# 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?

• There exist methods that calculate coupling terms between different PES. They are called sth like "nonadiabatic dynamics".. – Fl.pf. May 12 '17 at 8:28
• I'm talking about routine, "static" calculations. If two electronic states are almost degenerate, then the Born-Oppenheimer approximation breaks down. Do we take care of this when doing calculations on molecules with near-degenerate states? – Yoda May 13 '17 at 7:09
• afaik the BO doesn't really break down when two PES "come near each other", it just breaks down when you try to calculate the change from one to the other. I' not really an expert in BO, but I'll show this someone who is and maybe he can really help you ;) – Fl.pf. May 13 '17 at 8:03
• Okay, I'll see what I can come up with. – Fl.pf. May 18 '17 at 12:01
• Logically they must be independent, because the concept of "static correlation" occurs "only" (which is in practise almost always) when one builds multielectron wave functions from one-electron-wave functions. Purely mathematically this is not neccessary, so stat. corr. is a quasi-artefact. While the BO approximation is completely independent on the strategy you tackle to solve the Schrödinger eq. However a famous case comes to my mind where both (BO breakdown + stat. corr.) come togehter: NO3^. see here molspect.chemistry.ohio-state.edu/institute/2007/stanton/… . – Rudi_Birnbaum Jun 3 '17 at 19:49

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.