Formation of Energy Bands - due to electron interaction?

Question

I am reading Semiconductor Physics and Devices (specifically pg. 78 of the pdf in link).

The formation of energy bands is discussed, and the following two sentences appears:

The wave functions of the two atom electrons overlap, which means that the two electrons will interact. This interaction or perturbation results in the discrete quantized energy level splitting into two discrete energy levels...

What I don't understand:

1. What does it mean that the electrons interact? Does it mean that the potential of the electron is treated as a perturbation? Why is this the same as saying that their wavefunctions overlap?
   
   More importantly:
   
   
2. Is it really the perturbation caused by the presence of a second electron that causes the splitting of the energy (i.e. " the two electrons will interact"), or is it the perturbation due to a second proton? If there was just one electron and two protons would the energy spectrum not look the same? In the Kronig-Penney model only the periodic nuclei are taken into account; there is no reference to the other electrons in the system (at least none that I can find in the above book).

Related question

Answer 1

Electrons interact because they repulse each other electrostatically, and there is a specific interaction term in the hamiltonian for that. This acts regardless of whether the wavefunctions interact, but the effect is much stronger if they do.

However, your second point is fully correct: the splitting is due to the presence of the second nucleus, and a single electron in that potential will still have a splitting. Electron interaction has a strong effect on the spectrum (and if the interaction is strong enough, it can blow the single-electron structure right out of the water) but the splitting itself is a single-particle effect.