Attraction and repulsion between atoms

Question

In six east pieces Richard Feynman describes atoms as little particles that move around in perpetual motion attracting each other when they are a little distance apart but repelling upon being squeezed into one another. Can someone explain from where this attraction and repulsion comes from? I always thought that only specific atoms attract each other and I didn’t even think they repel. I’m really intrigued

Answer 1

There are several different interatomic potentials that can model the attraction and repulsion between atoms as a function of distance (and perhaps other parameters), but the Lennard-Jones potential, which looks like $$V_{LJ} = 4\varepsilon\left[\left(\frac{\sigma}r\right)^{12} -\left(\frac{\sigma}r\right)^6\right]$$ is one of the most straight-forward to understand.

In this model (to borrow words from Wikipedia), $\varepsilon$ is the potential depth, and $\sigma$ is the distance at which the the particle-particle potential energy $V$ is zero. The minimum value is reached at $r=r_m = 2^{1/6}\sigma$, at which point the potential energy reaches the value $V=-\varepsilon$.

You can see what it looks like below.

Note that I am not an expert in this area, but simply will briefly interpret (bolding is mine) the following passage (from the Wikipedia page for the Lennard-Jones potential):

The Lennard-Jones potential models the two most important and fundamental molecular interactions: The repulsive term ( $1 / r^{12}$ term) describes the Pauli repulsion at short distances of the interacting particles due to overlapping electron orbitals and the attractive term ( $1 / r^6 $ term) describes attraction at long ranged interactions (dispersion force), which vanish at infinite distance between two particles. The steep repulsive interactions at short distances yield the low compressibility of the solid and liquid phase; the attractive dispersive interactions act stabilizing for the condensed phase, especially the vapor-liquid equilibrium.

My pointers:

• Why do we have an effective Pauli repulsion "force"? This is because electrons are fermions, which cannot be in same quantum state. This is the Pauli exclusion rule, and leads to an exchange interaction that is not a true force, but still causes a real effect that can be modeled as an effective potential or force.
• Long range, attractive dispersion forces occur because the neutral atoms temporarily polarize each other (the "electron cloud" of one shifts the "electron cloud" of the other), and thus attracts on average. This temporary polarization causing a net attraction is an interesting, and I personally think non-trivial, effect.
• Note both terms are present at any distance, but the repulsion due to Pauli exclusion wins out tremendously at short distances even just slightly smaller than $r_m$ and dies very quickly past $r_m$. We can intuitively think that if the atoms are far apart, there is no way that we can confuse which electrons belong to which, and thus the Pauli exclusion force should be incredibly weak. But nonetheless there is thus always a competition between attractive and repulsive forces.

Answer 2

The simplest example of this type of interaction is the molecule of $H_2$. The quantum mechanics calculations of the distance of equilibrium between the 2 protons takes 4 pages in the book from Griffiths on QM. And to make even simpler, he consider the $H_2$ ion, that is: 2 protons and only 1 electron.

The purpose of the calculations is to verify that the stable state is the $H_2$ ion, instead of 1 atom of H (with the electron) and a free separated proton.

In my opinion, if we look classicaly at this situation: 2 equal positive charges and 1 negative charge, it seems that as soon as the (-) charge is closer to one (+) than the other, it is trapped there, and we have a neutral object and a free positive charge.

But the QM results using the Schroedinger equation and variational principle is that the $H_2$ ion is the stable configuration. The energy of the system has a minimum about 2.4 Bohr radius. As $F = -\nabla E$, if the protons becomes closer than that there is a separation force. And an atractive force if they are more distant.

They oscillate around that equilibrium position.