# Repulsive part of interatomic/intermolecular potentials: Pauli, Coulomb or both?

We know that if we take two atoms/molecules, their interaction energy shows a short-range attractive part and a medium-range attractive part.

One popular way to schematize this interaction is to employ the Lennard-Jones potential:

$$U(r) = 4 \epsilon \left[ \left( \frac{\sigma} r \right)^{12} - \left( \frac{\sigma} r \right)^{6}\right]$$

The exponent $6$ comes from the dipole-dipole van der Waals interaction, while the exponent $12$ has no theoretical justification whatsoever and only represents a very strong repulsion at short distances.

It is very common to find the statement that this repulsive term comes from the Pauli exclusion principle. For example, from Wikipedia:

The $r^{-12}$ term, which is the repulsive term, describes Pauli repulsion at short ranges due to overlapping electron orbitals...

However, in the book Statistical Mechanics by K. Huang (par. 2.3) we find the following statement about molecular potentials:

The attractive part of the potential energy originates from the mutual electric polarization of the two molecules and the repulsive part from the Coulomb repulsion of the overlapping electronic clouds of the molecules.

I also found this post on Quora which basically asks the same thing. In the first answer we can find the following statement:

In fact the repulsive part of the "electrostatic force" is really caused by the Pauli exclusion principle: when the atoms are pushed together too much, the distortion of the orbitals caused by the Pauli exclusion principle will result in the electrons being "squished" out of the region between the atoms - this will then allow the strong electrostatic repulsion between the two positively charged nuclei to push the atoms apart.

So the repulsive term would be explained by a combination of Pauli exclusion and Coulombic interaction.

What is the real origin of the repulsive term? Pauli exclusion principle, Coulombic repulsion or a combination of both?

PS: There are many questions on the site similar to this one and many of them are labeled as duplicates of this one. However, the answers to this question are (apart from maybe one) incredibly vague, and tend to address more the general problem of the impenetrability of solids (ex. "you don't fall towards the center of the Earth because the ground is solid and not liquid") than the more specific problem of the origin of the repulsive term in interatomic/intermolecular interaction. This is why I think that this post is not a duplicate of the aforementioned one.

• Have a look at Wouter's answer here and my answer here. Assuming I've understood you correctly these address what you are asking. – John Rennie Jul 14 '17 at 15:42
• @JohnRennie Thanks, this is useful. Especially your answer, which addresses the role of the electrostatic interaction explicitly. – valerio Jul 14 '17 at 16:16