  We investigate the lattice and electronic structures of the bulk and surface
of the prototypical layered topological insulators Bi$_2$Se$_3$ and
Bi$_2$Te$_3$ using ab initio density functional methods, and systematically
compare the results of different methods of including van der Waals (vdW)
interactions. We show that the methods utilizing semi-empirical energy
corrections yield accurate descriptions of these materials, with the most
precise results obtained by properly accounting for the long-range tail of the
vdW interactions. The bulk lattice constants, distances between quintuple
layers and the Dirac velocity of the topological surface states (TSS) are all
in excellent agreement with experiment. In Bi$_2$Te$_3$, hexagonal warping of
the energy dispersion leads to complex spin textures of the TSS at moderate
energies, while in Bi$_2$Se$_3$ these states remain almost perfectly helical
away from the Dirac point, showing appreciable signs of hexagonal warping at
much higher energies, above the minimum of the bulk conduction band. Our
results establish a framework for unified and systematic self-consistent first
principles calculations of topological insulators in bulk, slab and interface
geometries, and provides the necessary first step towards ab initio modeling of
topological heterostructures.
