Rzbs said:
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.
