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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 ΞΆ'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?

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  • $\begingroup$ +1. I've made a minor modification to the title: The question's body asks whether or not the sequences mathematically converge to exactly the CBS limit, not whether or not they were "designed" to do so :) $\endgroup$ Commented Sep 19, 2022 at 3:02

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"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.

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    $\begingroup$ First you accepted the answer and gave it an upvote, then you unaccepted it and downvoted it. Interesting. This answer can't be a "circular" argument because nothing is argued, it's just told. You asked whether or not it converges, not for a proof that it converges. $\endgroup$ Commented Sep 19, 2022 at 3:41
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    $\begingroup$ @wzkchem5 The cc basis sets are defined for arbitrary N, but in order to optimize the exponents such that they give the lowest possible CISD energy, you need to be able to do the integrals, which while theoretically possible to do, cannot practically be done when the number of orbitals gets too big. I touched on this a bit in my answer when I said that it's quite impractical to regularly go beyond N=10 for triatomics, or N=4 for most molecules of interest, and you're right that even optimizing the exponents of the basis set is very challenging to do beyond N=6, but it's theoretically possible. $\endgroup$ Commented Sep 23, 2022 at 17:40
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    $\begingroup$ @NikeDattani I see. Then the question reduces to: when we add more and more basis functions for each angular momentum and optimize them to give the lowest atomic CISD energy, does the resulting basis converge to the CBS? While numerically this seems to be true to extremely high accuracy, it is not mathematically obvious whether this is rigorously true. And I think the OP is more interested in mathematical rigor than practical feasibility. $\endgroup$
    – wzkchem5
    Commented Sep 23, 2022 at 18:33
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    $\begingroup$ @wzkchem5 good point again! Would you say that an infinite number of linearly independent 1D Gaussians would form a "basis" for constructing any 1D function with error less than any $\epsilon$? If the answer is yes, then it's not too hard to generalize to 3D or n-dimensions. I was originally thinking that each time we increase $N$ in cc-pV$N$Z we are introducing a new orthogonal basis function, but Susi's answer 4 hours ago says that they're not orthonormal when linear combinations (LCAO) are used for polyatomics. Luckily they don't have to be orthogonal, just linearly independent. $\endgroup$ Commented Sep 23, 2022 at 18:44
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    $\begingroup$ This answer to "Can any function be decomposed as sum of Gaussians?" on StackOverflow is a good place to start! $\endgroup$ Commented Sep 23, 2022 at 18:53
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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.

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