# When and Why does Density Functional Theory (DFT) fail?

Every method has its strengths and weaknesses. For instance, a strength of DFT is that is has HF like speeds, but can also account for electron-correlation and this is a pretty big feature since methods that account for electron correlation typically either require more than 1 Slater determinant (Configuration Interaction, Coupled Cluster etc.), or involve expensive perturbation about a HF reference system (Many Body Perturbation Theories).

A result of DFT being so pound-for-pound good is that its popularity is soaring.

Given the relative ease in which a DFT calcualtion can be performed, this makes it perfect for "turning the crank".

There are however instances where DFT fails or has caveats.

## What are the systems/phenomena/caveats users should be aware of when modelling materials with DFT?

For instance, one caveat I know of is that because of the complexity of the exchange/correlation functionals, they must be numerically integrated. This means a grid-size must be set, and while programs such as Gaussian allow the user to set the grid-size, generally, a default grid-size is used unbeknownst to the novice user.

• I can only reiterate: Density Functional Theory is in principle exact, what you are talking about are methods implementing it. I think for some specific functionals plenty of cases are already documented, take b3lyp you can find hundreds of papers describing its failures. All things considered, I think this question needs clarification and focus. – Martin - マーチン May 12 '20 at 9:52
• @Martin-マーチン What I am after is general tripping hazards. DFT in general can not do as a simple example H2+. Relative energies of states with different spin mulitiplicity are commonly poorly described. These are just two things from Frank Jensen's single page dedicated to DFT problems. Given that his book is introductory I imagine there are more. Further, even if there isn't, alot of novices don't even have his book, so this would be a decent spot just for the basics. I don't like seeding my own answers, hence why I asked but do not answer. – B. Kelly May 12 '20 at 17:46
• Since hybrids are the most common (B3LYP/PBE0/$\omega$B97XD) a response specifically aimed at hybrids would suffice. – B. Kelly May 12 '20 at 17:57

First, a word of caution: it is hard to generalize since there are so many different approximations to the exact exchange-correlation functional. Nonetheless, in my opinion:

1. The biggest weakness of all existing (and arguably all plausible) implementations of DFT is their limited predictive power. In practice, this means that you need to know a lot about your system to choose the right method (functional) and you can only sort of trust the answer (as evidenced by the variety of answers you get when using different functionals). More precisely, your degree of confidence decreases as you move to more "weird" systems, that are not usually used to inform the construction of the exchange-correlation functionals.

2. Density functionals are not systematically improvable. There is no guarantee that using the density functionals higher up on the Jakob's ladder will give a more accurate answer. This is fundamentally different from the coupled-cluster or configuration interaction approaches. For a recent long-read review check out this work from Martin Head-Gordon group

3. Finally, the grid convergence you mentioned is a big issue for an average user that is treating DFT implementations as black-box methods. Steven Wheeler has explored this recently.

It is very important to differentiate between Density Functional Theory (DFT) and Density Functional Approximation (DFA). DFT is an exact theory and if we know the exact formulation for exchange-correlation functional, we should get the exact solution. However we don't have the exact formulation and hence we choose different approximations for it, essentially making it a DFA. And the failures are that of DFA and not DFT.

Quoting Becke [1]

Let us introduce the acronym DFA at this point for “density-functional approximation.” If you attend DFT meetings, you will know that Mel Levy often needs to remind us that DFT is exact. The failures we report at meetings and in papers are not failures of DFT, but failures of DFAs.

The known failures of DFAs like the lack of long-range correlation or dispersion interactions and the spurious self-interaction error are addressed by specialized developments for specialized materials (DFT+U, DFT+vdW).[2]

References

1. Becke, Axel D. "Perspective: Fifty years of density-functional theory in chemical physics." The Journal of chemical physics 140.18 (2014): 18A301.

2. Maurer, Reinhard J., Christoph Freysoldt, Anthony M. Reilly, Jan Gerit Brandenburg, Oliver T. Hofmann, Torbjörn Björkman, Sébastien Lebègue, and Alexandre Tkatchenko. "Advances in density-functional calculations for materials modeling." Annual Review of Materials Research 49 (2019): 1-30.

Known failures of density functional approximations (DFAs) include anions, charge transfer systems and point defects (e.g. vacancy states). These are mainly due to self-interaction error, which can be mitigated to some level with hybrid functionals and range-separated hybrids.

DFAs are also generally unreliable for systems with strong correlation, like many transition metal complexes.

These are problems also when the calculations are done correctly, i.e. at the complete basis set limit. (The quality of the results at any level of theory will suffer enormously if there are coarse errors in the computational paradigm, e.g. insufficient basis set, insufficient quadrature, insufficient k point sampling, etc.)

• +1. For anions Density-Corrected DFT has done a nice job of correcting the error, since in that case the density error often dominates the functional error. – Nike Dattani Jun 8 '20 at 14:52

DFT is single effective correlated particle theory

Problems that can be described by single determinant theory DFT in principle should able to provide a good description given that exact form of xc functional is known. It is not the problem of DFT that it fails. Failure is due to approximate nature of xc functional. One should in KS-DFT (one that uses explicit density dependent form of xc), non-local potential is approximated my local form, it might work certain problems not for other problems. That is why optimized amount of non-local (HF) exchange help address some problem. Please have a look at article titled "Increasing the applicability of density functional theory. III. Do consistent Kohn-Sham density functional methods exist?" https://aip.scitation.org/doi/abs/10.1063/1.4755818

DFT can break down (like all numerical methods) if you want to model a too-large or too-complicated system. This is especially relevant if you want to study impurities, where periodic boundary conditions are less helpful.

The exchange correlation functionals are a key weakness for DFT, since they are empirical approximations. Therefore the method may encounter trouble when trying to model materials where electron-electron correlations are very important.

• This generalises beyond the useful. DFT is not a numerical method, it's a theory. What you mean are implementations of it. – Martin - マーチン May 12 '20 at 9:54
• Exchange-correlation functionals are not empirical as a rule, they are usually constructed to obey strict properties of the (unknown) exact functional and in most cases they are exact in at least one limit and/or model system. – Phil Hasnip May 13 '20 at 23:20