  The role of electron-nuclear entanglement on the validity of the
Born-Oppenheimer (BO) approximation is investigated. While nonadiabatic
couplings generally lead to entanglement and to a failure of the BO
approximation, surprisingly the degree of electron-nuclear entanglement is
found to be uncorrelated with the degree of validity of the BO approximation.
This is because while the degree of entanglement of BO states is determined by
their deviation from the corresponding states in the crude BO approximation,
the accuracy of the BO approximation is dictated, instead, by the deviation of
the BO states from the exact electron-nuclear states. In fact, in the context
of a minimal avoided crossing model, extreme cases are identified where an
adequate BO state is seen to be maximally entangled, and where the BO
approximation fails but the associated BO state remains approximately
unentangled. Further, the BO states are found to not preserve the entanglement
properties of the exact electron-nuclear eigenstates, and to be completely
unentangled only in the limit in which the BO approximation becomes exact.
