It is very well known that Kohn-Sham DFT underestimates bandgap. To get an accurate estimate of the bandgap, people often turn to Hybrid functionals (if they don't want to perform the actual quasiparticle calculation). I understand how the Hybrid functional, say for example, HSE works. It is an admixture (of a specific ratio) of Hartree Fock and KS-DFT. This can give a more reliable estimate of the bandgap in most cases.

But my question is as follows. Often, people perform KS-DFT calculations to study the electronic properties, the optical selection rules etc. Then, they just calculate the bandstructure with the hybrid functional turned on, to get a more agreeable bandgap. Does this mean the character of the bands is unchanged between KS-DFT and HSE? Is the electronic band gap the only thing that changes between these two calculations? In this context, I've come across a term called 'rigid shift' which just shifts the Hamiltonian without changing the wavefunctions. Is 'rigid shift' relevant here?

  • $\begingroup$ +1. However this might be seen as 2 questions in one, which can become hard for people to answer. People who want to make the best effort possible to answer the full question, may be reluctant to write anything if they know only about the effect on band structure of hybrids but nothing about rigid shifts. Also, if the band structure is a whole pattern of energy levels, and hybrids significantly effect the energy, I would be surprised if the structure remained unchanged and only the band gap changed. Maybe your question is: Why do you only use hybrids for the band gap but KS-DFT for properties? $\endgroup$ – Nike Dattani Jun 5 at 22:12
  • $\begingroup$ Nike, I agree with the title edit. That is the question I wanted to ask. Regarding your qualm about the second part, I am kind of divided. I feel that rigid shift is related to the main question. I think a separate question about rigid shift would be more appropriate for Phys stack exch. since it is a condensed matter concept. TLDR: I think the first part of the question is related to rigid shift, hence I feel we can keep it this way until someone refutes it, perhaps. $\endgroup$ – livars98 Jun 5 at 23:11
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    $\begingroup$ Essentially for those reasons, I didn't edit the rigit shift part and just wrote a comment instead. I do point out though, that you are asking whether rigid shift is relevant (whereas your comment claims that it is certainly related). About condensed matter: We have quite a lot of condensed matter physicists here too, and the tag "condensed-matter" is in our top 15 tags. If the question is about how rigid shift is related to KS-DFT and HSE band calculations, I think it's completely on topic here. (It can also be on topic on other SEs, and this overlapping is perfectly fine). $\endgroup$ – Nike Dattani Jun 5 at 23:20

The band gap problem in DFT is not just due to approximate exchange-correlation functionals--it is a reflection of the fact that the Kohn-Sham (K-S) orbitals are a mathematical construction of a non-physical, non-interacting system of electrons that yields the true ground state charge density of the real many-body system. In exact DFT, the derivative of the total energy vs. number of electrons added to the system is piece-wise continuous, linear for fractional numbers added and with a discontinuity at each integer number of electrons. This discontinuity is a significant contribution to the difference between the true gap and the K-S gap. Hybrid functionals improve the treatment of band gaps because they incorporate part of the derivative discontinuity into the K-S eigenvalue gap (DFT+U also does this). You can read up on many papers on this subject. A lot of fundamental work was done in this area by Sham, Perdew, Burke, Cohen, Levy and others. I've included a short bibliography. This is definitely not the same as "rigid shift" or "scissor operator" which you mentioned.

Hybrid functionals can definitely change the character of the bands. At the end of the day you are still removing some of the spurious self-interaction of DFT exchange functionals. This should also give energies (e.g. formation energy, adsorption energy, etc.) that are more accurate, especially when dealing with systems that have more localized states that exacerbate the self-interaction issue. This is in fact the original intention of hybrid functionals: to correct the self-interaction present in DFT approximate exchange and get better predictions of the thermochemical properties of molecules. It's worth reading Becke's original hybrid functional paper.

That being said, I'm sometimes surprised by how little the general picture changes sometimes. One nice paper to illustrate this is this one by He and Franchini, which studies first-row transition metal perovskites with HSE. HSE can help split up manifolds that are normally tangled together in PBE, but the general picture of bonding stays pretty similar overall. This is in contrast to using DFT+U empirically and just fitting to a desired quantity, which can also change the bonding character significantly. Of course, in materials where self-interaction error is very large, or in some more exotic situations like orbital ordering, etc. HSE can still have a big impact on the electronic structure (see the He and Franchini paper sections on LaTiO3 or LaVO3 as opposed to LaScO3, or LaFeO3-- the electronic structures are significantly different with HSE vs. PBE in the former cases, and not as much in the latter with the exception of the gap).

