Symmetry-based restrictions of band-to-band coupling can be gauged from the inner products ⟨ψV|π|ψi⟩ and ⟨ψC|π|ψi⟩. Since the π operator is vector-like, it is odd under inversion. Consequently, ⟨ψV|π|ψi⟩ ̸ = 0 only when ψV and ψi have opposite parity. This is because the inner product is an integral over the Brillouin zone, which is a symmetric interval around k0 assuming crystal inversion symmetry is preserved; therefore, the integral over an odd function, which occurs when ψV and ψi have the same parity, vanishes. The same reasoning holds for ⟨ψC|π|ψi⟩.