonal or not), velocity operator and angular momentum operator look like?

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I can say something about the exponential localization and the Chern number (Berry phase).

The first paper that realize the relation of the localization of Wannier functions and zero Chern number of the band is due to Thouless. A more rigorous proof and a generalization to multibands can be found, for example, here.

A simple observation is like that. If we have a set of exponentially localized Wannier functions, the Bloch states (or quasi-Bloch states for multibands case) constructed from Wannier states must be analytic and single valued in the whole Brillouin zone due to the properties of Fourier series. This is impossible if the vector bundle formed by the Bloch states is non-trivial, i.e., with nonzero Chern number.

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Wannier functions are a basis set that you create from an other calculation, usually Density-Functional theory. By definition, they span the same subspace of the Hilbert space as the Kohn-Sham orbitals they are generated from.

Depending on the method you use, the following calculation using Wannier orbitals can be more or less exact than the DFT calculation. For example, you can do DMFT using Wannier orbitals, which is a more exact method. If you do tight-binding using Wannier orbitals, the method is less exact than DFT, but you can describe millions of atoms instead of hundreds. But to answer your question: methods from quantum chemistry and solid state physics are never exact and always approximations.

A matter of terminology: "atomic orbital" means that the orbital is localized around an atom. A Wannier orbital is an atomic orbital, because it is (or at least should be) localized around an atom. Sometimes they are also localized between two atoms, e.g. an sp2 hybrid orbital which as more density between the atoms than on it - but I suggest not to split hairs about that.

A great program to obtain Wannier orbitals is wannier90.

To be more specific, Wannier orbitals are obtained by

1. Solving the system using an ab-initio method (like DFT)
2. Choosing a unitary transformation (the matrix $U$) from the DFT Kohn-Sham orbitals to localized orbitals according to the formula (1.1) in the aformentioned paper:

$$w_{n\mathbf R}(\mathbf r)=\frac{V}{(2\pi)^3}\int_{BZ}\left[\sum_m U_{mn}^{(\mathbf k)}\psi_{m\mathbf k}(r)\right]e^{-i\mathbf kR}d\mathbf k$$

For MLWFs (Maximally localized Wannier functions), $U$ is chosen so that the sum of spreads (=variance, as defined in statistics) of all orbitals is minimal.

Wannier orbitals are cool/useful because

1. You obtain a real space representation, see formula (1.1)
2. You obtain tight-binding matrix elements by transforming the (diagonal) Hamilton operator in the Kohn-Sham basis to the Wannier orbital basis using the matrix $U$
3. The tight-binding matrix elements exactly reproduce the DFT bandstructure (note: that doesn't make the method exact).
4. The way to obtain those elements is systematic, fast and convenient.

Before I talk about operators, a short review: What is the "normal" way to obtain TB parameters?

1. Create a matrix with parameters in it
2. Solve the Bloch eigenvalue problem (in short: find the eigenvalues)
3. See if the eigenvalues match the DFT bandstructure
4. Change parameters and repeat from 1. until 3. is true

This way, you only get information about the energy structure of the system. You don't use any information about the real space representation, so the TB parameterization doesn't include it.

On the other hand, Wannier orbitals are created using a unitary transformation. By applying this transformation to the real space representation of the Kohn-Sham orbitals from DFT, you obtain the real space representation of the Wannier orbitals.

2. With Wannier orbitals, you can evaluate $\vec r_{ij}=\left<\phi_i|\vec r|\phi_j\right>$ in real space. It is a matrix in the Wannier orbital basis. You can do the same for the angular momentum and the momentum operator. None of them is diagonal.