Note that, if we start with a U with det(U) = 1, then at every iteration, we flip the signs of a pair of columns in the U matrix in order to preserve the condition det(U) = 1. In Ref. 28, we demonstrated that this optimization approach yields the time-derivative coupling matrix elements that are as smooth as possible. Note that the optimization approach has the computational complexity O(N 2 s ), which is almost instantaneous in comparison to constructing the CIS state overlap matrix. By design, our hope has been that the OP protocol should allow for a larger time step dt c .