  We study the electronic properties of twisted bilayers graphene in the
tight-binding approximation. The interlayer hopping amplitude is modeled by a
function, which depends not only on the distance between two carbon atoms, but
also on the positions of neighboring atoms as well. Using the Lanczos algorithm
for the numerical evaluation of eigenvalues of large sparse matrices, we
calculate the bilayer single-electron spectrum for commensurate twist angles in
the range $1^{\circ}\lesssim\theta\lesssim30^{\circ}$. We show that at certain
angles $\theta$ greater than $\theta_{c}\approx1.89^{\circ}$ the electronic
spectrum acquires a finite gap, whose value could be as large as $80$ meV.
However, in an infinitely large and perfectly clean sample the gap as a
function of $\theta$ behaves non-monotonously, demonstrating
exponentially-large jumps for very small variations of $\theta$. This
sensitivity to the angle makes it impossible to predict the gap value for a
given sample, since in experiment $\theta$ is always known with certain error.
To establish the connection with experiments, we demonstrate that for a system
of finite size $\tilde L$ the gap becomes a smooth function of the twist angle.
If the sample is infinite, but disorder is present, we expect that the electron
mean-free path plays the same role as $\tilde L$. In the regime of small angles
$\theta<\theta_c$, the system is a metal with a well-defined Fermi surface
which is reduced to Fermi points for some values of $\theta$. The density of
states in the metallic phase varies smoothly with $\theta$.
