This is no simple answer to this question, however by considering it carefully, it should be possible to figure it. Since the gate introduces a rigid shift of the eigenvalues in the central region, you can study the eigenvalue spectrum, and move the levels (with the gate voltage) such that you push the HOMO above the Fermi level, or opposite (LUMO below).
It's a handwaving way, and might not work, but it's worth trying.
