The original account of the Born–Oppenheimer approximation is not mathematically secure because it is not legitimate to use perturbation theory in its development. It is necessary to use an asymptotic expansion based upon an electronic Hamiltonian defined in terms of a fiber bundle. Although with this approach it has been possible account for the traditional results for a diatomic molecule, rotational motion in the polyatomic case has not so far been accounted for. It is argued here that it is not generally possible to provide a mathematically secure account of the Born–Oppenheimer approximation for polyatomic molecules, in which rotation can be considered as a separable motion. © 2012 Wiley Periodicals, Inc.