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
- the orbital basis
- 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.
