  We consider the electrons of a molecule in the adiabatic time-dependent
density functional theory approximation. We establish the well-posedness of the
time evolution and its linear response close to a non-degenerate ground state,
and prove the appearance of resonances at relevant frequencies. The main
mathematical difficulty is due to the structure of the linearized equations,
which are not complex-linear. We bypass this difficulty by reformulating the
linearized problem as a real Hamiltonian system, whose stability is ensured by
the second-order optimality conditions on the energy.
