With the overlap matrix as constructed using Method #2, we now turn our attention to choosing the phase of the overlap matrix. Here we compare the electronic state population dynamics as obtained by integrating the electronic Schrodinger equation (Eq. ( )) using different phase protocols. At each time step, the phase of the overlap matrix U(dt c ) is determined following either the OP protocol or the MP protocol. Note that, when decreasing the classical time step dt c , one expects that both protocols should converge and agree with each other in the extremely small dt c limit.