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If I'm interested in getting an answer that is minimally correct what is the minimum basis set level I should use? (i.e. anything less would be much less reliable)

Obviously this depends on the methods, property of interest and opinion.

Let's say I'm just interested in relative energies (but feel free to chime in with information about properties if you are inclined). I am interested to hear how this depends on the different methods and the various arguments there are for minimum basis sets one should use.

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Different people will have slightly different opinions, so take my answer with a grain of salt. I'll try to give the minimum publishable basis sets, i.e. if you use worse basis sets than these, you'll have a strong reason why you do so, written in the article, otherwise you'll probably face suspicion from the reviewers.

DFT (excluding double hybrid) geometry optimizations: def2-SV(P) or 6-31G(d)

DFT (excluding double hybrid) single point: 6-311G(d,p) (some may argue it's not large enough) or def2-TZVP

Double hybrid functional: def2-TZVP (with geometry optimization one may get down to def2-TZVP(-f)? But definitely not down to double-zeta basis sets)

CCSD(T): def2-TZVP or cc-pVTZ

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