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This question arises from the need I have to prepare a lesson on the limitations of Density Functional Theory as a computational approach. I would like to know not only the limitations, but also reference texts I can use to prepare a more appealing and complete lesson.

Thanks

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    $\begingroup$ Do you mean Density Functional Theory or Discrete Fourier Transforms? You might edit the question body text to specify. $\endgroup$ – Bill Barth Dec 29 '11 at 17:43
  • $\begingroup$ Assuming DFT means Denisty Functional Theory, do you mean numerical limitations or physical limitations? $\endgroup$ – Dan Dec 29 '11 at 20:20
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    $\begingroup$ @Bill Density Functional Theory, sorry. $\endgroup$ – Stefano Borini Dec 29 '11 at 20:48
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    $\begingroup$ As a more general comment, though, it's not clear to whom you want to give this lesson? Is it a broad-spectrum audience? Are these computational science graduate students? Undergraduates? As a general rule, it's hard to say how much detail to go into (or not to go into) without more details. $\endgroup$ – aeismail Dec 29 '11 at 21:49
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    $\begingroup$ That is a wonderful question, however I feel that you need to clarify a few things: 1. What is the background of your audience? Chemists and physicists view different characteristics of DFT as limitations... CS people have an entirely different view on DFT computations altogether. 2. What is the breadth of the talk? 45 minutes is the ideal time for these sort of things. 3. Do you have any preference on the type of the limitations? i.e. computational feasibility or predictive power? $\endgroup$ – Dr_bitz Dec 29 '11 at 21:49
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One of the biggest limitations of density functional theory is that it doesn't correctly treat the exchange interaction. While it has been shown that there exists a functional that will treat exchange correctly, this functional is unknown and semi-empirical methods are used to make approximations to it.

The coulomb part of the functional that I see most often is also not exact, but I don't know whether the exact coulomb term is unknown or just has some unpleasant numerical property that keeps people from using it.

The only part of the Hamiltonian functional that is (as far as I know) exact is the term for the interaction with the external potential.

In either case, developing better approximations for the electron-electron coulomb and exchange terms is still an area of active research.

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    $\begingroup$ In what sense is the Coulomb part of DFT not exact? The only approximation I see is the use of a finite basis set. $\endgroup$ – Jeff Apr 13 '13 at 20:28
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    $\begingroup$ @Jeff The Coulomb integral used in DFT is non self-interaction free: it's calculated as the integral of the product of all the density at r with all the density at r'. The self-interaction correction term resembles the Hartree exchange term but is not the same. See, e.g. the Perdew-Zunger PRB paper from 1981. $\endgroup$ – Miguel Dec 22 '15 at 10:54
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An important limitation to density functional theory is DFT's characteristically poor treatment of long-range noncovalent interactions. Many functionals give very incorrect results for $\pi\cdots\pi$ stacking, hydrogen bonding and noble gas VdW dimerisation, however quite recent functionals such as the 'Minnesota' family of Trular and coworkers are specifically designed with ameliorating these drawbacks in mind, and they have been successfully used in modelling pathological cases like supramolecular complexation of buckyballs to corranulene-bearing 'buckycatchers'. In my (limited) experience, however, implementations of the meta-hybrid M06 functional are rather computationally expensive with respect to GGAs. Meanwhile, the pairwise-parameterised DFT-D methodologies of Grimme and coworkers represent another promising avenue.

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If you're looking for a general reference, I would recommend the book by Parr and Yang, Density Functional Theory of Atoms and Molecules. I used this book when I sat in on an advanced course in chemical kinetics which discussed how to use quantum chemistry to calculate energies and reaction rates. It's a reasonably complete overview of DFT, although it is somewhat outdated, as it was first published way back in the late 1980's.

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To specifically answer the question: The main shortcoming of Density Functional Theory is that even though it is a formally exact reformulation of quantum theory, in the current state of the theory, approximations are required for the Exchange-Correlation energy functional. All the Density-Functional approximations that we have so far fail to exactly reproduce the contributions from different phenomena to the Exchange- and Correlation-Energies.

