Second, after the H 2 collision (occurring at t ≈ 70fs), we continue to propagate the classical nuclear dynamics so that the H 2 molecule eventually moves far away from Ag 20 again. However, the silver cluster is now excited and vibrating; the excitation spectrum is now changing in time. In (Fig. indicate trivial crossings of two states where the electronic wavefunction is a superposition state and has non-zero coefficients for other CIS states. At (■), there is a complete electronic transition from state 75 (magenta) to state 74 (orange) while the coefficient of state 84 (blue) is nonzero with a relative phase of importance. The OP protocol preserves this relative phase and converges to the correct dynamics when dt c = 0.17 fs. However, the MP protocol disturbs this relative phase leading to incorrect long time dynamics and cannot converge even for dt c = 0.12 fs. In the end, using the OP protocol allows for using at least 40% larger classical time step than the MP protocol. Figure : The population dynamics for the OP protocol (solid lines) versus the MP protocol (dash-dot lines) using the smallest time step size tested (dt c = 0.12 fs). The OP and MP protocol agree with each other before t = 122 fs as labeled by (⋆). Note that the MP protocol does not yield the correct phase for the overlap matrix at (⋆), which leads to incorrect dynamics, suggesting that one would need an even smaller time step for the MP protocol to converge. emerged around 3.61 eV. Interestingly, by comparison with the spectrum before collision, the strongest peak now involves only single electron excitation, rather a collective excitation, whereas the second strongest peak now is a plasmon-like collective excitation (rather than a single electron excitation). These conclusions highlight the need to model both nuclear dynamics and plasmonic excitations at the same time.