than two states cross at the same time, one may find the diagonal element of the overlap matrix can be zero (especially when the classical time step is large), which makes parallel transport unstable and ambiguous; in such a case, there is no reason at all to presume that det(U) = 1. Therefore, changing the sign of the column with the smallest diagonal element (i.e. maximally positive) is a straightforward fix to ensure det(U) = 1. That being said, we emphasize that such a change does not minimize the norm Tr{| log U| 2 }, and the time derivative T matrix may not always be optimally smooth for a large time step. In principle, if one uses a very small time step and many grid points, the MP protocol can recover smooth wavefunction propagation and capture the correct electronic transition, but we expect that the need for such small time steps will make calculations more expensive than our optimized approach.