# Wannier function in tight-binding model

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## 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
Gold Member
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
Gold Member
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?

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
MathematicalPhysicist
Gold Member
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
Gold Member
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.

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.

Rzbs