  We study the effect of temporal correlation in a Langevin equation describing
non-adiabatic dynamics at metal surfaces. For a harmonic oscillator the
Langevin equation preserves the quantum dynamics exactly and it is demonstrated
that memory effects are needed in order to conserve the ground state energy of
the oscillator. We then compare the result of Langevin dynamics in a harmonic
potential with a perturbative master equation approach and show that the
Langevin equation gives a better description in the non-perturbative range of
high temperatures and large friction. Unlike the master equation, this approach
is readily extended to anharmonic potentials. Using density functional theory
we calculate representative Langevin trajectories for associative desorption of
N$_2$ from Ru(0001) and find that memory effects lowers the dissipation of
energy. Finally, we propose an ab-initio scheme to calculate the temporal
correlation function and dynamical friction within density functional theory.
