The integral is evaluated by numerical quadrature, using the DFT quadrature grid. The matrix W CAP does not depend on time and can be constructed once, at the beginning of a TDKS simulation, and is then added to the Fock matrix F(t) at each time step. Note that W CAP is a symmetric matrix and therefore -iW CAP in Eq. ( ) is skew-Hermitian, meaning that F is not Hermitian despite the fact that F 0 is real and symmetric at t = 0. This requires modification to the usual propagator, which is accomplished by approximate factorization of the relevant exponential, i.e., the split-operator technique: 55