Different approaches to creating (meta-GGA) DFT functionals

Contributed by David Bowler
Reposted from Atomistic Computer Simulations with permission

Within the DFT community, John Perdew’s idea of the Jacob’s ladder of accuracy[1] starts with LDA, moves to GGAs with the inclusion of the gradient of the electron density, and is further extended (to rungs three and four in the ladder analogy) with meta-GGA, where kinetic energy density is added, and hybrids, where exact exchange plays a role. Although meta-GGAs have been around for ten to fifteen years, they are only starting to become widely used. I will compare two examples which both seem promising, but also encapsulate the two most common approaches to functional creation.

Meta-GGAs are promising because the kinetic energy density allows some discrimination between areas with one or two electrons (in this way, they are similar in some ways to the electron localisation function, or ELF, which can be used to analyse bonding[2]). This gives some hope that they may be able to fit both strong and weak bonding as well as possibly mitigating the self-interaction error that plagues DFT.

The Minnesota meta-GGA MN15-L[3] takes a very complex functional form with 58 parameters and a non-separable form for the exchange-correlation (adding an extra correlation functional to this) and fits it to a portion of a very large database (I estimate that there are at least 900 entries in the database, covering many different chemical properties, specifically including solid state properties, transition barriers and weak interactions). The resulting functional is local (no hybrid or non-local van der Waals terms were included) and produces extremely small errors when compared to those parts of the database which were not used in fitting. Notably, it out-performs functionals with exchange and van der Waals included.

By contrast, the SCAN functional[4] uses only seven parameters (close to, if not at, the minimal number for a meta-GGA) and includes various physically motivated norms and constraints for the electron gas. The early progress in GGAs was made by satisfying important constraints, so this is seen as a good route to reliability. This functional was also tested on various databases, particularly focussing on solid-state and weak interactions. It has excellent agreement with these, though was published before the MN15-L functional so is not compared directly. In a follow-up paper[5] the performance of the SCAN functional for band gaps is found to be good, though it does not calculate Kohn-Sham gaps, but gaps within a generalized Kohn-Sham theory (a distinction which I don’t have time to discuss here; I may write another blog on this, as it is relevant to hybrid functionals among other things.)

What can we learn from this ? Both functionals perform well in the tests which are published. The functional that you choose will depend in part, as always, on which system you wish to study; however, both of these functionals show some promise in being widely applicable. Your choice will also depend on your attitude to fitting[6]: is a reasonable functional form with many parameters something that you trust, or do you prefer to be more prescriptive, and deal with fewer parameters ? Fifty years since its inception, DFT is still developing, communities are still somewhat divided in the approaches that they take to functional development, but there are an increasing number of ways to achieve efficiency and accuracy.

[1] My colleague, Mike Gillan, reckons that we should instead talk about wrestling Jacob when considering how to improve DFT functionals.
[2] J. Chem. Phys. 92, 5397 (1990) DOI: 10.1063/1.458517
[3] J. Chem. Theor. Comput. 12, 1280 (2016) DOI: 10.1021/acs.jctc.5b01082
[4] Phys. Rev. Lett. 115, 036402 (2015) DOI: 10.1103/PhysRevLett.115.036402
[5] Phys. Rev. B 93, 205205 (2016) DOI: 10.1103/PhysRevB.93.205205
[6] The oft-quoted maxim about fitting an elephant with four parameters has been put into practice (see the original paper here and a nice write-up and a python implementation here)