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It is understood that finding "the" universal functional is an NP-complete problem. However, progressively better functionals have been constructed decade after decade and fitted to more and more comprehensive datasets.

What is the closest humans have achieved, to a universal functional?

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I think this question somewhat comes down to what "camp" of DFT progression you subscribe to. I should specify upfront that this summary is mainly centered around molecular systems, so some of the recommendations likely vary for materials where the computational workload can often be much greater.

One side really emphasizes accuracy with respect to experiment and is somewhat less concerned with the physical interpretation of the functional form. These groups make efforts to directly improve the accuracy of functionals with respect to experiment by extensive fitting and parameterization. Some functionals that fit into this category would be the Minnesota Functionals from the Truhlar group, as well the ωB97X and ωB97M functionals from the Head-Gordon group. Based on fairly extensive benchmarking (see this excellent paper), these functionals are tough to beat for a wide variety of energetic metrics and types of molecules.

On the other side, the form of functionals is more physically motivated. This is done by ensuring the functional satisfies certain exact constraints of the "universal functional". A prominent example of this type is the SCAN functional from Perdew et al. While these types of functionals have not been able to achieve the same experimental accuracy as more heavily parameterized functionals, there is a chance that they are more robust and amenable to improvement, as they exactly match known properties of the "universal functional".

So it depends in what you are interested. If you want the closest functional form to the "universal functional", you would likely want something from the second camp. However, if your interest is in what will give you the best results for a wide range of complexes/materials right now, you will likely want to go with a functional that has been extensively parameterized on a large training set.

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    $\begingroup$ Great answer! To me an analogy is to compare this "parameterized" vs "not parametrized" paradigm to the machine learning concept of overfitting vs generalizability tradeoff. You may find this paper of interest "Density functional theory is straying from the path toward the exact functional" (and of course the Comments to the paper and the response to the comment ;)) $\endgroup$ – gogo May 22 at 12:48
  • $\begingroup$ One common criticism of the Minnesota functionals is the number of parameters. Some people accuse them of being over-fitted, though they give very good results. It is my understanding that the wB97 group of functionals use far fewer tuning parameters. So while both perform well, they do so by different means. And if my mind serves me right, my experience with wB97X was that they were slower than M06-2X when I was running calculations. $\endgroup$ – DetlevCM May 26 at 14:21
  • $\begingroup$ And it might be worth linking some benchmarks, Goerigkt et al. seem to be the most prolific, this is 2017, pubs.rsc.org/en/content/articlelanding/2017/CP/… here is 2011 pubs.rsc.org/en/content/articlelanding/2011/cp/… and I think there was an older one too... $\endgroup$ – DetlevCM May 26 at 14:24
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I'm not entirely sure what you mean by "universal".

If you mean a functional that can model a wide variety of materials with reasonable success, probably the closest we have are GGA functionals. They are not necessarily the most accurate, but they are used to regularly model metals and semiconductors. They get decent results, despite their known shortcomings (i.e. inaccurate band gaps). Hybrid functionals are increasingly becoming the norm, as we have fast computers that can deal with slow hybrid calculations.

If you mean a functional that is most accurate, there is no one functional that always works for every material. Presumably as you move up the DFT ladder, functionals get better. But this isn't always the case. For instance, hybrid functionals can get very different results for the same material. Some hybrid functionals may overestimate the band gap. Some underestimate. Some get the correct band gap.

My usual operating procedure is to look in the literature, see what has worked, and then use that as a jumping off point.

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    $\begingroup$ I believe he is using "universal" to mean the true density functional, which should in principle work for any system. $\endgroup$ – Tyberius Apr 28 at 20:59
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    $\begingroup$ @Tyberius is exactly correct. DFT in principle is "exact" provided that the "universal functional" is used. The problem is that we do not know what it is, so we have to approximate it, sometimes by fitting to empirical data. Often functionals are fitted for a particular purpose (like for transition metal chemistry, or for organic materials), so I would say a functional is "more universal" if it is closer to the "true" functional (meaning that it covers all scenarios). A caveat is perhaps what was pointed out in Tyberius's answer: there is usually a trade-off between accuracy and applicability. $\endgroup$ – Nike Dattani Apr 28 at 21:04
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    $\begingroup$ Fair enough. My answer addresses in part the idea of a functional working for any system. $\endgroup$ – che_kid Apr 28 at 22:56
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This paper published by Bikash Kanungo, Paul M. Zimmerman & Vikram Gavini provide an interesting solution to getting closer to a "universal functional"

Exact exchange-correlation potentials from ground-state electron densities

They have mapped very accurate electronic densities from ab initio full configuration interaction methods onto an exact exchange correlation functional.

The author's call this inverse DFT since it is going from electron densities to exchange potentials.

Specifically the authors believe the

inverse DFT problem to be instrumental in generating ${\rho(i),v(i)_{xc}}$ pairs, using $\rho(i)$’s from correlated ab-initio calculations. Subsequently, these can be used as training data to model $v_{xc}[\rho]$ through machine-learning algorithms which are designed to preserve the functional derivative requirement on vxc[ρ]. Furthermore, the xc energy ($E_{xc}[\rho]$) can be directly evaluated through line integration on $v_{xc}[\rho]$."

So maybe sometime soon we can expect a "Universal" DFT exchange functional which is born from exact ab-initio computations and an inverse-DFT generated functional.

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    $\begingroup$ This is definitely an interesting approach. There are already codes available out there to train neural networks from charge densities: Kolb, B., Lentz, L.C. & Kolpak, A.M. Sci Rep 7, 1192 (2017). doi.org/10.1038/s41598-017-01251-z $\endgroup$ – Kevin J. M. May 22 at 0:34

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