# How to construct a Tight-Binding Hamiltonian from first-principles computations?

Effective Hamiltonian approaches such as the Tight-Binding method played a central role in the reconciliation between chemistry and in physics in the solid state. A classical and complete treatment of the method is found in Walter A. Harrison's book "Elementary Electronic Structure" (an update to his prior text "Electronic Structure and the Properties of Solids: The physics of the chemical bond").

Tight-binding models continue to play a central role in condensed matter and materials physics. As they require certain model parameters, I ask:

• Is it possible to construct a Tight-binding model using Density Functional Theory or another ab initio method as a starting point?

• How is this achieved? Are there available codes that aid in this process?

• There are many 'density functional tight binding' (DFTB) programs and papers :-) For example DFTB.org and references therein – Geoff Hutchison May 2 at 1:11
• I knew about DFTB (DFTB+) but I see that as another computational method for modeling materials? Not sure if it is possible to then construct an effective Hamiltonian from DFTB thann can later be manipulated by hand or be used as input in codes such as Kwant for quantum transport. My feeling is that this might be closer to Wannier90 than DFTB+. – Etienne Palos May 2 at 1:20
• I think you can see the Hamiltonian from Wannier90 as a tight-binding Hamiltonian. For quantum transport you will need to describe the complete system including an external potential eventually. Therefore, you need to upscale the Hamiltonian to the device you want to simulate. – CKl May 2 at 10:26

Yes! It is definitely possible and it is useful for calculating other things like electronic transport and first-principles values of Hubbard U (i.e. the ACBN0 method, which I have used a bit). Some good papers to read are from Prof. Marco Buongiorno Nardelli's group at UNT:

1. L. A. Agapito, A. Ferretti, A. Calzolari, S. Curtarolo, and M. Buongiorno Nardelli, Physical Review B 88, 165127 (2013).
2. L. A. Agapito, M. Fornari, D. Ceresoli, A. Ferretti, S. Curtarolo, and M. Buongiorno Nardelli, Physical Review B 93, 125137 (2016).
3. L. A. Agapito, S. Ismail-Beigi, S. Curtarolo, M. Fornari, and M. Buongiorno Nardelli, Physical Review B 93, 035104 (2016).
4. P. D’Amico, L. Agapito, A. Catellani, A. Ruini, S. Curtarolo, M. Fornari, M. Buongiorno Nardelli, and A. Calzolari, Physical Review B 94, 165166 (2016).

The basic idea is that you take the Kohn-Sham (K-S) orbitals and project them on the atomic orbitals you want to use in your tight-binding model. A convenient choice is the pseudo-atomic orbitals present in most pseudopotential files. Some difficulties can arise, especially in projecting empty conduction band states onto the atomic orbitals, since they are not as well-described by this basis. The method from these authors filters out K-S orbitals that have a low projectability on the localized basis. These get replaced in the Hamiltonian as zero eigenvalues. Another step is actually "shifting" these eigenvalues to a non-zero energy to avoid interfering with "real" bands in the energy range of interest.

I believe the code WanT (Wannier-Transport) can do this with Quantum Espresso, though it might not be an "official" feature. You can probably look through the source code to find the relevant methods. I think the code will output a "prefix.ham" file for the TB Hamiltonian.

More recent related codes to look at would be PAOFLOW and AFLOW$$\pi$$ from the AFlow consortium.

• +1. Another great one from Kevin! – Nike Dattani May 23 at 5:33