# Does the accuracy of RI-MP2 follow the higher number of zetas of the main and fitting bases?

It has been answered to my previous question that RI-MP2/cc-pVTZ/cc-pVTZ gives results of the same quality as MP2/cc-pVTZ. I then started to think about RI-MP2/cc-pVTZ/cc-pVQZ.

The term "basis set" literally comes from Fourier analysis in mathematics, and in Fourier analysis the "errors" systematically "cancel out", for example when integrating a Fourier series expansion to find out the coefficients. This might somehow mean that the TZ errors cancel out and leave only the QZ errors, smaller than the original TZ errors behind; i.e. RI-MP2/cc-pVTZ/cc-pVQZ might be of the same quality as MP2/cc-pVQZ.

Is my logic (approximately of course, but still at least qualitatively) true?

• By RI-MP2/cc-pVTZ/cc-pVQZ, do you mean an RI-MP2 calculation with an orbital basis set of cc-pVTZ and an auxiliary basis set of cc-pVQZ/C? Oct 24, 2022 at 8:43
• @wzkchem5 yes I do Oct 24, 2022 at 9:00

The term "basis set" literally comes from Fourier analysis in mathematics, and in Fourier analysis the "errors" systematically "cancel out", for example when integrating a Fourier series expansion to find out the coefficients.

False. The term comes from linear algebra. We have a vector space spanned by our basis functions.

It has been answered to my previous question that RI-MP2/cc-pVTZ/cc-pVTZ gives results of the same quality as MP2/cc-pVTZ. I then started to think about RI-MP2/cc-pVTZ/cc-pVQZ.

I am not sure what you mean by your notation.

However, RI-MP2 is simply MP2 where one uses the RI approximation for the two-electron integrals, see Chem. Phys. Lett. 208, 359 (1993). The accuracy of the RI-MP2 calculation is determined by

1. the orbital basis
2. the auxiliary basis used to generate the integrals

If one uses an accurate enough auxiliary basis set, RI-MP2 recovers MP2 theory; for example, see J. Chem. Theory Comput. 17, 6886 (2021) for a recent demonstration with automatically generated auxiliary basis sets for any given orbital basis set.

The cost of RI-MP2 is mainly determined by the size of the orbital basis, as one needs to compute $$o^2v^2$$ excitations ($$o$$ and $$v$$ are the number of occupied and virtual orbitals). The cost is essentially linear in the auxiliary basis, excluding its setup.

Using a mismatched orbital/auxiliary basis combination gives you poorer results than a properly matched combination. You either spend unnecessary effort by using a too large auxiliary basis, or waste the calculation by compromising the results of the large-orbital-basis calculation by using an insufficient auxiliary basis set.