# Using Density Functional Theory to make a Tight Binding model?

1. ### VortexLattice

146
Hi all,

A professor asked me to do something, but I'm not quite sure what he means -- He asked me to use Density Functional Theory (DFT) calculations of the band structure of a certain crystalline metal and adjust the matrix elements of a Tight Binding (TB) model to make a "minimal" TB model.

(To be clear: He doesn't want me to do the DFT calculations myself, just to use already existing results!)

Since a quick search of the literature showed me that the band structure of this metal is very well known, he confirmed that DFT was probably used to calculate it. So I guess I'm just supposed to use a calculated band structure and that's the end of DFT's role?

I'm still not sure how to use the given band structure to get the minimal TB model, though. I know what the TB model is and I've used it in some simple examples. In my textbooks (Ashcroft and Mermin, Phillips), they simplify the Hamiltonian for electrons in a periodic potential until it is really simple, really just an equation of a couple unknowns (that I assume depend on parameters of the material), like this:

$$E(\vec{k}) = E_0 - t\sum_\delta e^{i \vec{k} \cdot \vec{\delta}}$$

Where the sum is over nearest neighbors of an atom in the lattice, ##E_0## is due to the background potential, and ##t## is the "hopping matrix element".

Something still doesn't make sense to me, though. Since I have the crystal structure and the band structure, I could try and fit the above equation, but it really only has two parameters, ##E_0## and ##t##. However, my textbook(s) had to make about a million approximations to reach that simple equation, so maybe it's possible one of those isn't valid here, and then there would be more parameters to fit?

Can anyone help me? Does anyone have any resources on this?

Thank you!

2. ### cgk

488
You would have to ask him what exactly he has in mind with the adjusted TB model. What you should do depends on the answer to that.

The fact that you can do a full DFT for the material does not mean that it is not still a sensible idea to construct a tight binding model. Such models can be used, for example, also in correlated calculations (like, DMFT/DMET/AFQMC or similar methods) or just to interpret processes. It might also be helpful in simulating other properties, like defects, which are not present in the full calculation. For purely covelantly bonded materials tight binding models give a very faithful representation of the electronic structure.

For a tight binding reference, I am not sure if the solid state textbooks are very helpful. Because, as you said, they tend to oversimplify things quite massively. What tight binding really means is only that you consider only Fock and Overlap matrix elements between covalently bonded atoms, and not between everything else, too. So, technically, one could make a DFT calculation, calculate Wannier orbitals, and transform the Fock matrix to this basis and just take whatever you get there as Fock matrix elements between the bonded atoms as hopping matrix elements (and afaik, this is what DMFT people actually do).

There is also something called density functional thight binding'' you might want to look into: http://dx.doi.org/10.1016/j.commatsci.2009.07.013 (Density-functional tight-binding for beginners'') . While I personally detest the "DFT is in principle exact, if we only knew the correct functional''-talk you see so often, DFTB is actually a quite good way to make tight binding models, and it might give you some insights into the approximations involved.

Btw: In the chemistry community, tight binding'' is known as extended Hückel'', so you might want to have look into this, too.

3. ### VortexLattice

146
Well, I found a paper that does literally exactly what he wanted me to do. It even includes a spin-orbit interaction term, like he requested. The paper maps a TB model to some experimental results of the band structure.

So that's good, but having read it, there was no way in hell I could've done that myself.