  A numerical scheme for solving the time-evolution of wave functions under the
time dependent Kohn-Sham equation has been developed. Since the effective
Hamiltonian depends on the wave functions, the wave functions and the effective
Hamiltonian should evolve consistently with each other. For this purpose, a
self-consistent loop is required at every time-step for solving the
time-evolution numerically, which is computationally expensive. However, in
this paper, we develop a different approach expressing a formal solution of the
TD-KS equation, and prove that it is possible to solve the TD-KS equation
efficiently and accurately by means of a simple numerical scheme without the
use of any self-consistent loops.
