# How does the mBJ potential give an accurate band gap as compared to LDA and GGA?

I am not able to understand the literature on how the modified Becke-Johnson(MBJ)$$^{[1]}$$ potential gives an accurate bandgap. Can someone please help? Thank you. The formula for the potential can be found here$$^{[2]}$$

References

1. Tran, Fabien, and Peter Blaha. "Accurate band gaps of semiconductors and insulators with a semilocal exchange-correlation potential." Physical review letters 102.22 (2009): 226401.
2. Jiang, Hong. "Band gaps from the Tran-Blaha modified Becke-Johnson approach: A systematic investigation." The Journal of chemical physics 138.13 (2013): 134115.
• Can you explain the acronym MBJ? May 18 '20 at 7:26
• It is the modified Becke-Johnson (mbJLDA) potential. May 18 '20 at 9:38
• Related and currently unanswered questions: materials.stackexchange.com/questions/154/…, materials.stackexchange.com/questions/391/…
– Tyberius
May 18 '20 at 15:39
• Just from the abstract: Seems like it is modifying the exchange part of the Kohn-Sham hamiltonian, which typically corrects for the self interaction error of DFT calculations that opens/widens up the band gap.
– gogo
May 22 '20 at 3:55

# Model potentials for exchange:

Exchange-correlation potentials like mBJ and GLLBSC have model orbital-dependent corrections included in the form of the functional. I'll first try to get the fundamental motivation across using GLLBSC as an example, and then attempt to go into how mBJ does this.

# GLLB/SC:

In the case of the GLLB work, the authors used a model exchange potential near the core region to represent the stepped structure of the short-range exchange potential. It is my understanding that this short-ranged stepped structured is challenging to reproduce using (semi)local approximations such as the LDA or the variety of GGA available. It appears that given a somewhat accurate description of the long-range tail of the exchange hole, the short range stepped structure can be approximated. This is so far my understanding.The approach has been improved to include a description for correlation: GLLBSC.

# mBJ:

In the work linked, equation 1 has a term $$t_{\sigma}$$ which is the KE density of the (KS) wavefunctions. This is your orbital-dependence. The $$\rho_{\sigma}$$ term does not have an orbital dependence. There are two remaining free parameters, $$\alpha$$ and $$\beta$$, which the authors choose these values that provide accurate gaps for a (not too large) list of solids. Whether this procedure for deciding these two parameters is valid for a larger range of solids can only be answered by extensive testing of the functional's performance. But the key point is this: The improved description of the fundamental gap comes in both of these approaches from a model potential used for the short-range part of the exchange potential instead of doing an exact-exchange or a GW calculation out. The gap is improved since the effect of exact-exchange that LDA/GGAs cannot describe quite well is described using a model orbital-dependent quantity. The benefit lies in the use of semilocal quantities, which prevent large numbers of Poisson equation solutions/other general calculations, as in the case of HSE/GW/any other exact-exchange-including approach.

# In summary:

The short-range part of the exchange potential has an improved description using (to the best of my knowledge, orbital-dependent) model potentials as compared to LDA/GGAs.

Note that in both the above cases, the common theme is the presence of orbital-dependent model corrections. These corrections may have free parameters (both of the above approaches do), and these free parameters need to be pinned down with some motivation. It can either be to recover well-known limits (GLLB), or to optimize against a test set of known observables (gaps in Table 1 in the case of mBJ or TB09, whichever name one prefers).

### References:

1. Gritsenko, O., van Leeuwen, R., van Lenthe, E., & Baerends, E. J. (1995). Self-consistent approximation to the Kohn-Sham exchange potential. Physical Review A, 51(3), 1944.
2. Kuisma, M., Ojanen, J., Enkovaara, J., & Rantala, T. T. (2010). Kohn-Sham potential with discontinuity for band gap materials. Physical Review B, 82(11), 115106.
3. Tran, F., & Blaha, P. (2009). Accurate band gaps of semiconductors and insulators with a semilocal exchange-correlation potential. Physical review letters, 102(22), 226401.