Long-range charge transfer excited states are notoriously badly underestimated in time-dependent density functional theory (TDDFT). This can have serious consequences. For example, as shown by Dreuw and Head-Gordon, charge-transfer quenching of excited-state fluorescence in light-harvesting bacteria is predicted, contrary to observation. We study how exact TDDFT captures charge transfer between two widely-separated open-shell species in a molecule: in particular the role of the step in the ground-state potential and the severe frequency-dependence in the exchange-correlation kernel. An expression for the latter is derived that becomes exact in the limit that the charge-transfer excitations are well-separated from other excitations. The exchange-correlation kernel has the task of undoing the static correlation in the ground state that is introduced by the step, in order to accurately recover the physical charge-transfer states.
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