  We present a detailed study of the exact-exchange (EXX) kernel of
time-dependent density functional theory with an emphasis on its discontinuity
at integer particle numbers. It was recently found that this exact property
leads to sharp peaks and step features in the kernel that diverge in the
dissociation limit of diatomic systems [Hellgren and Gross, Phys. Rev. A,
022514 (2012)]. To further analyze the discontinuity of the kernel we here make
use of two different approximations to the EXX kernel: the PGG approximation
and a common energy denominator approximation (CEDA). It is demonstrated that
whereas the PGG approximation neglects the discontinuity the CEDA includes it
explicitly. By studying model molecular systems it is shown that the so-called
field counter-acting effect in the density functional description of molecular
chains can be viewed in terms of the discontinuity of the static kernel. The
role of the frequency dependence is also investigated, highlighting its
importance for long-range charge transfer excitations as well as inner-shell
excitations.
