  The structural and electronic properties of twisted bilayer graphene are
investigated from first principles and tight binding approach as a function of
the twist angle (ranging from the first "magic" angle $\theta=1.08^\circ$ to
$\theta=3.89^\circ$, with the former corresponding to the largest unit cell,
comprising 11164 carbon atoms). By properly taking into account the long-range
van der Waals interaction, we provide the patterns for the atomic displacements
(with respect to the ideal twisted bilayer). The out-of-plane relaxation shows
an oscillating ("buckling") behavior, very evident for the smallest angles,
with the atoms around the AA stacking regions interested by the largest
displacements. The out-of-plane displacements are accompanied by a significant
in-plane relaxation, showing a vortex-like pattern, where the vorticity
(intended as curl of the displacement field) is reverted when moving from the
top to the bottom plane and viceversa. Overall, the atomic relaxation results
in the shrinking of the AA stacking regions in favor of the more energetically
favorable AB/BA stacking domains.
  The measured flat bands emerging at the first magic angle can be accurately
described only if the atomic relaxations are taken into account. Quite
importantly, the experimental gaps separating the flat band manifold from the
higher and lower energy bands cannot be reproduced if only in-plane or only
out-of-plane relaxations are considered. The stability of the relaxed bilayer
at the first magic angle is estimated to be of the order of 0.5-0.9 meV per
atom (or 7-10 K). Our calculations shed light on the importance of an accurate
description of the vdW interaction and of the resulting atomic relaxation to
envisage the electronic structure of this really peculiar kind of vdW bilayers.
