2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/144994We explain why the conventional argument for deriving the time-dependent Born-Oppenheimer approximation is incomplete and review recent mathematical results, which clarify the situation and at the same time provide a systematic scheme for higher order corrections. We also present a new elementary derivation of the correct second-order time-dependent Born-Oppenheimer approximation and discuss as applications the dynamics near a conical intersection of potential surfaces and reactive scattering.17 pages, no figuresMathematical Physics81Q05, 81Q15, 81Q70text