For completeness and to fix notation, we briefly recapitulate the LR-TDDFT formalism. This arises from considering the first-order response of Eq. ( ) to a weak perturbation, therefore LR-TDDFT may be regarded as the weak-field limit of TDDFT. Identical spectra are obtained with the TDKS approach when the perturbing field is small and the timedependent simulation is propagated for a sufficiently long time. The LR-TDDFT equation for excitation energies (ω) is