Is there any connection between "static correlation" and the Born-Oppenheimer approximation? Beware: High probability for that I am mixing two concepts that should not be mixed, just because the same term is used when explaining the two concepts!

For some reason I get really confused when I think about static, or nondynamical, electron correlation. I somehow want to make a connection to the Born-Oppenheimer (BO) approximation.

As we know, the BO approximation is no longer accurate when two electronic states come close in energy, due to the increase in the nonadiabatic coupling terms. These terms couple the electronic motion to the nuclear motion, and arise because the electronic and nuclear motion is not fully separable in the Schrödinger equation. 

Static correlation, sometimes called a "near-degeneracy effect", describes the correlated movement of electrons belonging in different electronic configurations. Certain systems are not described properly by just a single-determinant wave function. One example is the H-H molecule at long bond distances, for which the bonding and anti-bonding $\sigma$-orbitals become degenerate. (this was probably a bad explanation, as I struggle with this concept!)

So, as we have just seen, the BO approximation breaks down when two electronic states come close in energy (near-degeneracy), and the static correlation becomes extremely important in molecules with near-degenerate electronic states. Is there a connection here?

Imagine we are in the ground state at a point on some potential energy surface, and the first excited state is very close in energy. At this point, there should be a high coupling between the electronic and nuclear positions (which means what, exactly??). Also, at this nuclear geometry, the electronic wave function should be described as a linear combination of these two near-degenerate electronic states.

It is the BO approximation that introduces the idea of a potential energy surface (PES). The PES itself can be thought of as an approximation, as the nuclei should ideally be fully described by quantum mechanics. The nuclear positions should be described probabilistically by a wave function, and the very idea of a molecular geometry then becomes fuzzy. 

What is the connection, if there is any?
Credit to Rudi_Birnbaum for coming up in a comment to the question with what I think is the key here.

Static correlation is a difficult to grasp concept of which even a clear definition is beyond me. However, it seems clear that a full Configuration Interaction (FCI) calculation yields all correlation energy, thus, must correctly handle static correlation. From the discussion of the Born-Oppenheimer approximation (BO) by Jensen,$^{[1]}$ I gather that the BO approximation is essentially a modification of the Hamilton operator. This shortcoming cannot be healed by a FCI calculation that only considers the electronic problem. So, in this sense, the two concepts are unrelated.

(And now for the speculative part of my answer.)

However, I cannot speak as to whether they frequently show up together in chemical systems. Though I cannot find a reference for it, I recall that the beryllium atom requires a multi-reference treatment. Being a single atom that is rather heavy, the BO approximation could be acceptable for it. Thus, this would be a case where the two are not related.