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Existing recipes are ad hoc and fail in known test cases.","posed_since":"","precise":"Derive a mixed quantum-classical method in which nuclei follow independent trajectories and that (i) reproduces exact population dynamics within 0.05 absolute at all times for the three Tully one-dimensional models and the spin-boson model over the coupling and temperature ranges of standard benchmarks, (ii) yields Boltzmann-weighted state populations at long times (detailed balance), (iii) gives correct decoherence without adjustable parameters, and (iv) costs no more per trajectory than surface hopping (one electronic-structure call per nuclear time step). An answer is the method with its derivation from the exact equations, or a proof that independent trajectories cannot satisfy (i) to (iii) together.","problem_ref":null,"references":"","settled_by":"Derivation plus benchmarks against numerically exact multiconfigurational time-dependent Hartree results on standard models.","status_note":"","title":"A consistent trajectory method for coupled electron-nuclear 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