@Tyberius, @Thomas what do you think about this question? mattermodeling.stackexchange.com/q/953/5 At first I asked to re-open it because Susi seemed to have an answer in the comments, but at present there is very little about why there's no basis sets that minimize the energy, and instead it looks like there's an answer to a different question that hasn't been asked yet "What are the types of basis sets?"

@NikeDattani I think the question is fine and I think if Susi expands his answer, he seems to be answering that there isn't one. I'm interpreting his comment as saying that there wouldn't be consistent way to construct a basis set family that minimizes the energy. I think the Dunning example is to show that the goals of having a family (i.e. a consistent relationship between higher/lower sets) doesn't mesh with minimizing the energy (which requires a more ad hoc description for each new case)