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The potential energy between two atoms, in a molecule, is given by $$U(x)=\frac{a}{x^{12}} -\frac{b}{x^6}$$ where $a$ and $b$ are positive constants and $x$ is the distance between the atoms. The atom is in stable equilibrium when: (NEET 1995)

Differentiating and equating to zero gives a single solution, which must then be the stable equilbrium: $$\frac{dU}{dx}=0\Rightarrow x=(\frac{2a}{b})^\frac{1}{6}$$

Question: Is the potential energy function here an actual one for quantities like bond length, or just made up for the sake of this question?

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    $\begingroup$ Does this answer your question? Lennard-Jones Potential repulsion by nucleus nucleus or Pauli repulsion? $\endgroup$
    – Mithoron
    Commented Dec 18, 2023 at 15:28
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    $\begingroup$ I wanted to identify the function (which the comments have done for me), not ask about the properties of that function. In general, I do not think the other questions/answers answer my question. $\endgroup$
    – Starlight
    Commented Dec 19, 2023 at 12:12
  • $\begingroup$ I would think the Lennard-Jones potential is an empirical potential but based on solid theoretical grounds. Basically, the $C_{12}/r^{12}$ term describes the short range Pauli repulsion and the $C_6/r^6$ term is a long-range attractive potential that you can associate to dipole-dipole interactions. I think originally was developed to describe the interatomic interactions in gases but its used extended to many fields. $\endgroup$
    – PAEP
    Commented Dec 23, 2023 at 20:22
  • $\begingroup$ Although there are better interatomic potential functions, it gives you a good approximation of the isotropic part of the potential of Van der Walls complexes and in that constext an idea of their stability and bond length. You would find a more nuanced description in the interatomic potencial chpater of a Physical Chemistry textbook (e.g. Atkins, Engel, ...). $\endgroup$
    – PAEP
    Commented Dec 23, 2023 at 20:22

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The empirical potential

$$U(x)=\frac{a}{x^{12}} -\frac{b}{x^6}$$

is known as the Lennard-Jones (or 12-6) potential. It is ubiquitous in theoretical chemistry as an approximation of real interatomic potentials. It's "made up" but captures essential features of interatomic potentials, has been particularly useful because of its mathematical simplicity, as explained in another answer, and continues to be broadly implemented, for instance in MD simulation packages.

Is the potential energy function here an actual one for quantities like bond length, or just made up for the sake of this question?

Note that the real potential is "known" only to the interacting matter. As the number of interacting particles (nuclei, electrons) increases the number and complexity of interactions grows staggeringly fast. We do our best to approximate the interactions to facilitate predictions. Even QM simulations are slow to perform on all but the smallest systems. An L-J potential is a pair-wise mean-field potential and as such brushes under many of the complexities of real interactions. You might also find useful information on uses of the L-J potential in our sister site Matter Modeling.

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