Pair interaction potential with more than one mininum?

In summary, it is possible to have an interaction with more than one local minimum. This is how Rydberg molecules are formed.
  • #1
hilbert2
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TL;DR Summary
Would it be possible to create a molecule that interacts with other identical molecules with an interaction potential that has more than one local minimum value?
Pair interactions between atoms and molecules (in gas kinetic theory simulations or other applications) are described by empirical potential energy functions such as the Lennard-Jones potential:

##V_{LJ} (r) = 4\epsilon \left[ \left(\frac{\sigma}{r} \right)^{12} - \left(\frac{\sigma}{r}\right)^6 \right]##

these always seem to be functions where the energy reaches minumum value at some distance and then the interaction becomes repulsive at too short distance.

Is there known to be any molecular species where the pair interaction has more than one local minima? I believe that at least this doesn't happen in practice unless you specifically design a molecule that is easily deformed by the "tidal" effect of the intermolecular interactions. If that kind of molecule can't exist, I guess it would be really difficult to prove that impossibility mathematically from the Schrödinger equation.

If there exist that kind of molecules, my first guess would be that those compounds are more likely than usual to exist in more than one liquid or solid phase.
 
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  • #2
You did not specify what "identical" means exactly in your case. If different states of the same atom are allowed then: yes - it is possible to have an interaction with more than one local minimum. This is how Rydberg molecules are formed:

See, e.g., this paper on Ultracold Rydberg molecules by Shaffer et al.:
https://www.nature.com/articles/s41467-018-04135-6

In a nutshell: If one atom is in a highly excited Rydberg state and interacts with a ground state atom, one can approximate the interaction as scattering from the Rydberg electron (as the distance to the core of the Rydberg atom is large). This can be done using a pseudopotential approach (and indeed is the problem for which Fermi postulated this approach first).

You then find that as the wavefunction is oscillatory, so is the potential energy curve and you can get several local minima. See fig. 1)a) in the paper mentioned above.
 
  • #3
Thanks for the answer. The first what I going to say was that maybe the muonic helium atom, with a muon replacing one of the electrons, can also have that kind of a pair interaction potential because the muon in that atom is much closer to the nucleus than the electron. But the article you linked to says "The form of Eq. (4) suggests that the Born-Oppenheimer (BO) potential energy curves, ##U_{BO}(R)##, will be oscillatory functions of R, since the Rydberg wave function oscillates" so it's possible it requires that the negative particle further away from the nucleus is really in a high excited state and not just at longer distance because of smaller mass.

It would be quite difficult to invent a molecule where even in the ground state one of the electrons is much more delocalized than the others and much further away from the CMS of the molecule.

Another complication is that in intermolecular interactions also the rotation angle of the molecules affects the function ##V(R)##.

Edit: On a second thought, it could actually be possible that there are two versions of a bound state between two muonic helium atoms: one where they bind like two hydrogens and another that has a much smaller internuclear distance with a species similar to ##He_{2}^{2+}## molecular ion (but with two muons instead of two electrons) at the core and two electrons orbiting at much larger radius to balance the remaining electric charge.

There's also this related concept of "bond stretch isomerism", which in the 1980's was believed to exist in some ordinary chemical compounds, but now most experts don't seem to believe any example of that phenomenon has been found yet: https://www.sciencedirect.com/science/article/pii/S1631074802013802
 
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  • #4
hilbert2 said:
TL;DR Summary: Would it be possible to create a molecule that interacts with other identical molecules with an interaction potential that has more than one local minimum value?

Is there known to be any molecular species where the pair interaction has more than one local minima?
I think you need to be more specific. One can think of quite trivial cases, like a DNA strand that alternates GCGC, where there will be lots of local minima between a pair of identical molecules. Same with virtually any macromolecule.
 
  • #5
I'm mostly considering a situation where two molecules approach along a straight line with fixed orientation (rotation angle).

The system with a muon version of ##He_{2}^{2+}## as a "nucleus" and two electrons orbiting it is likely to be easy to calculate a ground state and equilibrium bond length for with quantum DMC. The core that is similar to helium molecular ion only has two muons, so there is no need to force the antisymmetry of the wave function as is required for systems of more than 2 fermions. After calculating the ground state energy and wave function for that, the wave function of the two orbiting electrons can be calculated separately by considering the nuclear part as a static charge distribution to account for finite-size effects. That system would be a kind of an extreme version of an atom that has an oblong, elliptic nucleus, enough to see its effects on the electronic spectrum.

Not sure how easily that type of bound systems could be formed in muon-helium collisions. They could be difficult to detect because of low formation probability and the short half-life of the muon.
 
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