Which is better for binding energies: DFT-D3 or RI-MP2?

Question

I am trying to calculate the binding energy of a small molecule with an underlying surface. I have used B3LYP-D3BJ (as implemented in ORCA) at def2-TZVP basis set to optimize the geometry of the VdW complex, and then performed a single point Binding Energy calculation using RI-MP2 at def2-TZVP basis set. However, I am concerned about the accuracy of the binding energy that I have obtained from RI-MP2 calculations. Should I calculate the binding energy using DFT-D3 or will the energies from RI-MP2 be more accurate?

Answer 1

Summary of my recommendation

If my choices are B3LYP-D3 and RI-MP2 for calculating a binding energy at a molecule whose geometry was optimized with B3LYP-D3, I would find it easier to justify using B3LYP-D3 for the binding energy. B3LYP-D3 was found in the study summarized below, to be more accurate than RI-MP2 for "basic properties" such as atomization energies, and that is without the RI approximation for MP2, and it's for extrapolated CBS estimates for MP2 rather than the single def2-TZVP basis set that you're using.

If somehow for your specific system's binding energy, RI-MP2/def2-TZVP is more accurate than B3LYP-D3/def2-TZVP and def2-TZVP is the biggest basis set that you can use for either of them, even if you were to do a (def2-TZVP,def2-DZVP → CBS) extrapolation for both RI-MP2 and B3LYP-D3, the results below show that other dispersion-corrected hybrid functionals with a similar cost to B3LYP-D3 are likely to beat MP2 by quite a lot (and therefore also RI-MP2 unless something fortuitous and unpredictable happens) in every category presented in the summarized results.

Results from Goerigk and Grimme (2011)

As summarized in my answer to What are some recent developments in density functional theory?, a 2011 study involving 841 relative energies and comparing more than 20 different hybrid functionals, for a variety of different properties, found that overall B3LYP-D3 was slightly worse than the average hybrid functional calculation (all hybrid calculations included the -D3 correction but the suffix was dropped for diagram), which had an average discrepancy of around 3.3 kcal/mol with the benchmark energy differences (mainly coming from experiments or CCSD(T)/CBS estimates):

                            

It would be great if we could know where MP2 lies on a histogram for exactly the same dataset. In fact the original paper has the following comparisons between MP2 and DSD-BLYP-D3:

DSD-BLYP-D3 is a "dispersion-corrected, spin-component-scaled, double-hybrid (DSD)" functional, and since you're using B3LYP-D3 which is a single-hybrid functional, it would be prudent for you to ignore the DSD-BLYP-D3 results in the above histogram. Instead let us compare the MP2 numbers in the above histograms with the B3LYP-D3 numbers in the histogram below (again, all calculations included a -D3 correction, but the suffix was dropped in the labeling of the functionals):

This leads me to the following summary for average discrepancies with the benchmark numbers:

Method	Basic Properties	Reaction Energies	Non-covalent interactions	Complete Set
MP2	5.7 kcal/mol	3.6 kcal/mol	0.90 kcal/mol	3.6 kcal/mol
B3LYP-D3	5.0 kcal/mol	4.7 kcal/mol	1.10 kcal/mol	3.7 kcal/mol

My first observation is that B3LYP-D3 is worse for everything except for "basic properties".

So what are "basic properties"? They are described in the paper as "atomization energies, electron affinities, ionization potentials, proton affinities, SIE related problems, barrier heights".

You seem to be interested in "binding energies", which are similar to "atomization energies", which is where B3LYP-D3 wins against MP2, and would probably win even more after we consider:

• that the MP2 results in the above histograms are for CBS estimates and your MP2 calculations are using just the def2-TZVP basis set with no extrapolation, and
• that the MP2 results in the above histograms are from full MP2 rather than the RI-MP2 approximation to full MP2 (and you are using the latter),
• that you optimized the geometries with B3LYP-D3, so your B3LYP-D3 binding energies would most likely have an inherent advantage over your MP2 ones.

