  Adiabatic approximations in time-dependent density functional theory (TDDFT)
will in general yield unphysical time-dependent shifts in the resonance
positions of a system driven far from its ground-state. This spurious
time-dependence is rationalized in [J. I. Fuks, K. Luo, E. D. Sandoval and N.
T. Maitra, Phys. Rev. Lett. {\bf 114}, 183002 (2015)] in terms of the violation
of an exact condition by the non-equilibrium exchange-correlation kernel of
TDDFT. Here we give details on the derivation and discuss reformulations of the
exact condition that apply in special cases. In its most general form, the
condition states that when a system is left in an arbitrary state, in the
absence of time-dependent external fields nor ionic motion, the TDDFT resonance
position for a given transition is independent of the state. Special cases
include the invariance of TDDFT resonances computed with respect to any
reference interacting stationary state of a fixed potential, and with respect
to any choice of appropriate stationary Kohn-Sham reference state. We then
present several case studies, including one that utilizes the
adiabatically-exact approximation, that illustrate the conditions and the
impact of their violation on the accuracy of the ensuing dynamics. In
particular, charge-transfer across a long-range molecule is hampered, and we
show how adjusting the frequency of a driving field to match the time-dependent
shift in the charge-transfer resonance frequency, results in a larger charge
transfer over time.
