that updates the density matrix from t N to t N +1 = t N + ∆t. The quantity F N +1/2 is the Fock matrix at t = t N + ∆t/2. For a molecule in a laser field, the Fock operator F consists of a field-free molecular part ( F0 ) augmented with an additional term E • r involving the electric field E(t). In addition, we add a CAP of the form -iW CAP (r), which is discussed in detail below. These additional terms added to F0 both take the form of real-space potentials and the total Fock operator including these additions is