Benefit of contracted basis?

[Nicholas Winner]

Hello all, a quick question:

When employing an auxiliary basis for a system, we have a choice of many such as FITx, cFITx, cpFITx... I understand that the "x" refers to the number of Gaussian exponents, and that the prefix indicates whether you are using uncontracted, contracted, or contracted with additional polarization functions, respectively. What I don't know is why you would choose contracted/uncontracted. Both end up having the same number of primitive basis functions in your calculation if I understand correctly. So what is the use of having one over the other?

Thanks for your help.

-Nick

[Lucas Lodeiro]

Hi Nick,

I am not an expert on CP2K, but this question is more general than CP2K implementation.

When you have a set of primitives, you can use each of them by itself, then you have one constant for each primitive to apply the variational principle, and they are independent between them (obviously they have the orthonormal restriction for the solutions).

If you contract some primitives, you have the "same" number of primitives in the set, but your variational constant are less, this is, when you contract some primitives, you constrain the constant of these primitives to be in a given proportion, and this primitive mix have only one variational constant, making more simple the "diagonalization" or solution for these basis set, but with a lower variational convergence.

In simple, if you have 3 primitives for a particular orbital, you can mix them with the constants a1,a2,a3 in any proportion, but if you constrain the second and the third, you only have now 2 constants for the variation, this is, a1 and a23.

Regards

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[Matt W]

I think Nick is talking about the Auxiliary Density Matrix Method, where the primary basis set is projected onto a smaller auxiliary basis to facilitate the hybrid functional part of the Kohn-Sham build. In this case there is no diagonalization involving the auxiliary basis as it gets merged back into the main KS matrix before diagonalisation / OT. There are a bunch of linear algebra operatations involved in the 'projections' from primary to auxiliary basis and back that can be more efficient with contracted functions but I am not sure there is a major advantage to using the contracted versions, I've never benchmarked.

Matt

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[hut...@chem.uzh.ch]

Hi

there are some gains in the linear algebra parts (density matrix
contractions), but the big difference is memory. You gain because
of the n^4 scaling of the number of integrals to store.

Juerg
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Date: 05/08/2020 11:40PM
Subject: Re: [CP2K:13264] Benefit of contracted basis?

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[Matt W]

Ha, OK, thanks Juerg.

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[Nicholas Winner]

Thank you Jurg, so the contracted basis functions take up less memory footprint? Even though there are the same number of them?

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[Matt W]

You would have to calculate all the primitive integrals, but you would only need to store their sum in memory as the contraction coeffs are fixed.

[Nicholas Winner]

Ah I see. Thank so much Matt, Jurg, and Lucas.

-Nick