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In Kohn-Sham DFT calculations, Van der Waals corrections are often implemented in the structure optimization calculations because the typical functionals such as LDA and GGA are found to not treat long-range dispersion forces faithfully. But even in the context of Van der Waals corrections, there are some options such as the DFT-D and DFT-D3 schemes (refer quantum espresso's documentation - under 'vdw-corr' tag).

What is the difference between these two corrections? Also, is it possible to conclude which one is more suited for a certain application?

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As you mention, there are many empirical dispersion corrections for density functional theory.

Generally, the term "DFT-D" refers to a generic dispersion-corrected density functional calculation, regardless of the specific method used for the dispersion correction used.

The D3 dispersion model is a specific dispersion correction method and is now something of a family, started by Grimme's 2010 article: "A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu" J. Chem. Phys. 132, 154104 (2010)

In most cases, it's recommended to use Becke-Johnson damping - D3BJ: J. Comput. Chem., 32: 1456-1465 (2011)

There have been a few efforts to improve these:

There are a wide variety of other methods, particularly those working to add many-body dispersion. Grimme wrote a review, although obviously it's been an active field:

"Dispersion-Corrected Mean-Field Electronic Structure Methods" Chem. Rev. 2016, 116, 9, 5105–5154

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    $\begingroup$ +1. We have a similar question that remains un-answered: materials.stackexchange.com/questions/63/…. I wonder what we can do about it: Is it something that you think can be answered without too much difficulty, or is its broadness (asking for a comparison for all dispersion corrections) something we should encourage the user to change (i.e. pick a few dispersion corrections and ask for comparison between those)? $\endgroup$ – Nike Dattani May 30 at 21:06

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