Although we have not been explicit about spin indices, the spin-orbital indices (i, a, etc.) could be limited to either α or β spin, e.g., to obtain a spin density matrix for either the particle (∆P elec α -∆P elec β ) or hole (∆P hole α -∆P hole β ). Whereas ∆P in Eq. (2.9) is available from x and y alone, calculation of Z requires solution of the coupledperturbed equations that are associated with the TD-DFT excited-state gradient. The density matrix P exc that includes Z is known as the "relaxed" density matrix, whereas