  In this work we have calculated excitation energies and photoionization cross
sections of Be and Ne in the exact-exchange (EXX) approximation of
time-dependent density functional theory (TDDFT). The main focus has been on
the frequency dependence of the EXX kernel and on how it affects the spectrum
as compared to the corresponding adiabatic approximation. We show that for some
discrete excitation energies the frequency dependence is essential to reproduce
the results of time-dependent Hartree-Fock theory. Unfortunately, we have found
that the EXX approximation breaks down completely at higher energies, producing
a response function with the wrong analytic structure and making inner-shell
excitations disappear from the calculated spectra. We have traced this failure
to the existence of vanishing eigenvalues of the Kohn-Sham non-interacting
response function. Based on the adiabatic TDDFT formalism we propose a new way
of deriving the Fano parameters of autoionizing resonances.
