What does it mean to say that the Fermi energy is equal to the hopping energy? I have a conceptual question regarding the relation between hopping energy and Fermi Energy.
In order for my question to make sense contextual background is needed:

So the hopping integral or hopping energy definition is given by $$t_{\alpha}\equiv-\int{u_L}^{*}\,V_L\,u_R \,dx=-\int{u_R}^{*}\,V_R\,u_L \,dx$$

The hopping integral $t_{\alpha}$ itself can be of either sign, depending on the potential and the orbital $\alpha$. For the double square well given above, we have an attractive potential between the electrons and the ions. $\epsilon_{\alpha}$ is the energy to put electron at a site and $−t_{\alpha}$ is the kinetic energy for hopping to neighbour.

Okay, that's enough background I think.
Now, here is the problem:

Looking at the form of the integral $\int{u_L}^{*}\,V_L\,u_R \,dx$, we can interpret $t_{\alpha}$ as the scattering of an electron in the right well to the left well. Similarly, the integral $\int{u_R}^{*}\,V_R\,u_L \,dx$ can be interpreted as an electron trapped in the left well seeing the potential due to the right well and hopping across into the right well.

If I'm to interpret the hopping integral, $t_{\alpha}$ as this, then I don't see how writing the Fermi-Level in terms of the hopping integral makes any sense:

So the Fermi-Level is defined to be:

Electrons obey Pauli exclusion principle. When we add electrons to a system, they
fill up successively higher energy states up to the Fermi energy or Fermi level $E_F$.
This is the kinetic energy of the most energetic electrons in the solid.

So the Fermi level is essentially the energy of the highest filled state. So does it seem that writing, say, '$E_F=-2t$' means that the Fermi level is comprised solely of the kinetic energy of moving electrons from one lattice site to another?
The reason this is confusing me is because I thought there were many, many electron states that were filled up much lower than the Fermi level (but still contribute to the overall Fermi energy), but not energetic enough to contribute to hopping. So why are we writing such a thing as $E_F=-2t$, ie. all the Fermi energy is hopping energy?

Edit
I forgot to mention that I have books by Kittel (7th and 8th edition), Ashcroft & Mermin, Hook & Hall and Rosenberg on Solid-State physics; but when I tried to look up 'hopping integral' in the index I found that it's not in any of those books.
 A: Well, this is called the Hubbard model, where you account for the low energy physics in a condensed matter system using lattice hopping energy (t) and on-site Coulomb energy (U).The dispersion relation you have mentioned can be obtained from applying second quantization on a tight-binding Hamiltonian in a cubic lattice. If you change the lattice, the dispersion relation will change. In conclusion, fermi energy, being a form of energy that is crucial to determining the number density of fermions present in a system, can be interpreted in many ways, as you can see in the first 2 paragraphs of the hyperlinked page (one for semiconductors, one for metals and so on). One such interpretation is the Hubbard model, where kinetic and potential energies of electrons are not explicitly accounted for. Rather, t and U are the main parameters where KE and PE are indirectly present.
And as a wise man once said, "All models are correct in their own sense, but some are useful". It so happens that the Hubbard Model explanation can help us understand a lot in condensed matter physics
