I was just wondering - in question 11.3 of the book, where the orthogonality assumption is dropped, can it be assumed that the off-diagonal matrix element B is real (i.e. B=B*)? The algebra looks pretty nasty otherwise... I currently have for the dispersion relation:
E = epsilonA -t(Bexp(-ika)+ B*exp(ika)) / [A + Bexp(-ika) + B*exp(ika)]
(where by epsilonA I mean epsilon multiplied by A)
And it's not clear to me how to simplify this to get cosines/sines rather than exponentials. I know the dispersion relation should just be the real part, so my idea to try and separate real and imaginary parts would be to multiply through top and bottom of the fraction by the complex conjugate of the denominator. But with B not equal B*, this doesn't stop the denominator being complex...