# Knowing that CI is variational for any basis set, why is coupled-cluster variational in the full correlation limit?

This is the question asked in Martin's comment to my answer to: Why is Coupled Cluster not variational?

In the comment I was also asked to provide a publication as a reference:

• This doesn't seem to be proper form. Questions are supposed to be generally useful. You can say it's a follow-up. but making a picture of comment was total waste of time. – Mithoron Jul 11 '18 at 14:36
• I agree with Mithoron. While I like the question and answer, I think including the comment itself is unnecessary; The link to the previous question is enough for a curious person. The question itself should be edited to be more general. – Tyberius Jul 11 '18 at 14:57
• Maybe I put too much effort making the screenshot and adding it. Martin asked this question in a comment to a different question, and then when I answered it he said to make it a question and answer. I am new to the chemistry stack exchange. – user1271772 Jul 11 '18 at 15:07

The CI (configuration interaction) energy, given by Eqs. 3.9-3.10 of this, is an upper bound to the exact ground state energy (i.e. CI is variational). Coupled cluster energies are not generally variational, as explained in the above linked question, except in the case of CC($N$) for an $N$-electron system. The reason is because CC($N$) for an $N$-electron system is equivalent to FCI (full configuration interaction) which is an example of CI, and all CI energies are variational.

Since I was asked for a publication as a reference, a good one is this, in which it is shown that CC and CI become the exact same in the full-correlation limit.