The energetics of the translational and rotational degrees of freedom of graphene layers are investigated using density functional theory. It is found that the sliding (translation) energy of a bilayer depends dramatically on whether such layers are mutually rotated or not. While for unrotated layers the sliding energy is large, with the AB stacked bilayer lowest in energy, for mutually rotated layers the sliding energy is zero. Turning to the rotational degree of freedom, we find that

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... e of energy on the relative rotation between layers is considerable, and that the lowest energy structure is that generated by 30 • ± 2.208 • . The impact of a perpendicular electric field on mutually rotated graphene layers is explored. The electronic decoupling of such layers ensures that the Dirac cones simply shift relative to each other to accommodate the charge transfer between the layers. Interestingly, this shift is approximately the same in magnitude as that of the field induced gap opened when an electric field is applied to an AB stacked bilayer.