  Born-Oppenheimer dynamics is shown to provide an accurate approximation of
time-independent Schr\"odinger observables for a molecular system with an
electron spectral gap, 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 a Hamiltonian system interpretation of the
Schr\"odinger equation and stability of the corresponding Hamilton-Jacobi
equation, bypasses the usual separation of nuclei and electron wave functions,
includes caustic states and gives a different perspective on the
Born-Oppenheimer approximation, Schr\"odinger Hamiltonian systems and numerical
simulation in molecular dynamics modeling at constant energy microcanonical
ensembles.
