where U(t+∆t, t) is a unitary time-propagation operator for the time step t → t + ∆t. Because both P and F in Eq. ( ) are time-dependent quantities, the definition of U(t + ∆t, t) involves time-ordering of the matrices F(t ) at points t along the integration (t ≤ t ≤ t + ∆t), or else a Magnus expansion of nested commutators. Various forms for U(t+∆t, t) have been discussed in previous work, including schemes that iterate the time propagation to self-consistency over the course of a single time step from t to t + ∆t. In the present work, we focus on the modified-midpoint algorithm. This approach corresponds to a propagator