  We analyze the time reversible Born-Oppenheimer molecular dynamics (TRBOMD)
scheme, which preserves the time reversibility of the Born-Oppenheimer
molecular dynamics even with non-convergent self-consistent field iteration. In
the linear response regime, we derive the stability condition as well as the
accuracy of TRBOMD for computing physical properties such as the phonon
frequency obtained from the molecular dynamic simulation. We connect and
compare TRBOMD with the Car-Parrinello molecular dynamics in terms of accuracy
and stability. We further discuss the accuracy of TRBOMD beyond the linear
response regime for non-equilibrium dynamics of nuclei. Our results are
demonstrated through numerical experiments using a simplified one dimensional
model for Kohn-Sham density functional theory.
