A less pragmatice question then. When we have solved the secular equation before (in 1-d) we have only even had the wavevectors labelled by k, k+G. Why do we not include all the (nearly infinite number of) possible contributions to the Bloch wavefunction? An unperturbed state k also mixes with k-G,k+G, k-2G, k+2G etc.
I know that any wavefunction with a label k, also must include k+G,k+2G etc by Bloch's theorem, but this has been ignored so far. Is there a good (physical rather than pragmatic) reason for this?