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
- Solving the system using an ab-initio method (like DFT)
- 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
- You obtain a real space representation, see formula (1.1)
- 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 $U$
- The tight-binding matrix elements exactly reproduce the DFT bandstructure (note: that doesn't make the method exact).
- 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?
- Create a matrix with parameters in it
- Solve the Bloch eigenvalue problem (in short: find the eigenvalues)
- See if the eigenvalues match the DFT bandstructure
- 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.
About the operators:
- If "normal" TB, you can't evaluate any of those operators.
- 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.
I can't tell you anything about the Berry phase.
