The origin of the phrase "nearly free" in the nearly free electron model comes from the fact that we introduce a small periodic potential in a metal lattice as a perturbation to free electrons, so that they are not quite "free". Likewise, what is the origin of the word "tight" in the tight binding model? Why is it called "tight"?

  • $\begingroup$ Hello Ismail and welcome to Physics Stack Exchange! This might ask too many questions, I don't know. But keep in mind that if this question is closed as too broad to try splitting up the questions. For example, the last sentence is an entirely new question, it seems. So maybe split that off into a new question. $\endgroup$ – heather Oct 11 '16 at 0:44

In essence, you are kind of on the right path. Going to Ashcroft & Mermin's Solid State Physics text (from the 2d edition):

In developing the tight-binding approximation, we assume that in the vicinity of each lattice point the full periodic crystal Hamiltonian, $H$, can be approximated by the Hamiltonian, $H_{at}$, of a single atom located at the lattice point. We also assume that the bound levels of the atomic Hamiltonian are well localized... (Chapter 10)

So, the crystal Hamiltonian is treated as the atomic Hamiltonian plus some correction term that accounts for the periodicity of the lattice.

Consider it as the opposite of the 'nearly free' case - you assume that everything is tied closely to the lattice atom, except for a small correction that lets some electrons be free enough to make the crystal.

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    $\begingroup$ Your answer about "tight-binding" being the opposite case of "nearly free" made it quite clear to me now. Thanks! $\endgroup$ – luminare Oct 12 '16 at 15:30
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    $\begingroup$ @Jon Custer Thanks for the information! Is there any way to construct the atomic Hamiltonian $H_a$? $\endgroup$ – Harsha Jan 23 '17 at 14:45

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