  A fundamental property of a quantum system driven by an external field is
that when the field is turned off the positions of its response frequencies are
independent of the time at which the field is turned off. We show that this
leads to an exact condition for the exchange-correlation potential of
time-dependent density functional theory. The Kohn-Sham potential typically
continues to evolve after the field is turned off, which leads to
time-dependence in the response frequencies of the Kohn-Sham response function.
The exchange-correlation kernel must cancel out this time-dependence. The
condition is typically violated by approximations currently in use, as we
demonstrate by several examples, which has severe consequences for their
predictions of time-resolved spectroscopy.
