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I am in high school taking AP Chemistry and I am wondering how accurate the curriculum is from a quantum-mechanical perspective--specifically, how accurate are potential energy curves?

Potential energy curves as taught in high schools

The way chemistry is taught in high schools, we use Coulombic forces to explain phenomena such as ionization energies and intermolecular forces. Potential energy curves such as the one above seem to be assuming that atoms in a chemical bond are attracted by a purely inverse-square-law force and experience some sort of angular momentum. This would be completely plausible at a macroscopic scale such as a binary star system, but I'm confused as to how accurate this approximation would be at such a small scale. I've searched for a more rigorous alternative to potential energy curves but as far as I have seen it's just a superficial analogy designed to develop an elementary understanding of bonds.

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    $\begingroup$ Why is the Coulombic interaction not plausible at small scales? $\endgroup$
    – Buck Thorn
    Feb 12 at 8:27
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    $\begingroup$ Search for the Morse potential, it has more or less the shape you show, approximates the harmonic potential near to the bottom of the well. It is widely used to define potential energies and is a solution to the Schroedinger equation. Actual potentials are more complex but have the same sort of shape. $\endgroup$
    – porphyrin
    Feb 12 at 8:30
  • $\begingroup$ buck thorn, while coulombic interaction is plausible at small scales, quantum mechanics is far favored at the level of, say, diatomic molecules, no? $\endgroup$ Feb 12 at 8:40
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    $\begingroup$ If you want to have a go at your own calculation see the code here chemistry.stackexchange.com/questions/174913/… $\endgroup$
    – porphyrin
    Feb 13 at 11:02
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    $\begingroup$ @RaphaelEsquivel One still uses the Coulombic potential for each single particle in (nonrelativistic) quantum mechanics. $\endgroup$ Feb 14 at 7:01

2 Answers 2

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That is far beyond what I learned in high school!

Those look like pretty accurate potential energy curves for diatomic molecules (as far as their shape is concerned—I cannot speak for the numbers). If you look at almost any book or review article discussing spectroscopy of diatomic molecules, those are the sorts of potential energy surfaces (PESs) you will see (see, for example, this question).

The cool thing is that these PESs can be calculated theoretically or determined experimentally (the classic experiment for undergraduate chemistry students is to analyse the UV-vis spectrum of $\ce{I2}$ and plot the potential energy surface based on the vibrational transitions observed for $\ce{I2}$ in several electronic states (here is an example of the experiment). In fact, just to prove it, here is my experimental PES for when I did the $\ce{I2}$ lab:

PES of iodine

The PES is not simply an inverse square rule, it is actually a result of something people cannot actually model accurately yet—anharmonicity. For a diatomic, the closest we can get to the PES is a variation on a Morse potential (as said by Buck Thorn), which looks like: $$V(r) = D_e(1-e^{-\beta(r-r_e)})^2$$ where $$\beta = \omega_e\sqrt{\frac{2\pi c \mu}{D_eh}}$$ where $r - r_e$ is the deviation from the equilibrium bond length, $D_e$ is the dissociation energy, $c$ is the speed of light, and $\mu = \frac{m_am_b}{m_a + m_b}$ is the reduced mass of the molecule. For polyatomic molecules, the equations become infinitely more complex.

In summary, you are not being misled by your teacher, they are just covering the initial concepts, which is likely all you need to know to pass your exams. To alleviate your concerns, I can say that the experimental PES above was derived from entirely quantum mechanical principles (observation of discrete vibrational and electronic transitions). And, while I am fuzzy on the maths, I suspect the angular momentum you are referring to in your question is called the vibrational angular momentum. If you want to look that up you can, but, again, it is probably not important for you to know about.

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    $\begingroup$ Angular momentum tends to be associated with rotational, not vibrational motions. There are rovibrational energy levels, however: chem.libretexts.org/Bookshelves/… But I agree that this is a more advanced topic than might be covered in HS. $\endgroup$
    – Buck Thorn
    Feb 12 at 18:48
  • $\begingroup$ @BuckThorn I know, I only really know about it in the context of rotational (or electronic) spectroscopy, but I am certain I've heard of 'vibrational angular momentum' somewhere as well. It looks like it is a thing, though, although I'm not good enough at the maths to know whether it is referring to the effect of rotation on the vibration, or the vibrations themselves: onlinelibrary.wiley.com/doi/10.1002/jcc.26762 $\endgroup$ Feb 13 at 2:29
  • $\begingroup$ I believe that VAM is a way of dealing with the QM equivalent of ro-vibrational coupling, or coriolis forces. The term VAM was foreign to me. $\endgroup$
    – Buck Thorn
    Feb 13 at 7:29
  • $\begingroup$ @BuckThorn that would make sense! $\endgroup$ Feb 13 at 17:29
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One approach to bridge classical and QM descriptions (consistent with the correspondence principle that these descriptions should merge as you change the energy) is to transform the classical potentials into operator form by substituting the classical variables describing position and momentum into equivalent QM operators. Beyond conceptual differences in interpreting the nature of the solutions provided by QM (quantization, delocalization, etc) and adding some details that account for the effect of relativity (spin), this approach leaves the classical form of the potentials mostly unaltered. In essence the position coordinates get little hats. The treatment of electromagnetic interactions, for instance by using a Coulombic potential, are very similar in the quantum realm and in the classical realm.

There are many aspects of "reality" brushed aside during introductory treatments, and sometimes they do involve the shape of the potential. For instance, atoms bonded in a diatomic molecule may be assumed to vibrate harmonically (like an ideal spring or pendulum, except one obeying quantum mechanics). That harmonic approximation however assumes that the potential between the atoms is quadratic (parabolic), whereas the reality is closer to another, since a quadratic potential does not allow atoms to dissociate (they are stuck in the parabola for all eternity). A better approximation is for instance a Morse potential.

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