# What quantity to perform CBS extrapolation on for adsorption/binding energies?

When calculating complete basis set extrapolations for quantities, such as adsorption energies or binding energies, which depend on the difference between three quantities (i.e. $$E_\textrm{prop} = E_\textrm{AB} - E_\textrm{A} - E_\textrm{B}$$), should the extrapolation be performed on $$E_\textrm{prop}$$ or the individual terms ($$E_\textrm{AB}$$, $$E_\textrm{A}$$ and $$E_\textrm{B}$$)?

Just for more information, the specific CBS extrapolation I am looking at are the default two-point extrapolation formulae, used in ORCA, for basis sets of cardinal numbers $$X$$ and $$Y$$: $$\begin{equation*} E^\textrm{CBS}_\textrm{HF} = E^X_\textrm{HF} - \dfrac{E^Y_\textrm{HF} - E^X_\textrm{HF}}{\exp(-\alpha \sqrt{Y}) -\exp(-\alpha \sqrt{X})}\exp(-\alpha \sqrt{X}), \end{equation*}$$ $$\begin{equation*} E^\textrm{CBS}_\textrm{corr} = \dfrac{X^\beta E^X_\textrm{corr} - Y^\beta E^Y_\textrm{corr}}{X^\beta - Y^\beta}, \end{equation*}$$ where $$\alpha$$ and $$\beta$$ have been parameterised for various basis sets in this paper

Extrapolating $$E_{\mathrm{prop}}$$ is completely equivalent to extrapolating the individual terms. This follows simply from the fact that all "sensible" basis set extrapolation formulas are a linear combination of the finite basis set energies used for extrapolating the CBS energy. You can easily see this fact from Eqs. (3-4) of the paper posted by Nike Dattani. If the extrapolation formula tells you, e.g. $$E_A(\mathrm{CBS}) = 1.5E_A(\mathrm{QZ}) - 0.5E_A(\mathrm{TZ})$$, for arbitrary $$A$$, then from you expression of $$E_{\mathrm{prop}}$$, it must be that $$E_{\mathrm{prop}}(\mathrm{CBS}) = 1.5E_{\mathrm{prop}}(\mathrm{QZ}) - 0.5E_{\mathrm{prop}}(\mathrm{TZ})$$.

And I said "all sensible basis set extrapolation formulas", because if there is a basis set extrapolation formula that is not a linear combination of the energies it uses, then the extrapolated energy is not size-consistent (although such kind of formulas do exist and are not uncommon - thank Tyberius for pointing out that!). This is because size-consistency dictates that the extrapolated energy of a system composed of two arbitrary non-interacting molecules $$A$$ and $$B$$ must be the sum of the extrapolated energies of $$A$$ and $$B$$. As the finite basis set energies of $$A$$ and $$B$$ can adopt any value, the only way that an extrapolation formula always respects size-consistency is that it does not contain any nonlinear and/or cross terms of $$E_A$$ and $$E_B$$.

• While size consistency is important, especially for solids, it is not necessarily essential. Many molecular calculations are still carried out with truncated forms of CI, which are not size consistent. Nonlinear extrapolation schemes are not uncommon either. An exponential extrapolation formula has seen use for computing CBS correlation energies.
– Tyberius
May 19, 2021 at 20:01
• +1 for the insight about extrapolation schemes often involving a linear combo, but non-linear ones do exist, and the comments by the OP on my answer, suggest that there's some extrapolation formula that they were considering using for $E_{\textrm{prop}}$, where $E_{\textrm{prop}}$ would be calculated with multiple basis sets and then the CBS limit would be calculated using these energies and what the OP calls "parameters" --- I think we would need to see that formula before saying that it's entirely equivalent to extrapolating the individual terms separately then doing $E_{AB} - E_A - E_B$. May 20, 2021 at 0:21
• Thank for for this clear answer. The extrapolation equations (now added to the question to make it clearer) which I am using for both HF and correlation energies are indeed a linear combination, so extrapolating $E_\textrm{prop}$ is equivalent to extrapolating the individual terms that make it up -- something I also double-checked using my own numbers. May 20, 2021 at 7:40
• @Tyberius Sorry I was wrong in saying that non-size-consistent extrapolation formulas cannot pass peer review at all (why did I say this...), although I personally still dislike them. I have edited my reply to reflect what you said May 20, 2021 at 8:17

A lot of basis sets such as the Dunning family of basis sets (e.g. cc-pVXZ) are designed with a specific goal in mind, which is not necessarily to give the lowest energy for a certain number of orbitals, but to achieve smooth extrapolations to the CBS limit for some properties.

