  Certain excitations, especially ones of long-range charge transfer character,
are poorly described by time-dependent density functional theory (TDDFT) when
typical (semi-)local functionals are used. A proper description of these
excitations would require an exchange-correlation response differing
substantially from the usual (semi-)local one. It has recently been shown that
functionals of the generalized gradient approximation (GGA) type can yield
unusual potentials, mimicking features of the exact exchange derivative
discontinuity and showing divergences on orbital nodal surfaces. We here
investigate whether these unusual potential properties translate into
beneficial response properties. Using the Sternheimer formalism we closely
investigate the response obtained with the 2013 exchange approximation by
Armiento and K\"ummel (AK13) and the 1988 exchange approximation by Becke
(B88), both of which show divergences on orbital nodal planes. Numerical
calculations for Na2 as well as analytical and numerical calculations for the
hydrogen atom show that the response of AK13 behaves qualitatively different
from usual semi local functionals. However, the AK13 functional leads to
fundamental instabilities in the asymptotic region that prevent its practical
application in TDDFT. Our findings may help the development of future improved
functionals, and corroborate that the frequency-dependent Sternheimer formalism
is excellently suited for running and analyzing TDDFT calculations.
