I Lennard-Jones potential and interatomic distance

[eneacasucci]

I've found this image online

(ref: https://edurev.in/t/188018/Origin-of-Energy-Bands ), it should be the graphical representation of the potential binding energy between two nearest-neighbor atoms.
I don't understand how it can be correlated to the Lennard-Jones pontential graph:

, in which we see the r_m (equilibrium distance) which is the most stable distance between the particles, that I assume being what in the previous figure is a.
1) Why in the first figure we see a minimum in the middle of the two atoms?
2) are the two potentials displayed related to different things?
3) does it mean that the distance to have the minimum potential is r_m and that V(r_m) is reached in the midpoint of the two particles?

 

[Charles Link]

I could be wrong, but I think the first curve represents the potential energy that the electron sees as a function of position as it moves around a couple of ions in a solid that is composed of positive nuclei plus an electron background.

Edit: It represents the periodic potential of an electron in a crystal in solid state physics. It can be used with the Schrodinger equation to solve for the wave function of an electron. Bloch's theorem is an important concept in this topic.

The Lennard-Jones potential is a graph of the potential energy of a diatomic molecule with off-setting electron charge around it as a function of the spacing of the nuclei.

Edit 2: If I remember my college studies of solid state physics correctly, in practice they use a pseudo-potential to represent approximately the periodic potential that the electron sees, (especially in the conduction band), as the nuclei are screened by the other electrons.