Born-Oppenheimer approximations in molecules

[kelly0303]

Hello! I am a bit confused about Born-Oppenheimer approximations in molecules (mainly diatomic ones). It seems that all the books where I read about it, it is presented as a major breakthrough (at least in the context of molecular physics), but when I look into it in more details, it seems to involve just the separation between the electronic and nuclear motion. Am I missing something? I am totally aware that this approximation is extremely useful and given that it was derived in the early days of quantum mechanics makes it even more impressive, but it just seems like the obvious, first thing to do, as a first approximation in this situation. So I feel like I am missing a deeper insight into the meaning of this approximation. Can someone enlighten me please? Thank you!

 

[Dr_Nate]

Here is their http://www.chm.bris.ac.uk/pt/manby/papers/bornop.pdf I believe in the early days many were still thinking with as they say the "old theory" and many weren't able to do the math. Have a look at the paper and let me know what you conclude!

 

[dRic2]

Disclaimer: I first studied the Born-Oppenheimer approximation in the contest of lattice dynamics, but I guess the reasoning applies as well to molecular dynamics. I do not have a reference for BO approximation in molecular dynamics but in Ziman's book "Principles of theory of Solids" p 200-203 there is a very elegant "proof" regarding BO approximation.

it seems to involve just the separation between the electronic and nuclear motion. Am I missing something?

But what might seem obvious requires a rigorous proof. In fact, treating the nuclear and the electron motion separately is the first thing that comes to mind, but how good of an approximation is that? How reliable is the result obtained? Born-Oppenheimer approximation basically proves that electron motion and nuclear motion can in fact be treated separately if you neglect second-order therms in perturbation theory. Those second order therms give rise to excitations of electronic levels due to nuclear motion (in a crystal lattice they are the phonon-electron interactions). So basically BO approximation tells us that if we are willing to neglect those terms it is correct to study the electronic motion separately from the nuclear one.

 

[hilbert2]

The BO approximation works because the timescale of electronic motion is much faster than that of nuclear motion.

A similar example is the muonic helium atom, with an electron and a muon orbiting a helium nucleus. It behaves much like an isotope of hydrogen, with the muon cancelling the charge of one of the protons. In that case it's the difference in distance scales that allows seeing it as a hydrogen-like atom - the muon orbit radius is much smaller than that of the electron.

 

[DrDu]

I doubt that is was considered a major breakthrough. The fact, that the electronic and nuclear motions decouple due to different masses and timescales was clear to most physicists and e.g. Slater published a paper with similar ideas already before Born and Oppenheimer. Nevertheless, the BO approximation is a very singular approximation and many techniques to make it watertight had still to be developed. The original BO perturbation theory involved both an adiabatic separation of the electronic degrees of freedom and a semiclassical treatment of the nuclear motion. Later on, it was seen that the adiabatic separation is much more useful as the semiclassical approximation. A first step in this direction was the so called Born Huang approximation. A more formal justification came with the paper by Weigert and Littlejohn: https://www-users.york.ac.uk/~slow500/reprints/P006MulticompWaveEqus1993.pdf