I was looking at problem 6.3 in The Oxford Solid State Basics, and have a couple of questions...
I assumed that there'd be two different Vcross terms if the atomic orbitals have unequal energies? But this makes the algebra pretty nasty - the expressions I have for the eigenvalues are kind of messy and not easy to interpret. Is this actually the case, and if so, is there some way to see from the expressions that the electron is definitely more localised on the lower energy atom? Or should the two Vcross terms be the same?
Also - the "extensions" to the simple cases taught in the course like this question, the 2d electrons of Q 4.5, the diatomic Einstein solid of Q 2.7 (just as examples) ... are these sorts of things examinable?