Wannier function in tight-binding model

[Rzbs]

TL;DR Summary

Relation between Wannier function and linear combination of atomic orbitals (LCAO) in tight-binding model

What is the relation between Wannier function and LCAO in tight-binding (TB) model? I know in TB we use LCAO but why we have an alternative approch using Wannier function? For example as said in this lecture note:

An alternative approach to the tight-binding approximation is through Wannier functions. These
are defined as the Fourier transformation of the Bloch wave functions

But I don't understand why we use wannier function? Or in general why we use a Fourier transform of a function?

 

[MathematicalPhysicist]

For your general question, usually calculations made on the Fourier transform give indication on how to find the energy levels of the system by using Parseval's theorem.

 

[Rzbs]

Thanks

 

[DrDu]

Wannier functions are a kind of localized functions which evidently can also be calculated in a LCAO approach. So I don't think this is an alternative method of calculation, but rather representation. Note that the full wavefunction in Hartree Fock is invariant under arbitrary unitary transformations of the occupied orbitals.

 

[Rzbs]

Thanks;
I need more explanation; we use both atomic orbitals and Wannier function in Tight-binding model; when we use atomic orbitals and when we use Wannier functions? Is it true to say Wannier functions is a better way because they are Fourier transform of Bloch wave functions?!

 

[MathematicalPhysicist]

Thanks;
I need more explanation; we use both atomic orbitals and Wannier function in Tight-binding model; when we use atomic orbitals and when we use Wannier functions? Is it true to say Wannier functions is a better way because they are Fourier transform of Bloch wave functions?!

Have you read Ashcroft and Mermin's discussion of Wannier functions on pages 187-189?

 

[Rzbs]

Have you read Ashcroft and Mermin's discussion of Wannier functions on pages 187-189?

Because I'm reading this book I have these questions. I've read these pages but I didn't understand what are Wannier functions is so I searched to find another book or lecture note to explain these functions in more details. But I became more confused:(

 

[MathematicalPhysicist]

Because I'm reading this book I have these questions. I've read these pages but I didn't understand what are Wannier functions is so I searched to find another book or lecture note to explain these functions in more details. But I became more confused:(

I can sympathize with your efforts.
I also tried doing a thesis based MSc in maths and physics but failed miserably...
2020 miserable year...

 

[MathematicalPhysicist]

BTW, is there anyone here who wants to solve the problems in Ashcroft and Mermin that don't have solutions in the web?

At least I didn't find solutions to all the problems in A&M.

 

[peterwang]

Once we have solved the electronic structure problem, we have the Bloch waves, then we are able to Fourier transform the Bloch to Wannier. But how can we solve the electronic structure problem? Using atomic orbitals as basis set for expansion of the Bloch waves is one of the solution.

 

[rkkyjw]

Summary:: Relation between Wannier function and linear combination of atomic orbitals (LCAO) in tight-binding model

What is the relation between Wannier function and LCAO in tight-binding (TB) model? I know in TB we use LCAO but why we have an alternative approch using Wannier function? For example as said in this lecture note:
But I don't understand why we use wannier function? Or in general why we use a Fourier transform of a function?

Although we can construct TB model from Wannier functions or LCAO, there are some differences between them.

First, how to get the basis wavefunctions. Atomic orbitals, just as the name pointing out, normally are the wavefunctions of isolated atoms and are directly distributed on the crystal lattice. There is no other procedure to do. On the other hand, Wannier functions are defined as a Fourier transformation of Bloch functions, which means we should get Bloch functions $\psi_k(r)$first, then we can define the Wannier functions $a_{R_i}(r)$ . Since there is an arbitrary phase for each Bloch function with pseudo momentum k, actually the Wannier functions are not unique. Namely, we may have infinite choices of Wannier functions depending on the gauge choice of Bloch functions.

Second, the orthogonality. Note, the atomic orbitals $\phi_{R_n}(r)$ at $R_n$ site are not necessarily orthogonal to orbital $\phi_{R_m}(r)$ which is at $R_m$ site and they can have finite overlap. When constructing TB model and calculate speturms, we can do some orthogonalization first, or using the variational method to get the eigenvalues, or even assume they are orthogonal and directly solve TB model. For Wannier functions, $a_{R_i}(r)$ is orthogonal to $a_{R_j}(r)$, which is related to the orthogonality of Bloch functions. So if atomic orbitals from different lattice sites are orthogonal to each other, they can be regarded as one special Wannier function.

Why should we use Wannier functions? First, the LCAO method is usually an empirical method. Since the hopping integral and overlap integral are usually fitted to the DFT calculations or experiments. However, the Wannier functions and corresponding TB models can be directly converted from DFT calculations, which makes them have more accurate descriptions of electronic properties. Second, some physics is closed related to local properties, for example, the defect in a solid or the surface states of topological materials, or some correlated physics. Then it is much convenient to write the Hamilton on Wannier basis. Third, one concept of Wannier functions called "center of Wannier functions " is a very fundamental concept, which is closely related to the topological properties of solid.