  We revisit Kohn-Sham time-dependent density-functional theory (TDDFT)
equations and show that they derive from a canonical Hamiltonian formalism. We
use this geometric description of the TDDFT dynamics to define families of
symplectic split-operator schemes that accurately and efficiently simulate the
time propagation for certain classes of DFT functionals. We illustrate these
with numerical simulations of the far-from-equilibrium electronic dynamics of a
one-dimensional carbon chain. In these examples, we find that an optimized 4th
order scheme provides a good compromise between the numerical complexity of
each time step and the accuracy of the scheme. We also discuss how the
Hamiltonian structure changes when using a basis set to discretize TDDFT and
the challenges this raises for using symplectic split-operator propagation
schemes.