I think it would be harder to justify using RI-MP2 for the binding energy after optimizing the geometry with B3LYP-D3 rather than maintaining consistency by using B3LYP-D3 for both the geometry optimization and the binding energy calculation. I guess you chose to use RI-MP2 because it's a more expensive method, with a formal scaling of O(N^5) as opposed to B3LYP-D3 whose cost is probably dominated by the HF part which has a formal scaling of O(N^4), but B3LYP-D3 according to this 2011 study, B3LYP-D3/(aug-)def2-QZVP is more accurate than MP2/CBS for "basic properties", and some of the other dispersion-corrected hybrid functionals that cost almost the same as B3LYP-D3, beat MP2 in every single category in this study.

Other findings

When I searched DFT vs MP2 on Google, the first page of results gave me the following 10 results, in order:

• Performance of Møller-Plesset second-order perturbation theory and density functional theory in predicting the interaction between stannylenes and aromatic molecules
• What are the principle differences between DFT/DFT-D and MP2? [ResearchGate forum]
• Evaluation of MP2, DFT, and DFT-D Methods for the Prediction of Infrared Spectra of Peptides
• Performance of Hybrid DFT Compared to MP2 Methods in Calculating Nonlinear Optical Properties of Divinylpyrene Derivative Molecules
• DFT functionals from MP2 methods [MMSE, this is the only "irrelevant" result from the first page of Google search results in my opinion, maybe Google just knew I love MMSE].
• https://chemistry.stackexchange.com/q/117844/27342 [Chemistry Stack Exchange]
• MP2, DFT and ab initio calculations on thioxanthone
• What are the principle differences between DFT/DFT-D and MP2? [ECHEMI forum, with no comments on the question yet].
• MP2 or dispersion-corrected DFT-functionals? [multiwfn forum, with answers that support my above conclusions].
• Why is MP2-Water "Cooler" and "Denser" than DFT-Water?

When I searched MP2 kcal/mol accuracy on Google, the first page of results gave me the following 10 results, in order:

• Can spin-component scaled MP2 achieve kJ/mol accuracy for cohesive energies of molecular crystals?
• Pragmatic ab initio prediction of enthalpies of formation for large molecules: accuracy of MP2 geometries and frequencies using CCSD(T) correlation energies
• Quantum chemical accuracy from density functional approximations via machine learning
• Quantum chemistry composite methods [This is the most irrelevant result in this list, in my opinion].
• Accurate Calculation of the Heats of Formation for Large Main Group Compounds with Spin-Component Scaled MP2 Methods
• Accurate Intermolecular Interaction Energies from a Combination of MP2 and TDDFT Response Theory [This one is the second-most irrelevant result in this list, in my opinion].
• This one was a repeat of "Accurate Calculation of the Heats of Formation for Large Main Group Compounds with Spin-Component Scaled MP2 Methods" but on ResearchGate instead of the ACS journal website.
• RMS Deviations (kcal/mol) for the MP2-F12 correlation component of cohesive [Table from "Conventional and Explicitly Correlated ab Initio Benchmark Study on Water Clusters: Revision of the BEGDB and WATER27 Data Sets"].
• New accurate benchmark energies for large water clusters: DFT is better than expected

Related MMSE posts

Ideally these would appear in the "Related" section in our sidebar, but at the moment they don't:

• Does the RI method actually accelerate basis convergence of MP2, "by one more zeta"?
• Does the accuracy of RI-MP2 follow the higher number of zetas of the main and fitting bases?
• Does NWChem support RI approximations in the MP2 part of double hybrids?
• Are frozen-core approximations for the elements of first or second row accurate for potentially extremely short bond distances present in the molecule

RI-B2PLYP-D3 is mentioned in the last of these, and in theory it would have a similar cost to RI-MP2 since the dominant-scaling part of B2PLYP-D3 is the MP2-like calculation. Since in this comment you were worried about the cost of B2PLYP, RI-B2PLYP-D3 might be a good option for you too.