Help. In tight binding 2d square lattice with two atoms basis

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I didn't understand your first question, but I can answer the second one: these kinds of systems tend to have highly localized states - not in Anderson sense - such that its electronic clouds rarely pay visits to its neighbors'. The nearest-neighbor approximation can be done safely a lot of the time, because they usually are the most important in a material and suffice to explain its low-energy physics. Please, check Ashcroft and Mermin's Solid State Physics chapter 10 for an enlightening exposition, or Wikipedia's page for a quick start.

Just out of curiosity: which book are these images from?