Do the cc/pc/def2 basis sets mathematically converge to the CBS limit, assuming exact CI/DFT?

Question

The cc-, pc- and def2- basis sets are often described as "systematic", in the sense that the results of these basis sets at different cardinal numbers (i.e. the number of ζ's) can be extrapolated, by a closed-form equation involving the cardinal number, to the CBS limit.

However, as a mathematics student, I began to wonder if these equations are mathematically exact. Of course, an exact CI is computationally prohibitively expensive, and an exact DFT is not even known in closed form, but the least of mathematical rigour is important to the minds of those who study mathematics.

My question is now clear: Assuming exact CI/DFT, can the sequences of the results of an ever increasing cardinal number of the cc/pc/def2 basis sets mathematically shown to converge exactly to the CBS limit, according to the "CBS extrapolation equations" known in the literature?

Answer 1

"Assuming exact CI/DFT, can the sequences of the results of an ever increasing cardinal number of the cc/pc/def2 basis sets mathematically shown to converge exactly to the CBS limit, according to the "CBS extrapolation equations" known in the literature?"

Given an $\epsilon > 0$, there exists an $N$ such that:

$$ \tag{1} |E_{\textrm{cc-pVNZ}} - E_{\textrm{Exact}}| <\epsilon. $$

So by the definition of convergence, it does converge.

This also means that pretty much any sensible function that you fit to the $E_{\textrm{cc-pVNZ}}$ sequence will give you an extrapolated energy that is within $\epsilon$ of $E_{\textrm{Exact}}$ provided that $N$ is large enough to satisfy Eq. (1).

However, if your $\epsilon$ is (for example) 1 micro-Hartree, the $N$ required will be huge, even for a single atom. I have done calculations with $N=10$ for triatomic molecules but beyond that it would be extremely impractical even to do the 2-electron integrals with currently existing software and hardware, so you won't likely be seeing CISD done with $N>10$ for more than 3 atoms, and for most molecules of interest you won't even be seeing $N$ go past 4.

"I began to wonder if these equations are mathematically exact."

If $N$ is large enough, almost any sensible "equation" will get you within $\epsilon$ of the exact energy. If you are asking whether or not the equations are "exact" in the sense that they will help you extrapolate correctly to the CBS limit for any chosen value of $N$ then the answer is no. People can't even agree on which formula to use! The notion that we typically see in the literature, that the extrapolation formula should involve $1/N^3$ comes from studies of the helium atom, where some mathematics is actually possible, but then these formulas are reused for modeling the rest of the universe (with some authors choosing to make minor tweaks, such as making $N=3.5$ or $N=4$; see my linked paper above for examples).

These formulas are far from being "exact" in terms of being able to predict "exact" energies for some fixed value of $N$. However they work extremely well, in that using these formulas with calculations that take a few seconds (for example $N=3$ and $N=4$) can give extrapolated energies that are better than what you'd get from calculations that take days (for example with $N=9$).

One final remark, the "exactness" of the CI is not so important. Based on what you're saying about "exact CI" being prohibitively expensive, you mean "full CI" not "exact CI", and whether its full CI or CISD doesn't matter for what I wrote above. Given a large enough $N$, the finite-basis-set CISD energies will converge to the CBS limit of the CISD energy, and likewise for CISDT, CISDTQ, ..., FCI.

Answer 2

To supplement Nike's answer, I would like to point out that the situation is even worse when you look at heavier atoms. While Gaussians do a pretty good job for the first periods of the periodic table, by the time you get to the heavy and superheavy elements you end up making microhartree errors even in Hartree-Fock atomic energies, as I demonstrated recently in J. Chem. Phys. 152, 134108 (2020). Gaussian basis sets have global support and do not allow one to freely refine the accuracy of the basis set representation in space. For polyatomic calculations you end up with even worse problems since the basis functions on different atoms are non-orthonormal. Fully numerical methods, reviewed here, are a better alternative, which can be mathematically shown to converge to the CBS limit. Modern multiresolution approaches allow reaching microhartree accuracy for self-consistent field calculations with small- to moderate-size molecules; post-Hartree-Fock methods are still a work in progress.