It would be very hard to optimize a basis set so that $$E_{AB} - E_A - E_B$$ extrapolates smoothly to the CBS limit, for every single $$(A,B)$$ pair: even if you consider just 50 elements in the periodic table, there's 2500 pairs, and if you consider the essentially countless number of possible molecules you can make with 50 elements, you can see why basis sets would be optimized at the atomic level. In most basis set families, there's one basis set for each atom, and it's optimized using calculations only on that atom. It would be extremely rare to see a basis set optimized for an entire molecule or collection of atoms.

If you extrapolate $$E_A$$ to the CBS limit, then extrapolate $$E_B$$ to the CBS limit, then take the difference between the extrapolated $$E_A$$ and the extrapolated $$E_B$$, it may be more convenient for you and others, since your extrapolated $$E_A$$ and $$E_B$$ are in some sense more "fundamental" than an extrapolated $$E_{AB} - E_A - E_B$$ value, and the same is true for extrapolating each of $$E_A,E_B$$ and $$E_{AB}$$ and calculating $$E_{AB} - E_A - E_B$$ rather than extrapolating the complicated expression of $$E_{AB} - E_A - E_B$$.

In fact, I'm not sure what formula you would use to extrapolate $$E_{AB} - E_A - E_B$$ itself, because most extrapolation formulas work on individual energies like $$E_{AB}$$, $$E_A$$ and $$E_B$$. That being said, it is indeed possible that you find yourself in a situation where it's easier to get a good extrapolation for a property than for individual energies.

What worked very well for us in this paper in which the aim was to get an extremely accurate value for the carbon atom ionization energy, was that did a CBS extrapolation for the $$\ce{C}$$ atom and for the $$\ce{C^+}$$ ion with (aug-cc-pCV5Z,aug-cc-pCV6Z), with (aug-cc-pCV6Z,aug-cc-pCV7Z), and with (aug-cc-pCV7Z,aug-cc-pCV8Z), then we extrapolated the extrapolation by fitting a curve through the 3 extrapolations: see Figure 1 of the supplementary data here. So indeed you can try to fit the $$E_A$$, $$E_B$$ and $$E_{AB}$$ values for multiple basis set sizes to a curve and extrapolate that way, but the simplest and most common way would be to extrapolate $$E_A$$ and $$E_B$$ using something like Eqs. 1-4 in the first link above, and then approximate the CBS limits of $$E_{AB}$$ and $$E_{AB} - E_A - E_B$$ based on those extrapolated values!

• In my case, system A is a molecule and system B is a surface, with system AB or more aptly A+B being the adsorbed molecule on the surface, so no atoms are involved. In this case, I presume it is still more 'fundamental' to extrapolate the individual energies of systems A (molecule), B (surface) and AB (molecule + surface)? May 19, 2021 at 14:18
• Yes it's more "fundamental" because if you extrapolate molecule A and surface B and system AB individually, then someone else who studies molecule A bound to surface C, or someone else who studies molecule B bound to surface D, may be able to learn something from the results you got with a similar system. Whereas if you just calculate AB-A-B for a bunch of basis sets and fit a curve through the results to extrapolate, your extrapolation can't be broken down so easily into fundamental components that can help other people. How were you planning to extrapolate this "collectively" anyway? May 19, 2021 at 14:22
• Okay yes, that makes sense. I have been performing my CBS extrapolations on the individual terms so far so I have not thought too deeply about how I'm planning to extrapolate it 'collectively'. I was thinking of using the same equation (parameters taken from a paper) as I had done for the individual systems but have not done any tests so far and don't think I'll pursue it any further after your informative answers. May 19, 2021 at 14:32
• Thanks. Maybe you can put that paper in the question and point us to the parameters, because it sounds like something that I'm not used to (which is using Eqs 1-4 of my paper, which means zero parameters at all ... only the "raw" energies!). May 19, 2021 at 14:35