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Comparing wavefunction-based methods and density-based methods for quantum mechanical calculations, how is "rank" determined?

For example, in wavefunction methods, CCSD(T) is considered a higher level than MP2, which is a higher level than HF.

  • What about if I'm comparing MP2 to B3LYP? Or if I add dispersion corrections to the DFT method to use something like B3LYP-D3BJ?
  • What if I just want to compare different DFT functionals? Is hybrid higher than pure? Is B3LYP higher than PBE0?
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  • $\begingroup$ DFT doesn't fit well into the scheme, because it uses different approach comparing to post-HF methods. Personally I would consider it betwee HF and MP2, for all common functionals. Functionals themselves are not 'better' or 'worse' because all of them are semiempirical. $\endgroup$ – permeakra Apr 27 '17 at 19:55
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    $\begingroup$ A higher level of theory in QM calculations is always the one the anonymous reviewer of your publication claims is superior to the one you chose. $\endgroup$ – Todd Minehardt Apr 27 '17 at 20:27
  • $\begingroup$ @ToddMinehardt, that's funny. By the way, I noticed that you're a data scientist who once did theory. Can I ask how the transition went? $\endgroup$ – halcyon Apr 28 '17 at 2:14
  • $\begingroup$ @halcyon - It was a discontinuity, and there were/are pluses and minuses. I'd be happy to slate a chat with you in a chem.SE "room" as time permits. $\endgroup$ – Todd Minehardt Apr 28 '17 at 3:13
  • $\begingroup$ @ToddMinehardt oh god I cried. Kinda true though :( $\endgroup$ – Fl.pf. Apr 28 '17 at 12:45
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For single reference wavefunction based methods, the comparison is relatively straightforward within method "families". This is due to the nature of how these methods are formulated, usually involving a very long series that has been terminated at some point to make the calculation feasible.

You start with Hartree-Fock, which ignores electron correlation. After that you can add perturbative corrections of increasing order (this will be the MP2, MP3, MP4, MPn family), do something called configuration interaction, truncated at increasing levels of excitation (CIS, CISD, CISDT,...,FCI), or something called the coupled cluster methods, again truncated at some level of excitation (CCSD, CCSDT, CCSDTQ,...,FCI).

In these cases the most straightforward interpretation of "higher level" would mean a less aggressive truncation of the series that includes more terms. In case of CI and CC, this almost always results in more accurate results, and both converge to the same limit, where all possible terms are included and correlation effects are exactly calculated. (of course there are errors from other approximations) The MPn perturbation series has no such strong guarantees, and this is one of the reasons why pretty much noone uses anything beyond MP4 these days. (even MP4 is seldom used)

For DFT you have something called Jacob's ladder, but things are generally more murky, as most DFT functionals have many empirical (fitted) parameters, and there is no straightforward way of comparing them.

In general, the only way to compare them, is to take a big database of either experimental or calculated (via CCSD(T) or similar high level method) data, and compare the average/maximum errors.

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If you want to solely know about DFT functionals, read about Jacob's ladder of functionals. There's a good recent review of a slew of DFT functionals: http://dx.doi.org/10.1063/1.4948636

I do not think hybrid functionals are higher in theory. All that is being done is mixing exact exchange with other functionals that have already been established. But I think some people may argue that is not the case.

Another thing that you may want to look at are the "plus methods": DFT + GW, DFT + U, DFT + DMFT. These methods add a term to the Hamiltonian to correct errors or capture behavior that cannot be observed with traditional DFT.

Most comparisons between DFT and Post-Hartree Fock Methods are done by comparing calculation results. And this comparison is not very straightforward. Usually physical values (e.g. binding energy, bond lengths) are calculated by different method and compared to experimental values. Usually, however, wavefunction methods are deemed to be more accurate or consistent. So they are often used as benchmarks for DFT calculations.

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  • $\begingroup$ Jacobs ladder is highly subjective though, since DFT-Functionals mostly tend to do one thing pretty good and another thing horribly bad. $\endgroup$ – Fl.pf. Apr 28 '17 at 12:44
  • $\begingroup$ In general, GGA yields better results than LDA. The problem is that most "higher level" functionals such as meta-GGA are not very general. In some systems, they will be very good. But at some, they will underperform. The journal in my answer brings attention to this. It also brings attention to new meta-GGA functionals that are generally better than GGA. Hence the significance of the journal $\endgroup$ – CoffeeIsLife Apr 29 '17 at 3:20
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Aside from the used method, you should also pay attention to the used basis set. Generally, the larger the basis set the better the results. (This is strictly speaking not true, as just having a "large basis set" does not guarantee the additional basis functions are helpful. But usually basis sets are optimized to achieve maximum quality with a given number of basis functions.)

When comparing methods of different levels of theory, you should use a basis set of same quality. This will ensure the differences are really due to the method. In extreme cases a cheap method with a large basis set may give better results than a higher level method with a small basis set.

However, one should be careful with the "basis set of same quality" formulation, especially for DFT where thing are generally more fuzzy.

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    $\begingroup$ I've seen plenty of examples where smaller basis sets with DFT actually can perform better than larger ones. The reason for this is because DFT is not a convergent theory unlike ab initio approaches. $\endgroup$ – LordStryker Apr 28 '17 at 16:28
  • $\begingroup$ I work more with the ab initio approaches, so I don't know too much about DFT. But I can image this very well. However, I would also expect that in these examples not all molecular properties are better described at the same time? $\endgroup$ – Feodoran Apr 28 '17 at 18:32

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