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Abstract. Born-Oppenheimer, Smoluchowski, Langevin, Ehrenfest and surface-hopping dynamics are shown to be accurate approximations of time-independent Schrödinger ob-servables for a molecular system, in the limit of large ratio of nuclei and electron masses, without assuming that the nuclei are localized to vanishing domains. The derivation, based on characteristics for the Schrödinger equation, bypasses the usual separation of nuclei and electron wave functions and gives a different perspective on initial and boundary conditions, caustics and irreversibility, the Born-Oppenheimer approximation, hopping, computation of observables and stochastic electron equilibrium states in molecular dynamics modeling