As discussed by Cohen, Mori-Sánchez, and Yang in a paper titled "Insights into Current Limitations of Density Functional Theory" in the journal Science 2008 most of the weaknesses can be traced back to two main errors of standard density-functionals: The delocalization error and the static correlation error.

One has to read the paper to understand the details, but in a hand-wavy explanation what they say is that when using DFT, the electron density (or electron cloud) is artificially spread-out due to an incorrect behavior of the standard functionals.

This problem has its root in the fact that when using DFT even if you have only one electron, the density of that electron (a non-local object) interacts with the electron itself (a local object) producing an artificial repulsion of the electron caused by itself. An analogous situation happens with spin-spin interaction.

This is an artifact on the formulation of the exchange-energy functional (exact) that the correlation-energy functional (approximated) cannot correct in any of the functionals that we have prepared so far, including the fanciest ones, i.e. the "Minnesota" family.

This is reflected in the underestimation of the barriers of chemical reactions, the band gaps of materials, the energies of dissociating molecular ions, and charge transfer excitation energies. Density-Functional approximations also overestimate the binding energies of charge transfer complexes and the response to an electric field in molecules and materials.

Another practical issue in DFT is that it is not variational, which is a fancy terminology to say that if you use one of the simplest functionals and you get a some answer, you are not guaranteed to improve it by using a more complicated functional. Choosing a functional is a matter of experience and sometimes, luck.

Even though all of this may sound really bad, it surprising how DFT works much better and/or faster than other computational quantum methods modelling many different properties important for physics, chemistry and materials sciences.

For more details, I would also recommend the book by Parr and Yang, Density Functional Theory of Atoms and Molecules.

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From Chapter 9 of Bob Gotwal's lecture notes on Computational Chemistry for Chemistry Educators:

Advantages

The most significant advantage to DFT methods is a significant increase in computational accuracy without the additional increase in computing time. DFT methods such as B3LYP/6-31G(d) are oftentimes considered to be a standard model chemistry for many applications.

Disadvantages

One of the main disadvantages of DFT methods is the challenge in determining the most appropriate method for a particular application. The practitioner should, prior to choosing a DFT method, consult the literature to determine the suitability of that choice for that particular problem or application. As such, DFT usage tends to favor the more sophisticated user. In general practice (including educational environments), the B3LYP/6-31g(d) model chemistry is considered by most to be a good general-purpose choice

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    $\begingroup$ Welcome to Scicomp.SE! The question explicitly asks for reference texts; could you edit your answer to include pointers to literature where one could read up on the comments you make? $\endgroup$ – Christian Clason Feb 16 '16 at 14:08
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    $\begingroup$ @ChristianClason it just so happens that this answer is copied verbatim from page 1 of this report ... $\endgroup$ – GoHokies Feb 18 '16 at 14:02
  • $\begingroup$ @GoHokies Oh, that's disappointing (and dishonest). I'll try to find a proper reference for this, and edit it in (feel free to do it yourself, if you already have this information). $\endgroup$ – Christian Clason Feb 18 '16 at 14:46
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The exact exchange functional is known, this is the HF exchange energy. The problem here is that it depends explicitly on the orbitals and not in the electronic density as the usual functionals. By means of the Optimized Effective Potential Method is possible to evaluate exactly the exchange energy in the frame of DFT. This has been successfully done for atoms and molecules and more recently for periodic systems. So I think the problem is not anymore in the exchange, it is in the correlation functional that is still unknown and we have to use approximations. But most of the time correlations effects are really small and the aproximations are pretty good.

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DFT creates a basis for multiscale simulations molecular dynamics, Monte Carlo methods. But it rigorous methods and extensive approaches to model a atomic system using quantum mechanics.

Main drawbacks would be the timescale FEMTO SECOND to PICO SECOND only and expensive. But these are helpful in providing insights at atomic scale.

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  • $\begingroup$ Welcome to SciComp.SE. I think that your post is not answering the main question that is addressed. $\endgroup$ – nicoguaro Feb 18 '16 at 21:11

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