# How trustworthy is the Tran-Blaha modified Becke-Johnson potential and how does it perform across basis sets?

When modeling solid-state materials and particularly semiconductors, one must go beyond LDA and GGA. One alternative is to use hybrid functionals or the $$GW$$ method. However, this can be very prohibitive for certain systems.

A popular alternative is to use the Tran-Blaha modified Becke-Johnson potential [Phys. Rev. Lett. 102, 226401 (2009)],

\begin{align} \mathbf{v}_{x,\sigma}^{TB-mBJ}(\textbf{r})=c \mathbf{v}_{x,\sigma}^{BR}(\textbf{r}) + (3c-2)\frac{1}{\pi}\sqrt{\frac{5}{12}}\sqrt{\frac{2t_\sigma(\textbf{r})}{\rho_\sigma(\textbf{r})}} \end{align} where $$\rho_\sigma$$ is the electronic density, $$t_\sigma$$ is the kinetic energy density and $$\mathbf{v}_{x,\sigma}^{BR}(\textbf{r})$$ is the original Becke-Roussel potential.

The authors propose the TB-mBJ (aka TB09) potential and implement it in Wien2K,a code based on the Augmented Planewave + local orbitals [APW+lo] method. Wien2K is an "all electron code". Over the years, most papers I have read that use the TB

How trustworthy is TB-mBJ (a.k.a. TB09) and how is it seen by the community today?

Does it yield accurate results with pseudo-potential codes like Quantum ESPRESSO or VASP? Are there any references that study the performance of TB-mBJ accross codes and basis sets?

The Tran-Blaha functional is not a generally useful functional, so I would not describe it as "trustworthy" at all. For example, it is not size-extensive so twice as much material does not have twice the energy, and hence you can't use it for any thermodynamics or phase stability or finite-displacement phonons etc. There isn't a proper potential either, so you can't use it self-consistently, and it's very fiddly to get pseudopotentials right since it couples the core and valence states via one of its parameters.

The usual claim is that the Tran-Blaha functional is "good for band-gaps". I think a single scalar quantity is a poor indicator of functional quality, and band-gaps are especially poor since it is not at all clear what the Kohn-Sham gap with the "true" functional would be. My experience is that this functional is usually very poor for band-structures compared to, say, PBE, although it does tend to open up the Kohn-Sham gap more.

I think the SCAN functional has far more potential (pun intended) to be generally useful, once its various pathologies have been addressed. It may be used self-consistently, obeys many exact conditions and it is possible to construct pseudopotentials for it -- which, incidentally, is crucial to getting reasonable results with it in a pseudopotential program.

• It's also worth noting that "is good for band gaps" has a million caveats too for any functional. While I'm not very familiar with mBJ, if you're studying a spin-polarized system and the functional can't describe the ground state spin state correctly, then the band gap is going to be garbage no matter what. This is one example of how a "trustworthy" functional necessarily requires that it do more than one thing correctly (because they are often interrelated). – Andrew Rosen Jul 5 '20 at 23:33
• +1 for writing the most detailed answer to this question so far. I do wonder how it got 3000+ citations if the only positive claim is that it's good for band gaps, which you and Rosen both say has many caveats. – Nike Dattani Jul 5 '20 at 23:35

I would like to state that I can't give you an answer to

How trustworthy is TB-mBJ (a.k.a. TB09) and how is it seen by the community today?

However, the reference presents the Quantum ESPRESSO's implementation of Meta GGA TB09, by extending the xc functionals of Libxc library in QE. There you will be able to find some useful benchmarks.

Anibal pointed out that the authors used a modified version of libxc; TB09 has actually been generally available in libxc since version 1.0.0, meaning that studies should be available in the literature.

Libxc is also supported interfaced to nowadays by over 30 programs relying on a variety of numerical approaches from plane waves to Gaussian basis sets and everything in-between, so it shouldn't be too difficult to assess the accuracy of TB09 if you have reliable reference data available.