  1. Perdew, J. P., Parr, R. G., Levy, M. & Balduz, J. L. Density-Functional Theory for Fractional Particle Number: Derivative Discontinuities of the Energy. Phys. Rev. Lett. 49, 1691–1694 (1982).
  2. Perdew, J. & Levy, M. Physical Content of the Exact Kohn-Sham Orbital Energies: Band Gaps and Derivative Discontinuities. Physical Review Letters 51, 1884–1887 (1983).
  3. Sham, L. & Schlüter, M. Density-functional theory of the band gap. Physical Review B 32, 3883–3889 (1985).
  4. Lannoo, M., Schlüter, M. & Sham, L. Calculation of the Kohn-Sham potential and its discontinuity for a model-semiconductor. Physical Review B 32, 3890–3899 (1985).
  5. Perdew, J. P. Density functional theory and the band gap problem. International Journal of Quantum Chemistry 28, 497–523 (1986).
  6. Becke, A. D. A new mixing of Hartree–Fock and local density‐functional theories. The Journal of Chemical Physics 98, 1372–1377 (1993).
  7. Seidl, A., Görling, A., Vogl, P., Majewski, J. A. & Levy, M. Generalized Kohn-Sham schemes and the band-gap problem. Physical Review B 53, 3764–3774 (1996).
  8. Perdew, J. P., Ernzerhof, M. & Burke, K. Rationale for mixing exact exchange with density functional approximations. The Journal of Chemical Physics 105, 9982 (1996).
  9. Cohen, A. J., Mori-Sánchez, P. & Yang, W. Fractional charge perspective on the band gap in density-functional theory. Phys. Rev. B 77, 115123 (2008).
  10. Yang, W., Cohen, A. J. & Mori-Sánchez, P. Derivative discontinuity, bandgap and lowest unoccupied molecular orbital in density functional theory. The Journal of Chemical Physics 136, 204111 (2012).
  11. Mori-Sánchez, P. & J. Cohen, A. The derivative discontinuity of the exchange–correlation functional. Physical Chemistry Chemical Physics 16, 14378–14387 (2014).
  12. Himmetoglu, B., Floris, A., de Gironcoli, S. & Cococcioni, M. Hubbard-corrected DFT energy functionals: The LDA+U description of correlated systems. International Journal of Quantum Chemistry 114, 14–49 (2014).
  13. Perdew, J. P. et al. Understanding band gaps of solids in generalized Kohn–Sham theory. Proceedings of the National Academy of Sciences 114, 2801–2806 (2017).
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  • $\begingroup$ +10. Thanks for this! So basically: "rigid shift" is irrelevant, and surprisingly little changes apart from the band gap, when switching to hybrids? $\endgroup$ – Nike Dattani Jun 6 at 2:52
  • $\begingroup$ @NikeDattani I think it depends on the material. Those where self-interaction energy is very important might show more differences. It can also be important in orbitally-ordered materials. It definitely introduces some changes--by "bonding character" I just mean the relative mixing of states in the bands that you see in the PDOS. Since it decreases self-interaction, it also would help remedy the over-delocalization in GGA. I could easily be wrong in general about any of this stuff though. Just my casual observations. $\endgroup$ – Kevin J. M. Jun 6 at 2:55
  • $\begingroup$ Okay. Maybe a small addition like "I suspect that more would be affected in materials where the self-interaction energy is very important, or in orbitally-ordered materials", would help, since the question asked if anything other than the band gap can change when switching to hybrids. I really think the "rigid shift" part of the question was a bit unfair since it made this a case of "2 questions in 1", but now you've heroically answered both parts of the question, so hopefully people will recognize this and upvote accordingly. $\endgroup$ – Nike Dattani Jun 6 at 3:01

To add to the comprehensive answer by Kevin J. M., an example of a class of systems in which the use of a hybrid functional can lead to radically different band structure characteristics compared to a semilocal DFT, is in the area of topological materials. In this paper the authors show that semilocal DFT incorrectly predicts whether a material is topologically ordered or not (which in this case essentially depends on the "ordering" of the bands), when compared to a quasiparticle $GW$ approach. They also include calculations with the hybrid HSE06 functional, and find that in most cases (but there are a few exceptions), the hybrid agrees with the quasiparticle approach, and therefore predicts a different topological order to that predicted by the semilocal DFT. This is an example in which a simple "rigid shift" would fail.

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    $\begingroup$ +1. Thanks for this and nice first answer! We hope to see more of you!! $\endgroup$ – Nike Dattani Jun 7 at 13:39
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    $\begingroup$ Thank you for your welcoming words, I look forward to contributing and learning from the comunity! $\endgroup$ – ProfM Jun 7 at 14:04

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