# Are dispersion correction methods for DFT (such as D3) useful for geometry optimization?

For what I know the DFT-D3 method or similar corrections that take into account dispersion effect, add this type of contribution through calculation of atom pairwise interactions at the end of an energy calculation by simple addition. I would like to know if I'm right, because in this case I think it cannot affect the final geometry in a geometry optimization. I'm asking this question because usually all these methods are benchmarked only for energy calculation and never for geometry optimization. I hope to have explained well my answer.

• If a method gives good energies, doesn't that mean it would give good geometries too? Because the geometry is found by minimizing the energy after all. May 7 '21 at 8:20
• Yes, but I don't understand if with these methods the additional energy is added at the end of each step for geometry optimization and then taken into account in the next step of geometry optimization. May 7 '21 at 8:28
• It depends on the implementation, really. In some codes, and with some dispersion corrections, you only get the energy added to the final energy. In some other codes, the energy is added to each optimization step. In many other cases, force contributions are analytically available from the dispersion method. It is also very possible that even within the SCF cycle you already have a dispersion term. So to know what to expect, you need to consult your specific method's documentation in your code.
– user430
May 7 '21 at 13:49
• That is what I'm afraid of, I would like to know if it's always implemented at the code level or it depends on the choice of the developers. Most of the time the code's documentation lack of these information that canake the difference. May 7 '21 at 15:23
• Yes! I think you should add that as an answer if you feel it is helpful. Another thing, if you are responding to someone in the comments, then make sure to ping them with @, otherwise they won't get any notification, and may not know you have responded. May 7 '21 at 21:22

Yes, vdW interaction affects energies but, also, geometries. This effect is the most pronounced if a DFT method predicts an interaction in a dimer to be repulsive: a geometry optimization will then simply dissociate the dimer. Normally, any vdW correction term does not only contribute energies, but als gradients, which in turn affect geometry optimizations (or MD simulations). Although I gather from other answers that some codes do not implement gradients, which I didn't know.

Proper benchmarking is hard for structures, because gas phase, zero Kelvin geometries (the thing you calculate) are hard to come by experimentally for somewhat larger molecules—unfortunately exactly the kind of systems where you would expect vdW interactions to be imporant. Nevertheless, certain benchmark sets have been proposed, one of them based on rotational constants (which can be measured, and also calculated from an optimized structure). Here's an example of such a benchmark calculation, from Grimme & Steinmetz, Phys. Chem. Chem. Phys., 2013, 15, 16031-16042. You can see that dispersion corrections tend to make molecules "smaller" or more compact:

Updated versions of these test sets have also been published, and are being used more routinely. Grimme likes to use these a lot for testing his "low cost" methods, which are intended for rapid geometry optimizations in large systems. One recent example is the testing of r2SCAN-3c.

Also, if you consider molecular crystals to be "big molecules," you can see that proper vdW treatment is important to get correct lattice constants, for example for ice.

Dispersion correction DOES affect the final geometry.

The dispersion interaction is a function of geometry. A crude approximation is to write the dispersion interaction between two neutral molecules as $$\Delta E_{\rm disp}(R)=-\frac{C_6}{R^6},$$ where $$R$$ the distance between the two entities.

In some special case, like the system in which two rare gas atoms interacts with each other, the potential energy curve given by functional like B3LYP has no minimum, which is quantitatively wrong. However, if a dispersion correction is used, an acceptable energy minimum can be calculated and the geometry optimization can be done.

More detailed theory is well written in Prof. Grimme's review.

• Yes, that's the ideal case. But in reality it very much depends on the implementation used - dispersion corrections are surprisingly all over the place between DFT codes.
– user430
May 7 '21 at 13:55
• I would note that the dispersion correction is also a function of the charge of the system (expecting more dispersion if there are more electrons) and probably also of the electronic state (with excited, more diffuse states providing more dispersions because they are probably more polarizable). May 7 '21 at 17:26

Optimization routines typically use the negative gradient of the energy in some form to determine the displacement during an optimization step and the dispersion correction changes the energy, which changes the gradient, and thus the optimization.

You may get the same result in some cases but those are special cases, in general dispersion correction does have an effect on the optimization. The effects are often very noticeable if you have large molecules or two or more monomers that are calculated together. In such cases dispersion correction can have a large effect on the optimized geometry and can be essential to get good results.

Energy is simply the most common metric for benchmarks and it is easier to compare a single number than a whole geometry. However, I am not a specialist in benchmarking, so there could be other reasons for the prevalent use of energy. But the prevalence of energy as metric in benchmarks certainly does not mean that dispersion correction has no effect on geometry.

• What about the layer distances in graphene that where better simulated using dispersion corrections?
– Camps
May 7 '21 at 16:43
• @Camps ? I am not sure if you are asking me something. But dispersion correction is certainly relevant for large systems like graphene, or other extended systems. Also for benzene crystals or organic liquid crystals. May 7 '21 at 17:11
• Right. I think that good energy evaluation need to start from good geometries. I'm really convinced that unfortunately there is not so much attention in geometries, but only in single point energy. This question arise from the fact that in one of my research for the moment only few functionals, together with high level dlpno-ccsdt, are able to predict the geometry of one molecule I'm working with. My problem is that I'm and independent scholar and I've not a lot of time to dedicate to this research. But in my free time I want to dig deep to my problem. May 7 '21 at 17:12

Additive dispersion correction methods such as D3, D3(BJ), D3(BJ)+ATM, TS, MBD@rsSCS, etc. are good for geometry optimizations. These methods find the London dispersion interactions governed interlayer distance lower than the mean relative error of 2%. On the other hand, it's a good idea to avoid using these methods for energy calculations. For example, these methods overestimate the interlayer binding and exfoliation energies of 2D materials between MRE 20% and 80%

There are just a few methods available for both reliable geometry and energy calculations. High-level ACFDT-RPA (Computationally very expensive), SCAN-rVV10, and PBE-rVV10.

tldr; Use dispersion correction for optimization

I asked a very similar question a few years ago. Many indications for dispersion corrections in density functional methods are for intermolecular interactions. Clearly these corrections are important in such intermolecular or long-range interactions (e.g., inter-layer spacing in a material.. far from the bonding regime).

Kristof Bal's answer gives one insight, based on molecular rotational constants.

Our group published a benchmark, considering ~6500 conformer geometries across ~650 molecules. Using high-level DLPNO-CCSD(T) / def2-TZVP energies, we found a huge increase in accuracy (from several metrics) when using dispersion correction at minuscule cost in time.

'Assessing conformer energies using electronic structure and machine learning methods' Int. J. Quantum Chem. 2020

For example, the mean $$R^2$$ correlation between B3LYP and DLPNO-CCSD(T) energies was 0.706 without dispersion and 0.920 with -D3BJ dispersion correction. Similar effects were seen with other functionals.

Our conclusion is that intramolecular non-bonded interactions are handled better, reflecting medium-to-long range behavior of standard chemical functionals.

• I just learned about your work a few days before. Nice work! In particular the linear correlation between median R2 and the logarithm of computational time, over such a diverse set of methods, is really inspiring May 9 '21 at 9:30