  When running time-dependent density functional theory (TDDFT) calculations
for real-time simulations of non-equilibrium dynamics, the user has a choice of
initial Kohn-Sham state, and typically a Slater determinant is used. We explore
the impact of this choice on the exchange-correlation potential when the
physical system begins in a 50:50 superposition of the ground and first-excited
state of the system. We investigate the possibility of judiciously choosing a
Kohn-Sham initial state that minimizes errors when adiabatic functionals are
used. We find that if the Kohn-Sham state is chosen to have a configuration
matching the one that dominates the interacting state, this can be achieved for
a finite time duration for some but not all such choices. When the Kohn-Sham
system does not begin in a Slater determinant, we further argue that the
conventional splitting of the exchange-correlation potential into exchange and
correlation parts has limited value, and instead propose a decomposition into a
"single-particle" contribution that we denote $v_{xc}^s$, and a remainder. The
single-particle contribution can be readily computed as an explicit
orbital-functional, reduces to exchange in the Slater determinant case, and
offers an alternative to the adiabatic approximation as a starting point for
TDDFT approximations.
